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History, Road trip, Travel
USA: get your kicks on Route 66
February 27, 2018 February 26, 2018 by bmalzardLeave a Comment on USA: get your kicks on Route 66
A pilgrimage is what a drive along Route 66 is today. The fabled road lives in songs and literature and just won't fade away. This is America's Main Street.
In the earliest days of television in Australia, the black and white screen flickered with many (mostly) American series to entertain the young minds of the babyboomers: Mickey Mouse Club (remember Annette Funnicello?); Cheyenne, Sugarfoot and Rawhide (cowboy cool dudes) and family entertainment – The Nelsons (oh Ricky!);The Donna Reed Show and Father Knows Best. But for us growing, restless teenagers of the time this was 'too square', no grunt at all. Bandstand was a constant and we got our rock'n'roll edge with Six O'Clock Rock but drama to get our motor running? No.
Then in the early 60s along came Route 66, edgy, and supremely cool; two guys in a sleek convertible Corvette heading from Chicago to LA . . . with incidents and episodes along the way. Tod and Buzz slicked back their hair, got moody, threw on their duffle coats and jumped in the car – to drive.
We may not have heard the early song by Nat King Cole – well, it was a recorded in 1946 but Get Your Kicks on Route 66 is embedded in everyone's musical memory. And our generation may not have that restless seed of youth blossoming as it did in the 60s but the Mother Road waits for the adventurous!
The romanticised highway represents a transition from dirt road to super-highway. Route 66 was the shining beacon as to the importance of the automobile as a new world achievement – and to the American people it symbolised the new freedom and mobility for every person who owned and drove their car. Route 66 was born in 1926, pieced together from a network of existing trails and anonymous roads. It's the Mother Road that carried life along the highway from Chicago to Los Angeles. The Interstate Highways System took it over in the 1950s and much of old Route 66 was decertified and became a secondary road, a highway of broken dreams and a mythical place that stranded small towns and businesses with cruel bypasses.
Today it's the road to a new adventure – a link to the past with an air of freedom. In the past decade the idea of Route 66 has been reinvigorated and has made a rock'n'roll kind of comeback – warts and all.
Driving to find the glory days you don't have to go far out from Los Angeles and for a couple of hundred miles discover what's happening along the way.
Route 66 and beyond can work nicely if you:
Head out of LA and visit Big Bear Lake, Yucca Valley. Set in the heart of the San Bernadino Mountains, Big Bear Lake and surrounds is startlingly beautiful country with much to do for fun: off-road adventure in a Pinzgauer 4WD – hot to trot and loads of fun; Big Bear Village has a rustic charm with a sophisticated edge – the coffee in town is damn good too!
Yucca Valley is reached after a drive that descends through a layered spread of heavily grassed land to almost paint-by-numbers clarity in the tapestry effect of the landscape. And in the distance a tiny sliver of snow is glimpsed on the top of the San Bernadino Mountains.
There's a hometown atmosphere and this 'High Desert' region is welcoming and eye opening for its nature and cultural heritage. And if you want to meet the locals – they are the friendliest people around, just drop by Pappy & Harriet's Pioneering Palace for mesquite barbecue chow and you'll enter an establishment plonked on the side of the road that looks deserted except for a few utes parked out front. Walk through the doors and it's packed, there are waitresses zooming around the tables and the beer is flowing and the music is about to start – hip and happening! Step out the backdoor and you are in the old west.
Pioneertown was founded by a group of Hollywood investors in 1946 with dreams of creating a living movie set. The 1870s 'frontier town' with authentic facades never did take off but folk had fun there and several of the properties are working sets with art and craft people doing their thing.
Hit the highway and drive through the empty, lonely landscape until you come to the sublime (if you love a desert park) Joshua Tree National Park.
In the park is the Desert Queen Ranch, a crumbling reminder of the toughness of this country and the remains here tell the life story of Bill Keys' family – the trials, triumphs and tribulations. Keys lived there from 1910 until his death in 1969.
Bill Keys was a resourceful man who was the Desert Donald Trump of his day – pursuing mining and ranching and taking every opportunity to make a killing – which he actually did. He lived on the isolated ranch with his wife and seven children, three of whom are buried there.
During a property dispute Keys shot and killed a man and was convicted of murder and spent a few years in the big house educating himself. (He was paroled in 1950 and was pardoned in 1956 through the efforts of Earle Stanley Garner, author of the Perry Mason novels.)
The original homestead is still there plus outbuildings, a schoolhouse, farm and mining equipment and old truck chassis. The elements are rusting and corroding the debris from Bill Keys' empire – to restore it or let it crumble back to the earth – that's still being debated.
Even though it's a remote and strange desert park, Joshua Tree has drawn some pretty interesting characters – including lots of alien/spaceship sightings. Oddly, the Andreas Fault Line is visible in parts of the park and juts out of the earth like a dragon's spine. Cattle rustlers are legendary and Keith Richards and Graham Harrison sat out here and took copious amounts of drugs while waiting to have a close encounter with a UFO . . .
In keeping with the mystical/hippy/left-over60s vibe of the area much of the interesting stuff of the area has survived and thrived.
The 'Integratron' is an acoustically perfect tabernacle dome and energy machine sited on a powerful geometric vortex in the magical Mojave Desert. (This was created by George Van Tassle, and is based on the design of Moses' Tabernacle drawings and the writings of Nikola Tesla and telepathic directions of extraterrestrials – that's the speil anyway.) Whatever floats your boat out here but climbing up to the platform to lie down while someone rims ceramic bowls is quite something – a resonating sound bath indeed.
The historic route continues to evolve as states repair and redirect the road. Entering Amboy, there's the classic Route 66 stamped on the road in front of a Roy's Motel sign – classic 60s and a poor old motel that is crying in the shadows for someone to bring back its retro coolness. Faded and atmospheric, the location is not as neglected at is seems. It's used for commercials and movies – the no-longer-a- town Amboy, Roy's Motel and Cafe has a certain staying power. A buyer paid cash (lots) for the site in 2005 and has promised to preserve the town and reopen Roy's.
(The writer cruised into Amboy and was excited to see about a dozen gleaming, shiny black and silver clad Harley Davidson's – this is where it's happening she thinks. Just the backdrop for the 'Wild Ones'; as she drew closer to the bikies – she noticed as they were decked out in denim and leather, chains, bandanas and snake-hipped girls by their sides – they were all drinking Coca-Cola. This was a meeting for the Christian Motorcyclists Association, riding Route 66 to Chicago – it was big, but these boys weren't going to be bad!)
Away from the bustling metropolis of 'town' we started to see the beauty and drama of the Mojave National Preserve – sand dunes lead to volcanic cones, craters became mirage-like salt pans and sand was accumulated in vast amounts everywhere the eye could see – where does it come from?
We cruise past the Whiting Brothers Gas Station, a relic that has been preserved and then stop for a milkshake (it had to be a milkshake) at the Bagdad Cafe (remember the film?). It is still basking in the glory of movie fame – albeit needing a bit of nip and tuck. The love this cafe draws is evidenced by notes stuck on the windows – the clientele is universal; all stopping by to take a peek at the defunct motel from the film and to order a chocolate malt!
We stop for some big town buzz and Barstow fits the bill. Amazing outlet malls here and for some real local hospitality we eat at Idle Spurs Steakhouse – a favourite in the High Desert region of Southern California. Built around the original house – these folk have been slinging steaks since 1950.
After a lazy sleep-in we keep moving west as we've backtracked. We stop at the extremely charming and colourful Victorville Route 66 Museum – and we're greeted by the local lady volunteers – they are the ants pants! There is so much going on in the museum, take an hour or so to let it seep in – from The Grapes of Wrathto the Mother Road's heyday as one of the greatest ways to experience America on driving holidays.
We wrap up our short Route 66 driving trip with a stop at the original Wigwam Motel. This motel is one of just a couple remaining of a great chain. The wigwams are neat as pins and there's the place beside them to park your car. This is one of two only (the other is in Holbrook, Arizona) Wigwam Motels still in business to offer hospitality to drivers on Route 66.
Now stationary, I ponder America's amazing history – this small part of the vast continent. Roads criss-crossing the country, frenzied drivers taking the freeways, highways and byways, toll roads, side lanes and flyovers – is the romance still there for them, behind the wheel does it still feel that it's the way to claim the ground under their feet or is it like Route 66, a fabled Main Street where driving trip from A to B is the ribbon that unites more than an idea?
www.eaglerider.com (to hire Motorcycles ); www.wigwammotel.com
Architecture, Food, History, Travel
How to explore Umbria, Italy
February 20, 2018 September 18, 2018 by bmalzard3 Comments on How to explore Umbria, Italy
If it wasn't for the Etruscans, there might not be olive oil or wine in this region of Italy. Let's head to the hilltop fortress towns of Umbria and enjoy the legacy of the ancient invaders.
It had been a carb-overload lunch hosted by chef and pastamaking teacher Lorenzo Polegri, a showman and a man of smiles and passion as he demonstrated how to perfect the art of pasta making. None of us in our group were very talented but who cares? We enjoyed our repast in Ristorante Zeppelin in the quiet midday ambience of the mediaeval Umbrian town of Orvieto.
This town, a natural fortress, is one of the many glorious fortified towns and cities of the Umbrian region, which includes the spiritually robust Assisi, the glorious mystery of Perugia and this elegant site of Orvieto, all founded by the very late, lamented Etruscans who disappeared into the emerging Roman empire in the third century BC.
After eating food fit for Etruscan epicureans, we scattered to waddle into the narrow curved streets of this city bathed in autumnal afternoon light.
Curving past small shops, drifting along, I lifted my eyes and saw the most confoundingly beautiful structure – a green and white striped cathedral – a bold statement against the stark blue sky – with intricate, delicate relief carvings on the capitals with sumptuous cornerstones. The bold marble panels adorn the façade and are respected as one of the masterpieces of the late Middle Ages. It may not be the biggest and the best in the world – but this striped beauty captured my heart.
Orvieto Cathedral is one of Italy's most celebrated cathedrals. The 14th century edifice was built between 1290 and 1500 and she shines as brightly as ever. Built under papal direction, the building is famous for its mosaic inlay facade.
Duomo di Orvieto is widely considered the most glorious example of Italian Gothic. A miracle is said to have occurred in 1263 in the nearby town of Bolsena, when a travelling priest who had doubts about the truth of transubstantiation found that his Host was bleeding so much that it stained the altar cloth. The cloth is now stored in the Chapel of the Corporal inside the cathedral.
Next stop along the way on our escourted journey is the jewel of Umbria, Perugia. Perugia is the capital city of the region of Umbria and covers a high hilltop around the area. Another Etruscan beauty, Perugia is known for its universities (the first founded in 1308) and is celebrated for its culture and artistic pursuits.
The city has centuries of tumultuous religious and political (same thing then) history and all of it immersed in the stone here.
The Rocca Paolina was the 'underground' city, built in 1373 at the then highest part of the town. The richest merchants of 16th century Perugia lived here but it was destroyed by local citizens in an uprising provoked by the Pope. The town disappeared but the streets have been uncovered and the mediaeval homes that were a platform for the new fortress are now on show.
The stone houses with Gothic doorways and tunnels look as if they are waiting for people to go about their daily business. The atmospheric route through the fortress by escalators take you through Rocca Paolina under the portico of Palazzo del Governo.
We headed out into the night and into the strange and curious labyrinthine streets underground. After ascending to 'uptown Perugia' to the historic centre, in a state of wonderment we found ourselves in the vigorous city of Perugia, with its night lights on and aromas enticing us into a 21st century pizza house.
Next stop across the Umbrian Valley is the mother ship of holy hilltop fortress cities: Assisi. Birthplace (in 1182) of Italy's favourite saint, Francis, the city is always buzzing with pilgrims.
Saint Francis and female favourite Saint Clare are the drawcards for the fans. The bodies of both saints were discovered in 1818 and luckily they hadn't been tampered with by grave robbers. For centuries, holy relics had done great business across Europe.
So what is left (bones of Saint Francis) and preserved remains of Saint Clare is on show as the faithful and curious pass by in snaking queues through Basilica di Santa Chiara (where Clare is) and the Basilica di San Francesco.
There are fine examples of Giotto's frescoes and Cimabue's painting to be viewed and other cultural attractions includes many little pottery statues of chubby monks and waving popes. And for fans of Norberto, the famous Umbrian painter, there's a small gallery with an excellent variety of fine prints to aquire.
It feels like we've ticked off the holy trinity of hill towns in Umbria and look forward to the next stop in Tuscany. More and more I appreciate the fact that we enjoy these splendid visits, and afterwards hop back on to a comfy coach which takes us to our next stop. So far we have been transported from Rome, taken to 'secret places', had intimate meals with welcoming local folk, and been invited into authentic experiences. Each night we have been put to bed, well fed and happy, in lovely hotels.
As we anticipate lunch and tastings of the local olive oil, and a meal of regional cooking in another handsome hill town – this time Spello, with its historic centre still enveloped by Roman walls. We stretch our legs in the coach, chat about the marvellous day we have had and agree that those Etruscans knew a thing or two about leaving an amazing legacy in Umbria.
Writer, Bev Malzard was hosted by Insight Vaations insightvacations.com.au and found the itinerary exciting and edifying. She recommends sampling gelato at every stop along the way. You will not be disappointed.
Food, Travel
Vienna's best breads
February 13, 2018 February 13, 2018 by bmalzard2 Comments on Vienna's best breads
I love bread. Sourdough, rye, Turkish, flatbread, brioche, French sticks, naan, roti, wholemeal, soft white rolls, Italian focaccia and am not ashamed to admit to a couple of slices of white Tip Top bread with butter and vegemite.
And part of travel is eating lots of bread . . . (sorry to the gluten intolerants). Naan to mop up curry sauce and dhal for breakfast in India and beautiful European breads with cheese washed down with coffee.
Years ago when I first travelled to Austria I was naive and hadn't been exposed to sliced heavy bread in the morning. My first taste (not toast) was accompanied by a cream cheese and a fat slice of ham. And that's when romancing the loaf began. No matter where I am, I'll try the bread on offer.
Europe sets the standard for good, wholesome, hearty breads. And Vienna is upping the 'brot' ante for the comeback of artisan bread. For too long Vienna has loafed around with heavy, commercially manufactured bread and now it looks for a slice of the crusty good life.
The renaissance of small bakeries in Vienna is full on – here are a couple of snaps taken around the back streets where the aroma of freshly baked bread drew me into the bakery for a bun or two.
I was taken around the city bakeries by 'Brot Andi', Andreas Djordjevic, an institution in Vienna. Andreas is in charge of the bread cart in the two Michelin-starred Steirereck, the best restaurant in Vienna. The restaurant is divine with the most beautiful ceiling, amazing food styling – and flavour of course – and then there's the brot . . . .
I rather fancied the name of Arthur Grimm's bakery – no relation of you-know-who.
This was a scattering of breadcrumbs to entice you to walk the streets of Vienna in the early morning, and . . . just follow your nose.
www.austria.info/au
Writer, Bev Malzard once joined with a work colleague to experiment with 'the sandwich as a meal' concept. Over a week of 21 meals they managed to put every meal on a sandwich. Eggs on toasts for brekkie, a salad and ham sando for lunch, steak and chips between two slices of bread for dinner. Great strides in gastronomic adventure were made when there was a pie and sauce sandwich, a fish and chips sandwich, a pea soup sandwich (this had to be made and eaten rather quickly), a baked dinner sandwich, with gravy and beef and black beans and fried rice sandwich and the surprise meal . . . an apple pie sandwich. That was an excellent week.
The divine image at the top of the page was taken by the gifted Monika Grabkowska.
Architecture, Travel
Scandinavia: Great Danes
February 6, 2018 February 6, 2018 by bmalzardLeave a Comment on Scandinavia: Great Danes
'Wonderful, wonderful Copenhagen, friendly old girl of a town' . . . continues to enchant visitors with tradition and its easygoing personality. But there's more to explore as its varied and eclectic architecture – old and new – stirs 'for' and 'against' controversy, eventually melding into the visual fabric of the city, just as it always has.
If you're not walking, boat or bicycle is the way to discover Copenhagen. Denmark's capital city is a true, natural beauty that has, over the centuries, been designed, primped and polished to within an inch of its life. Always on show, and stepping up to the mark as a pretty, chocolate-box-toy-soldier presentation, the city on the water has burst out of its uniform ambience and over the past decade presented a collection of new buildings which have changed the face of the harbour shoreline, canals and through some of the quiet streets.
When you think of Denmark, what comes to mind? Hamlet, pastries, a little mermaid, fine china, a certain princess, Vikings, beech forests, Hans Christian Andersen, herrings and very down-to-earth sensible people? All of the above, but what shines the light on modern Denmark, the ancient centre of Scandinavia, is design. Everything in the country, and especially the city of Copenhagen is about excellent design. Whether it's mediaeval churches, Renaissance castles and country homes, ordered streets and canals, or pretty, colourful harbour-side storied houses of the cities' that flourished in the 'golden age' from 1588-1648, the element of design surprise continues to grow and be embellished to mellow into the 21st century.
A leader in industrial design for the last century or so, Danish design conjures names such as Bang and Olufsen (audio and stereo brand), Bodum (coffee chic), Royal Copenhagen (fine china) and Geog Jensen (the famous Danish silverware brand).
The essence of the country's individual design is to be found in its timeless simplicity, quality materials and functionality.
Danish architecture, in Denmark and abroad is a standout on the international architectural scene. And it is often at the centre of public scrutiny and controversy. For example, take one of the world's most famous buildings, the Sydney Opera House designed by Jorn Utzon. This universally admired building was bankrolled by a public lottery, caused so much dissention within the ranks of the state government in the 1960s that the design was drastically modified, much to the architect's disappointment at the time.
Utzon designed beautiful buildings all over the world and in his own country. In the dock area of Copenhagen, the Paustian furniture store, also Utzon's brainchild is a place of inspiration. Contained in the stunning building is Denmark's largest collection of quality furniture and carpets, lighting and accessories and basically the best interior design from Scandinavia and the rest of the world. Perhaps not the best place to pick up some souvenirs but fine for picking up design and decorating ideas.
Two newish buildings that have had Copenhagen residents atwitter over the past few years are the amazing Copenhagen Opera House (designed by Henning Larsen), completed and opened in 2004 and the Royal Danish Library (known as the Black Diamond), a looming construction pressing the shoreline and designed by Danish architects Schmidt Hammer Lassen. It is named for its outside cover of black marble and glass.
The Black Diamond has a host of detractors and as many admirers. It leans dramatically over the waterfront and reflects the water and light, constantly changing its hue. This extraordinary building is an extension to the 19th century red brick Royal Library. It certainly catches the eye and the imagination as you glide by in the harbour.
Best view of these beauties before actually heading indoors to experience Opera and books is to take a canal boat trip and get your bearings of the harbour and all it offers. In fact it's a lovely way to start your discovery of Copenhagen.
In and around town by foot or bicycle you'll find beautiful old buildings from various eras that undoubtedly caused a stir in their time too.
The Ny Carlsberg Glyptotek, a fine art museum is an imposing grand period design standing on Hans Christian Andersen Boulevard. This museum is full of wonderful items including Etruscan art, 19th-century paintings from Denmark and France and many sculptures than span 5000 years. There are a few starts on show here – but two favourites are the work of Paul Gaugain and more than 30 pieces by Rodin. Take a rest from art here and you'll be having coffee and cake or maybe a meal at a beautiful tropical plant filled winter garden restaurant.
Also one of the oldies but goodies is the famous Tivoli Gardens, an amusement park that has charmed adults and thrilled children for more than 160 years. It's not Luna Park or Disneyland, but an area with old-fashioned gardens, a host of food pavilions and rides galore from the innocent, romantic rides of the past to the modern Demon – a corkscrewing roller coaster and the dead drop Golden Tower. The Tivoli is at its best at night when the magic of lights turn the park into a fairytale enclosure. Kitsch and schmaltz rule the day and from the behaviour of the visitors this is just the atmosphere desired.
Along the waterfront the Amalienborg Palace is an austere standout as there are four 'mansions' in the square making up the one home for the royal family. The buildings surround the centre square and the royal precinct is guarded by sentries. There is a ceremonial changing of the guard at noon daily. Amalienborg is one of Copenhagen best examples of Baroque architecture that didn't run riot with the curls and swirls.
Head inland to view the splendid Frederikskirken, the marble church with its glorious dome that can be seen from all over the city.
You can't visit Copenhagen without a thought for Hans Christian Andersen's Little Mermaid. It's odd that she sits patiently looking toward the shore and not the harbour, which could take her away from all the attention and fuss. Nostalgia aside it feels disrespectful to see tourists putting their arms around her and invading her tiny space on a rock.
If you still have a hankering for unusual architecture there's Holmen's red brick warehouses, barracks and foundries that were built on reclaimed land in the 17th century for the Danish military. Holmen is home to schools specialising in drama, film, architecture and music. For architecture fans the Royal Danish School of Architecture holds regular exhibitions in Meldahis Smedie in Holmen.
But brilliant design isn't just about buildings. Copenhagen's restaurant and café scene underwent an amazing transformation in the mid nineties. This modern Scandinavian revolution has produced, to mention the tip of the eatery iceberg, Café Victor, Dan Turells, Café Sommersko, Ultimo and Quote. These new and very fabulous eateries and wine bars made their name for their interpretation of Danish cooking as well as their good looks.
Window shopping shows the sophisticated design side of fashion, from men's and women's clothing to cutting edge shoes and accessories. Homeware shops have so many innovative kitchen implements and interior design items that the best advice you can get before travelling to Copenhagen is to pack your suitcase to half full, then take half out  – there's serious shopping to be done here.
There's good food, great accommodation, excellent transport and activities to keep you going for a week or so, but spare a thought and perhaps your appetite for what is considered by many to be Denmark's crowning design glory – the open-face sandwich. Forget wraps, baps, filled croissants, bagels and foccacia – this is the real Danish deal, cheap and more than cheerful.
There are 13 one-Michelin stared restaurants in Copenhagen (Google them) and if you want an amazing, three-starred Miclein meal – book (well before you visit) Geranium or AOC – mad if you don't!
(The featured image at the top of the page was taken by Pierre Chatel-Innocenti.)
'Wonderful, wonderful Copenhagen, friendly old girl of a town' song written by Frank Loesser for the film Hans Christian Andersen (1952) starring Danny Kaye.
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Twenty years from now you will be more disappointed by the things you didn't do. So throw off the bowlines,sail away from the safe harbour. Catch the trade winds in your sails. Explore. Dream. Discover. - Mark Twain
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Blend of beauty, mental balance, physical power, music, and an overwhelming sense of art and finesse.
''The dance is strong magic.
The dance is a spirit.
It turns the body to liquid steel.
It makes it vibrate like a guitar.
The body can fly without wings.
It can sing without voice.
The dance is strong magic.
° J'adore ce sentiment de transmission primaire de force et d'énergie !
° I love this primal feeling of transmission of strenght and energy!
° Mwen renmen fòs ak enèji zantray sila !
2019 © Copyright Turenne Joseph (Tilarenn). Fourni par Blogger.


Conflict is an inevitable characteristic of all work teams. Considering the complex dynamics of virtual teams, understanding and managing conflict becomes a big challenge. Conflict can be both - constructive and destructive. When members of the team present different viewpoints about a task and positively contribute to decision making, it leads to constructive conflict. Destructive conflict results when group members doubt each other's intentions behind difference in opinions. Virtual teams present a very paradoxical situation because on one hand virtual team members are more task-oriented and therefore do not get into any interpersonal issues, while on the other hand, due to the lack of face-to-face interactions, task related conflict transpires into interpersonal/relationship conflicts. This impacts collaborative efforts, hinders productivity and decrease morale.
It is the key responsibility of a virtual team leader to prevent task-related disagreement to devolve into interpersonal conflict. A strong leader who can anticipate team members' expectations and disagreements acts as the lifeguard when any problem arises.
The first and foremost task is to ensure a strong and well-planned orientation at the time of project initiation. This is the time when ground rules are set, expectation clarified and responsibilities delegated. Good team building activities and icebreakers should also form part of the virtual team orientation session. Members should be encouraged to share their backgrounds, past experiences and other informal information about themselves. This would help members to relate to each other.
Constant flow of communication should be encouraged. Different media can be used to share information and knowledge. Online discussion board in a shared virtual workspace is a great platform for virtual team members to discuss their opinions openly, voice their concerns and connect with each other in a relaxed and casual way. This generates transparency and builds trust among the members. Thus people do not attribute any hidden agenda behind a task related disagreement.
During the course of project, virtual team leader should promote team bonding activities outside work such as a 5 minute small talk session before project meetings etc. Such actions keep the members engaged as well as committed to the common goal. A daily updates from all the members about the four things - work completed, issues/challenges faced, assistance required and next day agenda - goes a long way in avoiding any seeping conflicts.
Virtual team leaders are mainly responsible for dealing with conflicts but a lot of time, effort and energy that goes in resolving the conflicts, can be saved if utmost care is taken during hiring & selection of virtual team members. Hiring people with right attitude along with their technical skills and knowledge is necessary. Right attitude in the context of virtual teams means intercultural competence, tolerance and understanding. This would reduce the situations of interpersonal conflicts among the members.
Another cause of conflicts in virtual teams is lack of non-verbal features of communication including gestures, body language and facial expressions, especially when communication is done through emails. For example, many a times a simple question/query sounds interrogatory and reproachful resulting in negative feelings and misunderstandings. Therefore, as much as possible the problem discussions should be done via videoconferencing and other rich synchronous media.
Thus we see by keeping in mind such simple and inexpensive action steps, virtual team leaders can increase the productivity and commitment of their dispersed work teams.


I think that the forum haven't got enought appearence on google.
Does the explanation of adding a "-" before a search item mean it will NOT be included in the results? If it does than the search engine may need some work.
Is there another technique that I'm missing or is something wrong with the instructions or search algorithm?
Can we ban "Thanks" posts? they fill up the forum and are very annoying..
Hello. I am newbie thank you all for sharing your knowledge.
I want to warn all forum users against services of the guy "Remapmaster".
This man does not know what honor and mere honesty is.
I am new in this forum edu.
Request to all experience members for suggestions about programmer devices names?
I have been on here for a while and I still cant send PM's or download.


text:
Ti=Tribunal

sec=The parties consent to the personal and exclusive jurisdiction of {Tribunal.cl}.

=[G/Z/ol/Base]

redpajama_set_name: RedPajamaGithub


text:
\section{Introduction}

This note is concerned with modular forms for the (dual) Weil representation $\rho^*$ attached to an even lattice $\Lambda$ with quadratic form $Q$. Letting $A = \Lambda'/\Lambda$ denote the discriminant group of $\Lambda$, this is a representation of the metaplectic group $Mp_2(\mathbb{Z})$ on $\mathbb{C}[A]$. Weil representations are a family of representations with better-than-usual arithmetic properties; for one thing, there are bases of modular forms in every weight with integral coefficients \cite{McG}. (Inducing the trivial representation from noncongruence subgroups to $SL_2(\mathbb{Z})$ gives plenty of examples of representations for which this fails.) These representations are important in the context of theta lifts such as \cite{B}. \par \

When $\Lambda$ is positive-definite it is well-known that modular forms for $\rho$ are essentially the same as Jacobi forms of lattice index $\Lambda$. This is a higher-dimensional version of the theta decomposition of chapter 5 of \cite{EZ}. One immediate advantage of Jacobi forms is their natural algebraic structure; multiplying Jacobi forms corresponds to an unintuitive operation on vector-valued modular forms. \par \

In the vector-valued setting, many objects that are usually indexed by integers $n \in \mathbb{Z}$ have to be replaced by pairs $(n,\gamma)$ with $\gamma \in A$ and $n \in \mathbb{Z} - Q(\gamma)$ (as in the final paragraph of the introduction of \cite{Br}.) In particular we might consider Jacobi forms of index $(m,\beta)$ where $\beta \in A$ and $m \in \mathbb{Z} - Q(\beta)$ is not generally integral. Such Jacobi forms of rational index have appeared in \cite{W}. \par \

The general theory of Jacobi forms indexed by a rational, symmetric matrix was worked out by Raum \cite{R}. Nonintegral indices force the Heisenberg group to act through a nontrivial representation. The irreducible representations that can arise were classified in \cite{R}, where theta decompositions that reduce all such Jacobi forms to vector-valued modular forms also appear. \par \

In the first half of this note we define a family of (generally not irreducible) representations $\sigma_B^*$ of the Heisenberg group that appear when one considers the Fourier-Jacobi expansion of Siegel theta functions. For these Jacobi forms the theta decomposition is particularly explicit and has a simpler proof; it is little more than applying an index-raising Hecke operator to reduce to the theta decomposition of integral-index Jacobi forms. \par \

In the second half of this note we suggest some applications. We remark that Eisenstein series and Poincar\'e series match up under this isomorphism (proposition 8) so computing bases of rather general Jacobi forms reduces to Bruinier's formulas (\cite{Br}, section 1.2). We point out that setting $z=0$ in the resulting Jacobi forms leads to some isomorphisms between modular forms for lattices of unrelated discriminants and work out an amusing example (example 11). Finally we give another interpretation of quasi-pullbacks of Borcherds products (as in \cite{M}) and construct some Borcherds products of small weight on lattices of signature $(4,2)$. \par

\section{Jacobi forms of lattice index}

For simplicity we will work with fixed Gram matrices, rather than abstract discriminant forms or even lattices. Therefore let $\mathbf{S}$ be a Gram matrix (that is, a nondegenerate, symmetric integer matrix with even diagonal) of size $e \in \mathbb{N}$ and let $\rho^* = \rho^*_{\mathbf{S}}$ denote the (dual) Weil representation of the lattice $\mathbb{Z}^{e}$ with quadratic form $Q(x) = \frac{1}{2}x^T \mathbf{S}x$ and bilinear form $\langle x,y \rangle = Q(x+y) - Q(x) - Q(y)$ (as in \cite{B},\cite{Br}). If $A = A(\mathbf{S})$ denotes the discriminant group $\mathbf{S}^{-1} \mathbb{Z}^{e} / \mathbb{Z}^{e}$ and $\mathfrak{e}_{\gamma}$, $\gamma \in A$ denotes the natural basis of the group ring $\mathbb{C}[A]$ then on the usual generators $$S = \Big( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \sqrt{\tau} \Big), \quad T = \Big( \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, 1 \Big)$$ of $Mp_2(\mathbb{Z})$, $\rho^*$ is given by \begin{align*} &\rho^*(S)\mathfrak{e}_{\gamma} = \frac{\mathbf{e}(\mathrm{sig}(\mathbf{S}) / 8)}{\sqrt{|A|}} \sum_{\beta \in A} \mathbf{e}\Big( \langle \gamma, \beta \rangle \Big) \mathfrak{e}_{\beta}, \\ &\rho^*(T) \mathfrak{e}_{\gamma} = \mathbf{e}\Big( - Q(\gamma) \Big) \mathfrak{e}_{\gamma}.\end{align*} Here $\mathbf{e}(\tau) = e^{2\pi i \tau}$ and $\mathrm{sig}(\mathbf{S}) = b^+ - b^-$ if $b^+,b^-$ are the numbers of positive and negative eigenvalues of $\mathbf{S}$. \par \

For $N \in \mathbb{N}$, let $\mathcal{H}_{N}$ denote the Heisenberg group of degree $N$: $$\mathcal{H}_{N} = \Big\{ (\lambda,\mu,t) \in \mathbb{Z}^{N} \times \mathbb{Z}^{N} \times \mathbb{Z}^{N \times N}: \; t - \lambda \mu^T \; \text{symmetric} \Big\}$$ with group operation $$(\lambda_1,\mu_1,t_1) \cdot (\lambda_2,\mu_2,t_2) = (\lambda_1 + \lambda_2,\mu_1 + \mu_2, t_1 + t_2 + \lambda_1\mu_2^T - \mu_1 \lambda_2^T).$$ For any matrix $B \in \mathbb{Q}^{e \times N}$ for which $\mathbf{S}B$ is integral, we can define a unitary representation $\sigma_B^*$ of $\mathcal{H}_{N}$ on $\mathbb{C}[A]$ by $$\sigma_B^*(\lambda,\mu,t) \mathfrak{e}_{\gamma} = \mathbf{e}\Big( -\gamma^T \mathbf{S} B \mu - \frac{1}{2} \mathrm{tr}(B^T \mathbf{S} B (t - \lambda \mu^T)) \Big) \mathfrak{e}_{\gamma - B \lambda},$$ and this depends only on $B$ mod $\mathbb{Z}^{e \times N}$. (When $N = 1$ this reduces to the representations $\sigma_{\beta}^*$ of \cite{W}.) These representations are compatible with $\rho^*$ in the sense that $$\rho^*(M)^{-1} \sigma_B^*(\zeta) \rho^*(M) = \sigma_B^*(\zeta \cdot M),$$ where $Mp_2(\mathbb{Z})$ acts on $\mathcal{H}_{N}$ from the right by $$(\lambda,\mu,t) \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = (a \lambda + c \mu, b \lambda + d \mu, t),$$ so they combine to a representation $\rho_B^*$ of the degree $N$ Jacobi group: $$\rho_B^* : \mathcal{J}_{N} = \mathcal{H}_{N} \rtimes Mp_2(\mathbb{Z}) \longrightarrow \mathrm{GL}\, \mathbb{C}[A],$$ $$\rho_B^*(\lambda,\mu,t,M) = \rho^*(M) \sigma_B^*(\lambda,\mu,t).$$ All of this can be shown directly but it is easier to prove by restricting the Weil representation of Siegel modular groups to their parabolic subgroups. \par \

\begin{defn} A \textbf{modular form} of weight $k \in \frac{1}{2}\mathbb{N}_0$ for the Gram matrix $\mathbf{S}$ is a holomorphic function $f : \mathbb{H} \rightarrow \mathbb{C}[A]$ that satisfies the functional equations $$f\Big( \frac{a \tau + b}{c \tau + d} \Big) = (c \tau + d)^k \rho^*(M) f(\tau), \quad M = \Big( \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \sqrt{c\tau + d} \Big) \in Mp_2(\mathbb{Z})$$ and is bounded in $\infty$. The space of modular forms of weight $k$ will be denoted $M_k(\mathbf{S})$.
\end{defn}

Any modular form can be expanded as a Fourier series $f(\tau) = \sum_{\gamma \in A(\mathbf{S})} \sum_{n \in \mathbb{Z} - Q(\gamma)} c(n,\gamma) q^n \mathfrak{e}_{\gamma}$ where $c(n,\gamma) = 0$ unless $n \ge 0$. It follows from the functional equation under $Z = (-I,i)$ that $M_k(\mathbf{S}) = 0$ unless $k + \mathrm{sig}(\mathbf{S})/2$ is an integer. Moreover the coefficients $c(n,\gamma)$ satisfy $$c(n,-\gamma) = (-1)^{k + \mathrm{sig}(\mathbf{S})/2} c(n,\gamma)$$ so we refer to $k$ as a \textbf{symmetric} or \textbf{antisymmetric weight} as $k + \mathrm{sig}(\mathbf{S})/2$ is even or odd, respectively. \par \

\begin{defn} Let $\mathcal{M} \in \mathbb{Q}^{N \times N}$ be a symmetric positive-definite matrix and let $B$ be a matrix as above. A \textbf{Jacobi form} of weight $k \in \frac{1}{2} \mathbb{N}_0$ and index $(\mathcal{M},B)$ for the Gram matrix $\mathbf{S}$ is a holomorphic function $\Phi : \mathbb{H} \times \mathbb{C}^{N} \rightarrow \mathbb{C}[A]$ that satisfies the functional equations $$\Phi \Big( \frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d} \Big) = (c \tau + d)^k \mathbf{e}\Big( \frac{c}{c \tau + d} z^T \mathcal{M} z \Big) \rho^*(M) \Phi(\tau,z), \quad M = \Big( \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \sqrt{c \tau + d} \Big) \in Mp_2(\mathbb{Z})$$ and $$\Phi(\tau, z + \lambda \tau + \mu) = \mathbf{e}\Big(-\tau \lambda^T \mathcal{M} \lambda - \mathrm{tr}\, \mathcal{M}(2 \lambda z^T + t + \mu \lambda^T) \Big) \sigma_B^*(\lambda,\mu,t) \Phi(\tau,z)$$ for all $(\lambda,\mu,t)$ in the Heisenberg group (in particular, the left side of the above equation should be independent of $t$) and for which the usual vanishing condition on Fourier coefficients holds (see below). We denote the space of Jacobi forms as above by $J_{k,\mathcal{M},B}(\mathbf{S})$.
\end{defn}

This transformation law is often expressed through the slash operator $|_{k,\mathcal{M},B}$: $$\Phi \Big|_{k,\mathcal{M},B} M(\tau,z) = (c \tau + d)^{-k} \mathbf{e}\Big( -\frac{c}{c \tau + d} z^T \mathcal{M} z \Big) \rho^*(M)^{-1} \Phi(M \cdot \tau,z)$$ and $$\Phi \Big|_{k,\mathcal{M},B} (\lambda,\mu,t)(\tau,z) = \mathbf{e}\Big( \tau \lambda^T \mathcal{M} \lambda + \mathrm{tr}\, \mathcal{M}(2 \lambda z^T + t + \mu \lambda^T) \Big) \sigma_B^*(\lambda,\mu,t)^{-1} \Phi(\tau,z + \lambda \tau + \mu).$$

Jacobi forms have Fourier expansions in both variables; we write this out as $$\Phi(\tau,z) = \sum_{\gamma \in A(\mathbf{S})} \sum_{n \in \mathbb{Q}} \sum_{r \in \mathbb{Q}^{N}} c(n,r,\gamma) q^n \zeta^r \mathfrak{e}_{\gamma},$$ where $q^n = e^{2\pi i n \tau}$ and $\zeta^r = e^{2\pi i r^T z}$. The vanishing condition referred to above is then $c(n,r,\gamma) = 0$ whenever $4n - r^T \mathcal{M}^{-1} r < 0$. The indices $(n,r,\gamma)$ and $(\mathcal{M},B)$ that can actually occur are also restricted as follows: \par \

\noindent (i) the transformation under $T \in Mp_2(\mathbb{Z})$ implies $c(n,r,\gamma) = 0$ unless $n \in \mathbb{Z} - Q(\gamma)$; \\ (ii) the transformation under $(0,0,e_i e_j^T + e_j e_i^T) \in \mathcal{H}_{N}$ implies $c(n,r,\gamma) = 0$ unless $\mathcal{M} + \frac{1}{2}B^T \mathbf{S}B$ is half-integral, with integral diagonal; \\ (iii) the transformation under $(0,\mu,0) \in \mathcal{H}_{N}$ implies $c(n,r,\gamma) = 0$ unless $r \in \mathbb{Z}^{N} - B^T \mathbf{S} \gamma$. \par \

Moreover, the familiar restriction on weights holds: $J_{k,\mathcal{M},B}(\mathbf{S}) = 0$ unless $k + \mathrm{sig}(\mathbf{S})/2$ is integral, in which case $c(n,-r,-\gamma) = (-1)^{k + \mathrm{sig}(\mathbf{S})/2} c(n,r,\gamma)$ for all Jacobi forms $\sum c(n,r,\gamma) q^n \zeta^r \mathfrak{e}_{\gamma}$. \par \

\begin{rem} Vector-valued modular forms and Jacobi forms for a fixed Gram matrix $\mathbf{S}$ do not have natural ring structures although they are modules over the ring of scalar-valued modular forms $M_*$ in a natural way. However we will make reference to the operation of tensoring at several points. If $\mathbf{S}_1$ and $\mathbf{S}_2$ are two Gram matrices (possibly of different sizes) then tensoring their Weil representations gives $$\rho^*_{\mathbf{S}_1} \otimes \rho^*_{\mathbf{S}_2} \cong \rho^*_{\mathbf{S}_1 \oplus \mathbf{S}_2}, \quad \mathbf{S}_1 \oplus \mathbf{S}_2 = \begin{pmatrix} \mathbf{S}_1 & 0 \\ 0 & \mathbf{S}_2 \end{pmatrix}$$ and therefore the tensor product defines a bilinear map $$\otimes : M_{k_1}(\mathbf{S}_1) \times M_{k_2}(\mathbf{S}_2) \longrightarrow M_{k_1 + k_2}(\mathbf{S}_1 \oplus \mathbf{S}_2)$$ for all valid weights $k_1,k_2$. Similarly if $(\mathcal{M}_1,B_1)$ and $(\mathcal{M}_2,B_2)$ are valid indices for Jacobi forms with the same number of elliptic variables (i.e. the same $N$) then it follows from $$\sigma_{B_1}^* \otimes \sigma_{B_2}^* \cong \sigma_B^*, \quad B = \begin{pmatrix} B_1 \\ B_2 \end{pmatrix}$$ that the tensor product defines a bilinear map $$\otimes : J_{k_1,\mathcal{M}_1,B_1}(\mathbf{S}_1) \times J_{k_2,\mathcal{M}_2,B_2}(\mathbf{S}_2) \longrightarrow J_{k_1+k_2,\mathcal{M}_1+\mathcal{M}_2,B}(\mathbf{S}_1 \oplus \mathbf{S}_2).$$
\end{rem}

\begin{rem} We recover the familiar definition of Jacobi forms of lattice index (e.g. \cite{G}) by restricting to the case that $B=0$ and $\mathrm{det}(\mathbf{S}) = 1$ and by letting $2\mathcal{M}$ be the Gram matrix of the index lattice with respect to any basis.
\end{rem}

\section{Hecke $U$-operators}

The Hecke operators $U_{\ell}$, $\ell \in \mathbb{N}$ of Eichler and Zagier \cite{EZ} generalize immediately to Jacobi forms of lattice index: we define $$U_{\ell} : J_{k,\mathcal{M},B}(\mathbf{S}) \rightarrow J_{k,\ell^2 \mathcal{M}, \ell B}(\mathbf{S}), \quad U_{\ell}\Phi(\tau,z) = \Phi(\tau,\ell z).$$ The map $U_{\ell}$ is injective and $M_*$-linear and its image of $U_{\ell}$ consists of those Jacobi forms whose Fourier coefficients $c(n,r,\gamma)$ vanish unless $\ell^{-1} r \in \mathbb{Z}^{N} - B^T \mathbf{S} \gamma.$ \par \

There is a corresponding operator $U_{\ell}$ that acts on modular forms for the Weil representation attached to a (not necessarily direct) sum of two quadratic forms. Let $\mathbf{S}_1 \in \mathbb{Z}^{e \times e}, \mathbf{S}_2 \in \mathbb{Z}^{N \times N}$ be Gram matrices and let $U \in \mathbb{Z}^{e \times N}$. We define $$U_{\ell} = U_{\ell}^{e,N} : M_k \Big( \begin{pmatrix} \mathbf{S}_1 & U \\ U^T & \mathbf{S}_2 \end{pmatrix} \Big) \longrightarrow M_k \Big( \begin{pmatrix} \mathbf{S}_1 & \ell U \\ \ell U^T & \ell^2 \mathbf{S}_2 \end{pmatrix} \Big)$$ as the pullback $\uparrow_{\mathbb{Z}^{e} \times \mathbb{Z}^{N}}^{\mathbb{Z}^{e} \times \ell \mathbb{Z}^{N}}$ in the sense of lemma 5.5 of \cite{Br}; that is, $$U_{\ell} \Big[ \sum_{n,\gamma} c(n,\gamma) q^n \mathfrak{e}_{\gamma} \Big] = \sum_{n,\gamma} c(n,\gamma) q^n \times \sum_{p(\delta) = \gamma} \mathfrak{e}_{\delta},$$ where $p$ is a projection map that amounts to multiplying the second block component by $\ell$. \par \

\begin{ex} The weight three Eisenstein series for the Weil representation attached to the lattice with Gram matrix $\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ is $$E_3(\tau) = (1 + 72q + 270q^2 + 720q^3 + ...) \mathfrak{e}_{(0,0)} + (27q^{2/3} + 216q^{5/3} + 459q^{8/3} +...) (\mathfrak{e}_{(1/3,1/3)} + \mathfrak{e}_{(2/3,2/3)}).$$ Letting $e = N = 1$, its image under the Hecke map $U_2$ is \begin{align*} U_2 E_3(\tau) &= (1 + 72q + 270q^2 + ...) (\mathfrak{e}_{(0,0)} + \mathfrak{e}_{(0,1/2)}) + \\ &\quad +  (27q^{2/3} + 216q^{5/3} + 459q^{8/3} + ...) (\mathfrak{e}_{(1/3,2/3)} + \mathfrak{e}_{(2/3,1/3)} + \mathfrak{e}_{(1/3,1/6)} + \mathfrak{e}_{(2/3,5/6)}), \end{align*} which is a weight three modular form for the Weil representation attached to the Gram matrix $\begin{pmatrix} 2 & 2 \\ 2 & 8 \end{pmatrix}$.
\end{ex}

\begin{rem} One can define $U_{\ell}$ on arbitrary (holomorphic) functions $f(\tau,z)$ by $U_{\ell} f(\tau,z)= f(\tau,\ell z)$. Then $U_{\ell}$ respects the Petersson slash operator in the sense that $$U_{\ell} \Big( f|_{k,\mathcal{M},B} M \Big) = (U_{\ell} f) \Big|_{k,\ell^2 \mathcal{M}, \ell B} M$$ and $$U_{\ell} \Big( f |_{k,\mathcal{M},B} (\ell \lambda, \ell \mu, \ell^2 t) \Big) = (U_{\ell} f) \Big|_{k,\ell^2 \mathcal{M}, \ell B} (\lambda, \mu, t).$$ A similar statement holds for the operator $U_{\ell}$ on modular forms.
\end{rem}

\section{Theta decomposition}

Let $\mathbf{S}_1$ and $\mathbf{S}_2$ be Gram matrices of sizes $e,N$ where $\mathbf{S}_2$ is positive-definite and set $\mathcal{M} = \frac{1}{2}\mathbf{S}_2$. There is an isomorphism of modules $$M_{k-N/2}(\mathbf{S}_1 \oplus \mathbf{S}_2) \longrightarrow J_{k,\mathcal{M},0}(\mathbf{S}_1)$$ over the ring $M_*$ of scalar modular forms which is essentially given by tensoring with the theta function $\Theta_{\mathbf{S}_2}(\tau,z)$ of $\mathbf{S}_2$ and applying a trace map (see the next section for more details) and which identifies the modular form $$F(\tau) = \sum_{\gamma \in A(\mathbf{S}_1 \oplus \mathbf{S}_2)} \sum_n c(n,\gamma) q^n \mathfrak{e}_{\gamma}$$ with the Jacobi form $$\Phi(\tau,z) = \Theta(F)(\tau,z) = \sum_{\gamma \in A(\mathbf{S}_1)} \sum_{n,r} c(n,(\gamma,\mathbf{S}_2^{-1} r)) q^{n + r^T \mathbf{S}_2^{-1} r / 2} \zeta^r \mathfrak{e}_{\gamma}.$$

In this section we record a simple generalization of this. Let $\mathbf{S}$ be a Gram matrix of size $e$ and let $B \in \mathbb{Q}^{e \times N}$ be a matrix for which $\mathbf{S}B$ is integral. Also let $\mathcal{M} \in \mathbb{Q}^{N \times N}$ be a symmetric positive-definite matrix for which $2\mathcal{M} + B^T \mathbf{S}B$ is a Gram matrix, i.e. integral with even diagonal, and set $$\mathbf{\tilde S} = \begin{pmatrix} \mathbf{S} & \mathbf{S} B \\ B^T \mathbf{S} & 2 \mathcal{M} + B^T \mathbf{S} B \end{pmatrix}.$$

\begin{prop} There is an isomorphism of $M_*$-modules $$\Theta : M_{*-N/2}(\mathbf{\tilde S}) \longrightarrow J_{*,\mathcal{M},B}(\mathbf{S})$$ that sends a modular form $$F(\tau) = \sum_{\gamma \in A(\mathbf{\tilde S})} \sum_{n \in \mathbb{Z}^{e + N} - \tilde Q(\gamma)} c(n,\gamma) q^n \mathfrak{e}_{\gamma}$$ to the Jacobi form $$\Phi(\tau,z) = \sum_{\gamma \in A(\mathbf{S})} \sum_{n \in \mathbb{Z} - Q(\gamma)} \sum_{r \in \mathbb{Z}^{N} - B^T \mathbf{S}\gamma} c\Big(n, (\gamma - \frac{1}{2} B \mathcal{M}^{-1} r, \frac{1}{2} \mathcal{M}^{-1} r)\Big) q^{n + \frac{1}{4} r^T \mathcal{M}^{-1} r} \zeta^r \mathfrak{e}_{\gamma}.$$
\end{prop}
\begin{proof} First we should clarify that if $\gamma \in \mathbf{S}^{-1} \mathbb{Z}^e$ and $r \in \mathbb{Z}^N - B^T \mathbf{S} \gamma$, then $$\begin{pmatrix} \mathbf{S} & \mathbf{S} B \\ B^T \mathbf{S} & 2 \mathcal{M} + B^T \mathbf{S} B \end{pmatrix} \begin{pmatrix} \gamma - \frac{1}{2} B \mathcal{M}^{-1} r \\ \frac{1}{2} \mathcal{M}^{-1} r \end{pmatrix} = \begin{pmatrix} \mathbf{S} \gamma \\ B^T \mathbf{S} \gamma + r \end{pmatrix} \in \mathbb{Z}^{e+N}$$ and therefore $$\left(\gamma - \frac{1}{2}B \mathcal{M}^{-1} r, \frac{1}{2} \mathcal{M}^{-1} r\right) \in \mathbf{\tilde S}^{-1} \mathbb{Z}^{e+N};$$ moreover this is a well-defined element of $A(\mathbf{\tilde S})$ when $\gamma$ is interpreted as an element of $A(\mathbf{S})$. \par

Choose $\ell \in \mathbb{N}$ such that $\ell \cdot B \in \mathbb{Z}^{e \times N}$. The Gram matrix $\begin{pmatrix} \mathbf{S} & \ell \mathbf{S} B \\ \ell B^T \mathbf{S} & \ell^2 (2 \mathcal{M} + B^T \mathbf{S} B) \end{pmatrix}$ is equivalent to $\mathbf{S} \oplus 2 \ell^2 \mathcal{M}$ so for every weight $k$ we have an injective map $$M_{k-N / 2}(\mathbf{\tilde S}) \stackrel{U_{\ell}}{\longrightarrow} M_{k-N / 2}(\mathbf{S} \oplus 2 \ell^2 \mathcal{M})\longrightarrow J_{k,\ell^2 \mathcal{M},0}(\mathbf{S}).$$ In terms of Fourier expansions this sends $F(\tau) = \sum_{(\gamma_1,\gamma_2) \in A(\mathbf{\tilde S})} \sum_n c(n,(\gamma_1,\gamma_2)) q^n \mathfrak{e}_{(\gamma_1,\gamma_2)}$ first to the modular form $$U_{\ell} F(\tau) = \sum_{\gamma_1 \in A(\mathbf{S})} \sum_{\gamma_2 \in A(2 \ell^2 \mathcal{M})} \sum_n c(n,(\gamma_1 - \ell B\gamma_2, \ell \gamma_2)) q^n \mathfrak{e}_{(\gamma_1, \gamma_2)}$$ and then to the Jacobi form $$\Phi(\tau,z) = \sum_{\gamma \in A(\mathbf{S})} \sum_{n,r}  c\Big(n,\left(\gamma - \frac{1}{2\ell} B \mathcal{M}^{-1} r , \frac{1}{2\ell} \mathcal{M}^{-1} r\right)\Big) q^{n + \frac{1}{4\ell^2} r^T \mathcal{M}^{-1} r} \zeta^r \mathfrak{e}_{\gamma}.$$ These coefficients vanish unless $$\begin{pmatrix} \mathbf{S} & \mathbf{S}B \\ B^T \mathbf{S} & 2 \mathcal{M} + B^T \mathbf{S} B \end{pmatrix} \begin{pmatrix} \gamma - \frac{1}{2\ell} B \mathcal{M}^{-1} r \\ \frac{1}{2\ell} \mathcal{M}^{-1} r \end{pmatrix} = \begin{pmatrix} \mathbf{S} \gamma \\ B^T \mathbf{S} \gamma + \ell^{-1} r \end{pmatrix}$$ is integral, or equivalently $\ell^{-1} r \in \mathbb{Z}^{N} - B^T \mathbf{S} \gamma$, which is the condition for $\Phi$ to be in the range of $U_{\ell} : J_{k,\mathcal{M},B}(\mathbf{S}) \rightarrow J_{k,\ell^2 \mathcal{M},0}(\mathbf{S})$. Therefore we define the theta decomposition by requiring the diagram

\begin{center}
\begin{tikzpicture}[node distance=2cm, auto]
\node (A) {$M_{k-N/2}(\mathbf{\tilde S})$};
\node(AA) [right of = A] {$ $};
\node (B) [right of = AA] {$M_{k-N/2}(\mathbf{S} \oplus 2 \ell^2 \mathcal{M})$};
\node (C) [below of = A] {$J_{k,\mathcal{M},B}(\mathbf{S})$};
\node (D) [below of = B] {$J_{k,\ell^2 \mathcal{M},0}(\mathbf{S})$};
\draw[dashed,->] (A) to node {$ $} (C);
\draw[->] (A) to node {$U_{\ell}$} (B);
\draw[->] (C) to node {$U_{\ell}$} (D);
\draw[->] (B) to node {$ $} (D);
\end{tikzpicture}
\end{center} to commute. Its inverse can be constructed in the same way: starting with a Jacobi form, apply $U_{\ell}$ and the usual theta decomposition, then identify the result in the image of $U_{\ell}$ on modular forms. That this is independent of $\ell$ can be seen from the coefficient formula for $\Phi$ in the claim.
\end{proof}

\section{Eisenstein and Poincar\'e series}

Some of the simplest Jacobi forms that can be constructed are the Poincar\'e series: $$\mathcal{P}_{k,(\mathcal{M},B),(n,r,\gamma)}(\tau) = \sum_{(\lambda,0,0,M) \in \mathcal{J}_{N,\infty} \backslash \mathcal{J}_{N}} q^n \zeta^r \mathfrak{e}_{\gamma} \Big|_{k,\mathcal{M},B} (\zeta,M),$$ where $(\lambda,0,0,M)$ runs through cosets of the Jacobi group $\mathcal{J}_{N}$ by the subgroup generated by all $(0,\mu,t) \in \mathcal{H}_{N}$ and by $T = (\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, 1)$ and $Z = (-I,i)$ from $Mp_2(\mathbb{Z})$. The series $P_{n,r,\gamma}(\tau)$ is well-defined as long as $n \in \mathbb{Z} - Q(\gamma)$ and $r \in \mathbb{Z}^{N} - B^T \mathbf{S}\gamma$ and $k > 2 + N/2$ and it always transforms like a Jacobi form by construction. The vanishing condition on Fourier coefficients is satisfied when $4n - r^T \mathcal{M}^{-1} r \ge 0$. If $n=r=0$ then we call $\mathcal{E}_{k,\mathcal{M},\gamma}(\tau) = \mathcal{P}_{k,\mathcal{M},0,0,\gamma}(\tau)$ the \textbf{Eisenstein series}. See also \cite{M2} which considers the scalar-valued case in more detail. \par \

The point of this section is to show that the Jacobi Eisenstein and Poincar\'e series as above correspond to the usual Eisenstein and Poincar\'e series as in chapter $1$ of \cite{Br}: $$P_{k,n,\gamma}(\tau) = \sum_{M \in \Gamma_{\infty} \backslash \Gamma} q^n \mathfrak{e}_{\gamma} \Big|_{k,\rho^*} M;$$ i.e. any coefficient formula for Jacobi Eisenstein and Poincar\'e series even in the generality considered here (which includes the author's computations in \cite{W}) will reduce to theorem 1.3 of \cite{Br}. \par \

\begin{prop} In the notation of the previous section, the theta decomposition satisfies $$\Theta \Big( P_{k-N / 2, n, (\gamma - \frac{1}{2} B \mathcal{M}^{-1} r, \frac{1}{2} \mathcal{M}^{-1} r)} \Big) = \mathcal{P}_{k,(\mathcal{M},B),(n + r^T \mathcal{M}^{-1} r / 4, r, \gamma)}.$$
\end{prop}
Poincar\'e series extract Fourier coefficients with respect to an appropriately defined Petersson scalar product so this correspondence is suggested by the formula in proposition 7; one could give a proof along these lines but we use a more direct argument here.
\begin{proof} Suppose first that $B=0$ so the theta decomposition maps $F \in M_{k-N/2}(\mathbf{S} \oplus 2\mathcal{M})$ to $\pi(F \otimes \Theta)$ where $\Theta(\tau,z) = \sum_{\lambda \in \mathbb{Z}^{N}} q^{\lambda^T \mathcal{M}^{-1} \lambda / 4} \zeta^{\lambda} \mathfrak{e}_{(2\mathcal{M})^{-1} \lambda}$ and where $\pi$ is the linear map $$\pi : \mathbb{C}[A(\mathbf{S})] \otimes \mathbb{C}[A(2\mathcal{M})] \otimes \mathbb{C}[A(2\mathcal{M})] \rightarrow \mathbb{C}[A(\mathbf{S})], \quad \pi(\mathfrak{e}_{\beta} \otimes \mathfrak{e}_{\gamma} \otimes \mathfrak{e}_{\delta}) = \begin{cases} \mathfrak{e}_{\beta}: & \gamma = \delta; \\ 0: & \text{otherwise}. \end{cases}$$ Here $\Theta$ is a Jacobi form of weight $N / 2$ and index $\mathcal{M}$ for the (non-dual) Weil representation $\rho_{2\mathcal{M}}$ by Poisson summation. One can check that $\pi$ satisfies $\pi \circ (\rho_{\mathbf{S} \oplus 2 \mathcal{M}}^* \otimes \rho_{2\mathcal{M}}(M)) = \rho_{\mathbf{S}}^*(M) \circ \pi$ for all $M \in Mp_2(\mathbb{Z})$ on the generators $M = S,T$. It follows that if $F(\tau) = \sum_{M \in \Gamma_{\infty} \backslash \Gamma} \phi(\tau) \Big|_{k - N / 2, \rho_{\mathbf{S} \otimes 2 \mathcal{M}}^*} M$ for some seed function $\phi$ then

\begin{align*} &\quad \sum_{M \in \Gamma_{\infty} \backslash \Gamma} \pi\Big( \phi(\tau) \otimes \Theta(\tau,z) \Big) \Big|_{k,\mathcal{M}, \rho_{\mathbf{S}}^*} M \\ &= \pi\sum_{M \in \Gamma_{\infty} \backslash \Gamma} \Big(\phi(\tau) \otimes \Theta(\tau,z)\Big) \Big|_{k,\mathcal{M}, \rho_{\mathbf{S} \oplus 2 \mathcal{M}}^* \otimes \rho_{2\mathcal{M}}} M \\ &= \pi \sum_{c,d} (c \tau + d)^{-k+N/2} \rho_{\mathbf{S} \oplus 2 \mathcal{M}}^*(M)^{-1} \phi(M \cdot \tau) \otimes (c \tau + d)^{-N/2} \mathbf{e}\Big(-\frac{c}{c\tau + d} z^T \mathcal{M} z \Big) \rho_{2\mathcal{M}}(M)^{-1} \Theta\left(\frac{a\tau + b}{c \tau + d},\frac{z}{c\tau + d}\right) \\ &= \pi \sum_{c,d} (c \tau + d)^{-k+N/2} \rho_{\mathbf{S} \oplus 2 \mathcal{M}}^*(M)^{-1} \phi(M \cdot \tau) \otimes \Theta(\tau,z) \\ &= \pi \Big(F(\tau) \otimes \Theta(\tau,z)\Big) \end{align*} is the Jacobi form corresponding to $F$. When $\phi(\tau) = q^n \mathfrak{e}_{\gamma} \otimes \mathfrak{e}_{(2 \mathcal{M})^{-1} r}$ for some $\gamma \in A(\mathbf{S})$ then

\begin{align*} \pi\Big( \phi(\tau) \otimes \Theta(\tau,z) \Big) &= \sum_{\mu \in r + 2 \mathcal{M} \mathbb{Z}^{N}} q^{n + \mu^T \mathcal{M}^{-1} \mu / 4} \zeta^{\mu} \mathfrak{e}_{\gamma} \\ &= \sum_{\lambda \in \mathbb{Z}^{N}} \mathbf{e}\Big( \tau \lambda^T \mathcal{M} \lambda + 2\lambda^T \mathcal{M} z \Big) q^{n + r^T \mathcal{M}^{-1} r / 4 + r^T \lambda} \zeta^r \mathfrak{e}_{\gamma} \quad (\mu = r + 2 \mathcal{M} \lambda) \\ &= \sum_{\lambda \in \mathbb{Z}^{N}} q^{n + r^T \mathcal{M}^{-1} r / 4} \zeta^r \mathfrak{e}_{\gamma} \Big|_{k,\mathcal{M}} (\lambda,0,0)\end{align*} and therefore \begin{align*} \pi \Big( P_{k-N/2,n,(\gamma,(2 \mathcal{M})^{-1} r)}(\tau) \otimes \Theta(\tau,z) \Big) &= \sum_{M \in \Gamma_{\infty} \backslash \Gamma} \pi\Big( \phi(\tau) \otimes \Theta(\tau,z) \Big) \Big|_{k,\mathcal{M}, \rho_{\mathbf{S}}^*} M \\ &= \sum_{(\lambda,0,0,M) \in \mathcal{J}_{\infty} \backslash \mathcal{J}} \Big( q^{n + r^T \mathcal{M}^{-1} r / 4} \zeta^r \mathfrak{e}_{\gamma} \Big) \Big|_{k,\mathcal{M}} (\lambda,0,0) \Big|_{k,\mathcal{M},\rho_{\mathbf{S}}^*} M \\ &= \mathcal{P}_{k,\mathcal{M},(n+r^T \mathcal{M}^{-1} r / 4, r, \gamma)}(\tau,z). \end{align*}

For the general case of theta decomposition in the Gram matrix $\mathbf{\tilde S} = \begin{pmatrix} \mathbf{S} & \mathbf{S} B \\ B^T \mathbf{S} & 2 \mathcal{M} + B^T \mathbf{S} B \end{pmatrix}$ we need to check that the images of $P_{k-N/2,n,(\gamma-\frac{1}{2}B \mathcal{M}^{-1} r, \frac{1}{2} \mathcal{M}^{-1} r)}$ and $\mathcal{P}_{k,(\mathcal{M},B),(n + r^T \mathcal{M}^{-1} r / 4, r, \gamma)}$ match up correctly under $U_{\ell}$. On Jacobi forms we find \begin{align*} &\quad U_{\ell} \mathcal{P}_{k,(\mathcal{M},B),(n + r^T \mathcal{M}^{-1} r/4,r,\gamma)} \\ &= U_{\ell} \sum_{M \in \Gamma_{\infty} \backslash \Gamma} \sum_{\lambda \in \mathbb{Z}^N} (q^{n + r^T \mathcal{M}^{-1} r / 4} \zeta^r \mathfrak{e}_{\gamma}) \Big|_{k,\mathcal{M},B} (\lambda,0,0) \Big|_{k,\mathcal{M},B,\rho_{\mathbf{S}}^*} M \\ &= \sum_{M \in \Gamma_{\infty} \backslash \Gamma} \sum_{\lambda \in \mathbb{Z}^N} \Big[ U_{\ell} \Big( \sum_{\mu \in \mathbb{Z}^N / \ell \mathbb{Z}^N} q^{n + r^T \mathcal{M}^{-1} r / 4} \zeta^r \mathfrak{e}_{\gamma} \Big|_{k,\mathcal{M},B} (\mu,0,0) \Big) \Big] \Big|_{k,\ell^2\mathcal{M},\ell B} (\lambda,0,0) \Big|_{k,\ell^2 \mathcal{M},\ell B,\rho_{\mathbf{S}}^*} M \\ &= \sum_{\mu \in \mathbb{Z}^N / \ell \mathbb{Z}^N} \mathcal{P}_{k, \ell^2 \mathcal{M}, (n + (r + 2 \mathcal{M}\mu)^T \mathcal{M}^{-1}(r + 2 \mathcal{M}\mu)/4, \ell r + 2\ell \mathcal{M}\mu, \gamma + B \mu)}, \end{align*} while on modular forms, abbreviating $\tilde \gamma = (\gamma - \frac{1}{2}B \mathcal{M}^{-1} r, \frac{1}{2} \mathcal{M}^{-1} r)$ and applying $\uparrow$ termwise gives \begin{align*} U_{\ell} P_{k-N/2,n,\tilde \gamma} &= U_{\ell} \Big[ \sum_{M \in \Gamma_{\infty} \backslash \Gamma} q^n \mathfrak{e}_{\tilde \gamma} \Big|_{k,\rho_{\mathbf{\tilde S}}^*} M \Big] \\ &= \sum_{M \in \Gamma_{\infty} \backslash \Gamma} \Big( q^n \sum_{\substack{\delta \in A(\mathbf{S} \oplus 2 \ell^2 \mathcal{M}) \\ p(\delta) = \tilde \gamma}} \mathfrak{e}_{\delta} \Big) \Big|_{k,\rho_{\mathbf{S} \oplus 2 \ell^2 \mathcal{M}}^*} M \\ &= \sum_{\mu \in \mathbb{Z}^N / \ell \mathbb{Z}^N} P_{k-N/2,n,(\gamma - \frac{1}{2} \ell^{-1} \mathcal{M}^{-1} r, \frac{1}{2}\ell^{-1} \mathcal{M}^{-1} r + \ell^{-1} \mu)}, \end{align*} which correspond as they are expected to.
\end{proof}

\section{Kohnen plus space}

Since Jacobi forms of very small index tend to be determined completely by their value at $z=0$, applying the theta decomposition and setting $z=0$ in the resulting Jacobi forms gives isomorphisms between modular forms of apparently unrelated levels. The simplest example is an identification of the Kohnen plus spaces of half-integral weight modular forms of level $4$ with modular forms of level $3$ and odd integral weight. \par \

Let $k \in \mathbb{N}_0$. The Kohnen plus space of weight $k + 1/2$ is the subspace $M_{k+1/2}^+(\Gamma_0(4))$ of modular forms of level $\Gamma_0(4)$ in which the Fourier coefficient $a_n$ is zero unless $(-1)^k n \equiv 0,1$ mod $4$. This was introduced by Kohnen in \cite{K} (also for more general levels) as a natural setting for the Shimura correspondence: in particular there are Hecke-equivariant isomorphisms $M_{k+1/2}^+ \cong M_{2k}$ to classical modular forms of integral weight and level $1$. \par \

\begin{prop} For every odd integer $k$, there is an isomorphism $$M_k(\Gamma_1(3)) = M_k(\Gamma_0(3),\chi) \cong M_{k-1/2}^+(\Gamma_0(4)) \times M_{k+1/2}^+(\Gamma_0(4)) .$$ Here $\chi$ is the nontrivial Nebentypus modulo $3$. More precisely, there are isomorphisms $$M_k^+(\Gamma_0(3),\chi) \cong M_{k-1/2}^+(\Gamma_0(4)), \quad M_k^-(\Gamma_0(3),\chi) \cong M_{k+1/2}^+(\Gamma_0(4)),$$ where $M_k^+(\Gamma_0(3),\chi)$ is the subspace of forms whose coefficients $a_n$ vanish when $n \equiv 2$ mod $3$; and $M_k^-(\Gamma_0(3),\chi)$ is the subspace of forms whose coefficients vanish when $n \equiv 1$ mod $3$.
\end{prop}
These isomorphisms are not Hecke-equivariant but they are simple to express in terms of the coefficients, and more generally they are isomorphisms of graded modules over the ring $M_*$ of classical modular forms of level one; see the remark below.
\begin{proof} Using a more general result of Bruinier and Bundschuh \cite{BB}, all of the spaces here can be interpreted as spaces of vector-valued modular forms: $$M_k^+(\Gamma_0(3),\chi) \cong M_k \Big( \begin{pmatrix} -2 & -1 \\ -1 & -2 \end{pmatrix} \Big), \quad M_k^-(\Gamma_0(3),\chi) \cong M_k\Big( \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \Big),$$ $$M_{k-1/2}^+(\Gamma_0(4)) \cong M_{k-1/2}\Big( \begin{pmatrix} -2 & -1 &1 \\ -1 & -2 & 0 \\ 1 & 0 & 0 \end{pmatrix} \Big), \quad M_{k+1/2}^+(\Gamma_0(4)) \cong M_{k+1/2}((2)),$$ since all of the Gram matrices above have prime determinant. These identifications are essentially given by summing all components; such that for example the weight three Eisenstein series $$E_3(\tau) = (1 - 90q - 216q^2 - ...) \mathfrak{e}_{(0,0)} + (-9q^{1/3} - 117q^{4/3} - ...) (\mathfrak{e}_{(1/3,1/3)} + \mathfrak{e}_{(2/3,2/3)}) \in M_3\Big( \begin{pmatrix} -2 & -1 \\ -1 & -2 \end{pmatrix} \Big)$$ and $$E_3(\tau) = (1 + 72q + 270q^2 + ...) \mathfrak{e}_{(0,0)} + (27q^{2/3} + 216q^{5/3} + ...) (\mathfrak{e}_{(1/3,1/3)} + \mathfrak{e}_{(2/3,2/3)}) \in M_3 \Big( \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \Big)$$ correspond to the scalar modular forms $$1 - 18q - 90q^3 - 234q^4 - 216q^6 - ... \in M_3(\Gamma_0(3),\chi)^+$$ and $$1 + 54q^2 + 72q^3 + 432q^5 + 270q^6 + ... \in M_3(\Gamma_0(3),\chi)^-,$$ respectively. The theta decomposition gives identifications $$M_k\Big( \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} \Big) \cong J_{k+1/2,3/4,(1/2)}((2))$$ and $$M_{k-1/2}\Big( \begin{pmatrix} -2 & -1 & 1 \\ -1 & -2 & 0 \\ 1 & 0 & 0 \end{pmatrix}\Big) \cong J_{k,1/3,(-2/3,1/3)} \Big( \begin{pmatrix} -2 & -1 \\ -1 & -2 \end{pmatrix} \Big)$$ and setting $z=0$ in the resulting Jacobi forms gives the maps $$M_k\Big( \begin{pmatrix}2 & 1 \\ 1 & 2 \end{pmatrix} \Big) \rightarrow M_{k+1/2}((2)), \; M_{k-1/2}\Big( \begin{pmatrix} -2 & -1 & 1 \\ -1 & -2 & 0 \\ 1 & 0 & 0 \end{pmatrix} \Big) \rightarrow M_k\Big( \begin{pmatrix} -2 & -1 \\ -1 & -2 \end{pmatrix} \Big)$$ in the claim. Both maps are injective: one can use theorem 1.2 of \cite{EZ} to see that any Jacobi form $\Phi$ of index strictly less than one half for which $\Phi(\tau,0)$ vanishes, or of index less than one for which $\Phi(\tau,0)$ and $\frac{d}{dz}\Big|_{z=0} \Phi(\tau,z)$ both vanish, must be identically zero. (In the second case, $\frac{d}{dz}\Big|_{z=0} \Phi(\tau,z)$ is a modular form of antisymmetric weight for the Gram matrix $(2)$ so it vanishes identically regardless of $\Phi$.) Theorem 1.2 of \cite{EZ} considers only integer-index Jacobi forms but we can reduce to that case by tensoring $\Phi$ with itself enough times to produce something of integer index. \par \

To see that these are isomorphisms we compare the dimensions of both sides. One way to show that the dimensions are the same is to check the first few weights and then use the fact that for $k \ge 3$, the dimensions in all four cases increase by $2$ when we replace $k$ by $k + 12$ (since $2$ is the number of elements $\gamma$ in each discriminant group up to equivalence $\gamma \sim -\gamma$).
\end{proof}

\begin{rem}
Unraveling this argument in terms of scalar-valued modular forms, the isomorphisms above are $$M_k^-(\Gamma_1(3)) \rightarrow M_{k+1/2}^+(\Gamma_0(4)), \quad f(\tau) \mapsto \frac{1}{18} \begin{pmatrix} f(\frac{4\tau}{3}) & f(\frac{4 \tau + 1}{3}) & f(\frac{4 \tau + 2}{3}) \end{pmatrix} \begin{pmatrix} 4 & 1 & 1 \\ 1 & 4 & 1 \\ 1 & 1 & 4 \end{pmatrix} \begin{pmatrix} \vartheta(\frac{\tau}{3}) \\ \vartheta(\frac{\tau + 1}{3}) \\ \vartheta(\frac{\tau + 2}{3}) \end{pmatrix}$$ and $$M_{k-1/2}^+(\Gamma_0(4)) \rightarrow M_k^+(\Gamma_1(3)),$$ $$f(\tau) \mapsto \frac{1}{4} \sum_{a \in \mathbb{Z}/4\mathbb{Z}} f\Big( \frac{3 \tau + a}{4} \Big) \vartheta\Big( \frac{\tau - a}{4} \Big),$$ where $\vartheta(\tau) = 1 + 2q + 2q^4 + 2q^9 + ...$ One could probably prove directly that these are well-defined isomorphisms but using the argument above this is more or less automatic.
\end{rem}

\begin{ex} The first part of remark 10 means that to compute the Kohnen plus form corresponding to $f(\tau) = \sum_n a_n q^n \in M_k^-(\Gamma_1(3))$, we divide all coefficients $a_n$, $n \not \equiv 0 \, (3)$ by two, change variables $q \mapsto q^{4/3}$, multiply by $\vartheta(\tau/3)$, and restrict to integer exponents. For example, to compute the Kohnen plus form corresponding to $1 + 54q^2 + 72q^3 + 432q^5 + 270q^6 + ...$ we multiply \begin{align*} &\quad \Big( 1 + 27q^{8/3} + 72q^4 + 216q^{20/3} + 270q^8 + ... \Big) \Big( 1 + 2q^{1/3} + 2q^{4/3} + 2q^3 + ... \Big) \\ &= 1 + 2q^{1/3} + 2q^{4/3} + 27q^{8/3} + 56q^3 +  126q^4 + 144q^{13/3} + 146q^{16/3} + 54q^{17/3} + 216q^{20/3} + 576q^7 + ... \end{align*} and restrict to integer exponents to find the Cohen Eisenstein series $$1 + 56q^3 + 126q^4 + 576q^7 + ...$$ of weight $7/2$ \cite{C}. \par \

The second part of remark 10 means that to compute the level three form corresponding to a Kohnen plus form $f(\tau)$, we multiply $f(3\tau/4)$ by $\vartheta(\tau/4)$ and restrict to integer exponents. For example, the Cohen Eisenstein series of weight $5/2$ is $$f(\tau) = 1 - 10q - 70q^4 - 48q^5 - 120q^8 - 250q^9 - 240q^{12} - 240q^{13} - ...$$ and since \begin{align*} &\quad \Big( 1 - 10q^{3/4} - 70q^3 - 48q^{15/4} - 120q^6 - 250q^{27/4} - 240q^9 - ... \Big) \Big( 1 + 2q^{1/4} + 2q + 2q^{9/4} + 2q^4 + ... \Big) \\ &= 1 + 2q^{1/4} -10q^{3/4} - 18q - 20q^{7/4} + 2q^{9/4} - 90q^3 - 140q^{13/4} - 48q^{15/4} - 234q^4 - ...\end{align*} we see that the corresponding level three form is $$1 - 18q - 90q^3 - 234q^4 - ... \in M_3^+(\Gamma_1(3)).$$ We could also point out that the theta function $\vartheta(\tau) = 1 + 2q + 2q^4 + 2q^9 + ... \in M_{1/2}^+(\Gamma_0(4))$ itself corresponds to the theta function $\theta(\tau) = 1 + 6q + 6q^3 + 6q^4 + 12q^7 + ... \in M_1^+(\Gamma_0(3),\chi)$ of $x^2 + xy + y^2$ under this isomorphism.

\end{ex}

\section{Hermitian modular products of small weight}

Recall that the Borcherds lift takes vector-valued modular forms for the (non-dual) Weil representation attached to a lattice of signature $(b^+,2)$ with singularities at cusps to orthogonal modular forms with product expansions and known divisors (\cite{B},\cite{Br}). Given such an automorphic product one can produce automorphic forms on subgrassmannians essentially by restriction, with a renormalization process to deal with a possible zero or pole there. This operation is called the quasi-pullback (see \cite{M}, section 3) and it can be used to construct infinite families of Borcherds products. For example, Gritsenko and Nikulin constructed a paramodular form of weight $12$ in every level $t$ except $t=1$ using this kind of argument for the lift of $\Delta^{-1}$ to the Grassmannian of $II_{26,2}$ (\cite{GN2}, remark 4.4). It is also possible to find automorphic forms on subgrassmannians by tensoring the original input form by various theta functions and taking the Borcherds lift of this. Ma showed in \cite{M} that these processes lead to the same result, at least when Koecher's principle applies. \par \

Theta decompositions allow one to identify the nearly-holomorphic input functions with weak Jacobi forms, and in the case of integral index, setting $z=0$ in those Jacobi forms has the same effect as tensoring with theta functions as in the last paragraph. Taking theta decompositions into Jacobi forms of nonintegral index and setting $z=0$ gives a similar interpretation of the $\Theta$-contraction of \cite{M}. \par \

We give an example here by showing that for even discriminants there always exist Hermitian modular products of small weight. These are essentially the same as orthogonal modular forms on Grassmannians of signature $(4,2)$ lattices of the form $\Lambda_K = \mathcal{O}_K \oplus II_{2,2}$, where $\mathcal{O}_K$ denotes the ring of integers in an imaginary-quadratic number field $K$ together with its norm-form $N_{K/\mathbb{Q}}$ (see \cite{Dern} for details). This is based on an observation of A. Krieg who suggested this application to me (see \cite{Kr}). \par

\begin{prop} Let $K$ be an imaginary-quadratic field of even discriminant $d_K$ and set $m = -\frac{d_K}{4}$. \\ (i) If $m$ is a sum of three nonzero squares then there is a holomorphic product for $\Lambda_K$ of weight two. \\ (ii) If $m$ is a sum of two nonzero squares then there is a holomorphic product for $\Lambda_K$ of weight three. \\ (iii) If $m=1$ then there is a holomorphic product for $\Lambda_K$ of weight four.
\end{prop}
The cases $m=1,2$ are well-known and were important in finding the structure of the rings of Hermitian modular forms for $K = \mathbb{Q}(i)$ and $K = \mathbb{Q}(\sqrt{-2})$ in \cite{DK1},\cite{DK2}. One can also find the cases $m=5,6$ in the tables of appendix B of \cite{W1}. Those tables also show that no such statement is possible for odd discriminants; for example, the holomorphic products for $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{-7})$ of smallest weight have weight $9$ and $7$, respectively. Note that (i) and (ii) can both occur for a single $m$, e.g. $m = 17 = 1^2 + 4^2 = 2^2 + 2^2 + 3^2$, where we get holomorphic products of both weights.
\begin{proof} The products in the claim arise as quasi-pullbacks of a singular-weight product on a Grassmannian of type $O(6,2)$; we will show how these arise as Borcherds lifts using the theta decomposition. Suppose first that $m$ has a representation of the form $m = 1 + a^2 + b^2$ with $a,b \in \mathbb{N}_0$. Then the matrix $$\mathbf{S} = \begin{pmatrix} 2 & 0 & 0 & 0 \\ 0 & 2m & 2a & 2b \\ 0 & 2a & 2 & 0 \\ 0 & 2b & 0 & 2 \end{pmatrix}$$ is the Gram matrix of $x_1^2 + x_2^2 + x_3^2 + x_4^2$ with respect to the basis $(1,0,0,0),(0,1,a,b),(0,0,1,0),(0,0,0,1)$ of $\mathbb{Z}^4$. There is a nearly-holomorphic modular form $F$ of weight $-2$ with principal part $$F(\tau) = 4\mathfrak{e}_{(0,0,0,0)} + q^{-1/4} (\mathfrak{e}_{(\frac{1}{2},0,0,0)} + \mathfrak{e}_{(0,\frac{1}{2},0,0)} + \mathfrak{e}_{(0,0,\frac{1}{2},0)} + \mathfrak{e}_{(0,0,0,\frac{1}{2})}) + ...$$ (Under the Borcherds lift, this produces the singular weight product on the lattice $4A_1 \oplus II_{2,2}$. Up to an involution, i.e. taking the dual lattice and rescaling, this is the case $m=4$ of theorem 8.1 of \cite{Sch}.) Through the theta decomposition we can interpret $F$ as a weak Jacobi form $\Phi(\tau,z)$ of index $\mathcal{M} = \begin{pmatrix} \frac{1+b^2}{m} & -\frac{ab}{m} \\ -\frac{ab}{m} & \frac{1 + a^2}{m} \end{pmatrix}, \, B = \begin{pmatrix} 0 & 0 \\ a/m & b/m \end{pmatrix}$ for the Weil representation attached to the Gram matrix $\begin{pmatrix} 2 & 0 \\ 0 & 2m \end{pmatrix}$, in which we then set $z=0$. By proposition 7 the coefficient of $q^0 \mathfrak{e}_0$ in $\Phi(\tau,0)$ (i.e. twice the weight of the Borcherds lift) is $$4 + \#\left\{r \in \mathbb{Z}^2: \; \frac{1}{4}r^T \mathcal{M}^{-1} r = \frac{1}{4}\right\} = 4 + \#\{(r_1,r_2) \in \mathbb{Z}^2: \; (1+a^2) r_1^2 + 2ab r_1 r_2 + (1+b^2)r_2^2 = 1\};$$ this is $4$ if $a,b$ are both nonzero, and $6$ if one of $a,b$ is zero, and $8$ if both of $a,b$ are zero. \par \

The same argument works in the general case, but it is somewhat messier to write out: we choose a (primitive) vector $v = v_1 \in \mathbb{Z}^3$ with $v^T v = m$, extend it to a basis $\{v_1,v_2,v_3\}$ of $\mathbb{Z}^3$ and apply theta decomposition to the Gram matrix of $x_1^2 + x_2^2 + x_3^2 + x_4^2$ with respect to the basis (in block form) $\{(1,0),(0,v_1),(0,v_2),(0,v_3)\}$ of $\mathbb{Z}^4$.
\end{proof}

\begin{ex} Using proposition 7, the principal parts of these input functions can be read off the principal part of $$F(\tau) = 4\mathfrak{e}_{(0,0,0,0)} + q^{-1/4} (\mathfrak{e}_{(\frac{1}{2},0,0,0)} + \mathfrak{e}_{(0,\frac{1}{2},0,0)} + \mathfrak{e}_{(0,0,\frac{1}{2},0)} + \mathfrak{e}_{(0,0,0,\frac{1}{2})}) + ...$$  We list the first few examples below: \par \

\begin{longtable}{|l|l|l|}
\caption{Principal parts for Hermitian modular products of small weight} \\
\hline
         $m$            & Weight &              Principal part                                                          \\ \hline
$1$ & $4$ & $8 \mathfrak{e}_{(0,0)} + q^{-1/4} (\mathfrak{e}_{(1/2,0)} + \mathfrak{e}_{(0,1/2)})$ \\ \hline
$2$ & $3$ & $6 \mathfrak{e}_{(0,0)} + 2q^{-1/8} (\mathfrak{e}_{(0,1/4)} + \mathfrak{e}_{(0,3/4)}) + q^{-1/4} \mathfrak{e}_{(1/2,0)}$ \\ \hline
$5$ & $3$ & $\begin{aligned}&6 \mathfrak{e}_{(0,0)} + q^{-1/20} (\mathfrak{e}_{(0,1/10)} + \mathfrak{e}_{(0,9/10)} + \mathfrak{e}_{(1/2,2/5)} + \mathfrak{e}_{(1/2,3/5)}) + \\ &\quad + q^{-1/5} (\mathfrak{e}_{(0,1/5)} + \mathfrak{e}_{(0,4/5)}) + q^{-1/4} (\mathfrak{e}_{(1/2,0)} + \mathfrak{e}_{(0,1/2)}) \end{aligned}$ \\ \hline
$6$ & $2$ & $\begin{aligned} &4 \mathfrak{e}_{(0,0)} + 2q^{-1/24} (\mathfrak{e}_{(0,1/12)} + \mathfrak{e}_{(0,5/12)} + \mathfrak{e}_{(0,7/12)} + \mathfrak{e}_{(0,11/12)}) + \\ &\quad + q^{-1/6} (\mathfrak{e}_{(0,1/6)} + \mathfrak{e}_{(0,5/6)}) + q^{-1/4} \mathfrak{e}_{(1/2,0)} \end{aligned}$ \\ \hline
$10$ & $3$ & $\begin{aligned} &6 \mathfrak{e}_{(0,0)} + q^{-1/40} (\mathfrak{e}_{(0,1/20)} + \mathfrak{e}_{(0,9/20)} + \mathfrak{e}_{(0,11/20)} + \mathfrak{e}_{(0,19/20)}) + \\ &\quad + q^{-3/20} (\mathfrak{e}_{(1/2,3/10)} + \mathfrak{e}_{(1/2,7/10)}) + \\ &\quad + q^{-9/40} (\mathfrak{e}_{(0,3/20)} + \mathfrak{e}_{(0,7/20)} + \mathfrak{e}_{(0,13/20)} + \mathfrak{e}_{(0,17/20)}) + \\ &\quad + q^{-1/4} \mathfrak{e}_{(1/2,0)} \end{aligned}$ \\ \hline
$13$ & $3$ & $\begin{aligned} &6 \mathfrak{e}_{(0,0)} + q^{-1/13} (\mathfrak{e}_{(0,1/13)} + \mathfrak{e}_{(0,12/13)}) + \\ &\quad + q^{-9/52} (\mathfrak{e}_{(0,3/26)} + \mathfrak{e}_{(0,23/26)} + \mathfrak{e}_{(1/2,5/13)} + \mathfrak{e}_{(1/2,8/13)}) + \\ &\quad + q^{-3/13} (\mathfrak{e}_{(0,4/13)} + \mathfrak{e}_{(0,9/13)}) + \\ &\quad + q^{-1/4} (\mathfrak{e}_{(0,1/2)} + \mathfrak{e}_{(1/2,0)}) \end{aligned}$ \\ \hline
$14$ & $2$ & $\begin{aligned} &4 \mathfrak{e}_{(0,0)} + q^{-1/56} (\mathfrak{e}_{(0,1/28)} + \mathfrak{e}_{(0,13/28)} + \mathfrak{e}_{(0,15/28)} + \mathfrak{e}_{(0,27/28)}) + \\ &\quad + q^{-1/28} (\mathfrak{e}_{(1/2,5/14)} + \mathfrak{e}_{(1/2,9/14)}) + \\ &\quad + q^{-1/14}(\mathfrak{e}_{(0,1/14)} + \mathfrak{e}_{(0,13/14)}) + \\ &\quad + q^{-1/7} (\mathfrak{e}_{(0,2/7)} + \mathfrak{e}_{(0,5/7)}) + \\ &\quad + q^{-9/56} (\mathfrak{e}_{(0,3/28)} + \mathfrak{e}_{(0,11/28)} + \mathfrak{e}_{(0,17/28)} + \mathfrak{e}_{(0,25/28)}) + \\ &\quad + q^{-1/4} \mathfrak{e}_{(1/2,0)} \end{aligned}$ \\ \hline
$17$ & $2$ & $\begin{aligned} &4 \mathfrak{e}_{(0,0)} + 2q^{-1/17} (\mathfrak{e}_{(0,1/17)} + \mathfrak{e}_{(0,6/17)} + \mathfrak{e}_{(0,11/17)} + \mathfrak{e}_{(0,16/17)}) + \\ &\quad + q^{-9/68} (\mathfrak{e}_{(0,3/34)} + \mathfrak{e}_{(0,31/34)} + \mathfrak{e}_{(1/2,7/17)} + \mathfrak{e}_{(1/2,10/17)}) + \\ &\quad + q^{-1/4} (\mathfrak{e}_{(1/2,0)} + \mathfrak{e}_{(0,1/2)}) \end{aligned}$ \\ \hline
$17$ & $3$ & $\begin{aligned} &6 \mathfrak{e}_{(0,0)} + q^{-1/68} (\mathfrak{e}_{(0,1/34)} + \mathfrak{e}_{(0,33/34)} + \mathfrak{e}_{(1/2,8/17)} + \mathfrak{e}_{(1/2,9/17)}) + \\ &\quad + q^{-2/17} (\mathfrak{e}_{(0,6/17)} + \mathfrak{e}_{(0,11/17)}) + \\ &\quad + q^{-4/17} (\mathfrak{e}_{(0,2/17)} + \mathfrak{e}_{(0,15/17)}) + \\ &\quad + q^{-13/68} (\mathfrak{e}_{(0,9/34)} + \mathfrak{e}_{(0,25/34)} + \mathfrak{e}_{(1/2,4/17)} + \mathfrak{e}_{(1/2,13/17)}) \\ &\quad + q^{-1/4} (\mathfrak{e}_{(1/2,0)} + \mathfrak{e}_{(0,1/2)}) \end{aligned}$ \\ \hline
\end{longtable}

\end{ex}

\textbf{Acknowledgments:} I thank Shaul Zemel for pointing out some errors in the published version of this note. The author is supported by the LOEWE research unit Uniformized Structures in Arithmetic and Geometry.

\bibliographystyle{plainnat}

redpajama_set_name: RedPajamaArXiv


text:
Top News - US Photos
Updated: March 15, 2019 at 10:26 AM
Students march in Washington to protest gun violence(14 images)
Students from the Washington, D.C., area stage a walkout and march in protest of gun violence on Thursday. The group rallied at the White House and marched to the U.S. Capitol.
A young woman shouts with "Don't Shoot" written on her palms as students from the Washington, D.C., area protest gun violence. Photo by Pat Benic/UPI
A crowd of students sit in protest on Capitol Hill. Photo by Pat Benic/UPI
A young woman with "enough" written on her face participates in the protest. Photo by Pat Benic/UPI
Rep. Ted Deutch, D-Fla., who represents the district in Florida where the Marjory Stoneman Douglas High School shooting occurred in February 2018, speaks at a rally in front of the Capitol. Deutch tweeted before the event in support of legislation that would require background checks to buy gun ammunition. Photo by Pat Benic/UPI
Sen. Richard Blumenthal, D-Conn., speaks at the rally in front of the Capitol. Photo by Pat Benic/UPI
Participants in a demonstration pose with their signs in front of the Capitol. Photo by Pat Benic/UPI
The students primarily came from schools in the Washington, D.C., area, including Montgomery Blair High School. It was organized via social media and Montgomery County Students For Change in Maryland. Photo by Pat Benic/UPI
Large snake caught crossing Minnesota road

redpajama_set_name: RedPajamaCommonCrawl


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text:
November 29, 2021 No comments General
Iconic Indigenous actor David Gulpilil dies after battle with lung cancer
WARNING: Aboriginal and Torres Strait Islander readers are advised this story contains images of a person who is deceased.
David Gulpilil Ridjimiraril Dalaithngu, a "once-in-a-generation" artist who became an Australian film icon over a decades-long career, has died.
The passing of the Yolŋu actor, dancer, singer and painter, who starred in dozens of films, including his renowned debut Walkabout, Storm Boy and Ten Canoes, was confirmed on Monday night by the South Australian Premier's office.
Gulpilil was born in 1953 and raised in Arnhem Land, where he learned the traditional skills and knowledge of the Mandhalpingu clan of the Yolŋu people.
Some of those, such as traditional dancing, he would later take to the screen and the world.
He was just 16 when British film director Nicholas Roeg cast him in Walkabout after seeing him perform, according to his biography on the National Portrait Gallery website.
"His haunting, moving performance was equal parts devastating as it was electric," Mr Marshall said, before noting that it was Storm Boy that truly made him a star.
Gulpilil was made a Member of the Order of Australia in 1987, recognised for his services to the arts.
He received a NAIDOC Lifetime Achievement Award in 2019, recognition of his "outstanding contribution to the representation of Aboriginal and Torres Strait Islander people nationally and internationally".
"His breakthrough performance in Walkabout remains one of the most iconic and compelling performances of all time," the committee noted.
The Indigenous actor was diagnosed with lung cancer in 2017, and had not been expected to survive until the premiere of his last film, My Name is Gulpilil, let alone attend the premiere, Steven Marshall said.
"Yet it was no surprise to anyone that he was front and centre on opening night, where he would receive his final standing ovation," the Premier said, of the Member of the Order of Australia.
"He was a man who loved his land and his culture, and he was a man who took it to the world.
"My thoughts are with his family, and his dear friend and carer Mary Hood."
Mr Marshall described Gulpilil as an "iconic, once-in-a-generation artist who shaped the history of Australian film and Aboriginal representation on screen."
"An actor, dancer, singer and painter, he was also one of the greatest artists Australia has ever seen," the Premier said.
But the famed life of the famed actor, who would eventually move to the SA town of Murray Bridge, was "not without its struggles".
"He encountered racism and discrimination, and lived with the pressures of the divide between his traditional lifestyle and his public profile," Marshall said.
The National Film and Sound Archive of Australia describes Gulpilil as a "screen legend", crediting him with changing the way the Australian screen represents Indigenous peoples and cultures.
"In Walkabout (1970), Gulpilil embodied the stereotypical image of a 'traditional' Aborigine, untainted by western civilisation," Barnba woman Liz McNiven writes in an NFSA "portrait".
"Over the course of his career, he transformed this constructed identity into a more nuanced and accurate representation of Aboriginality."
The Premier's office said Gulpilil's family had advised his name and image may continue to be used following his passing, in line with his wishes.
Source: 9News
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With 4,529 fatalities, India records highest single-day COVID-19 deaths
New Delhi, May 19 (ANI): Relatives arrives to perform the last rites of the COVID19 patient at Nigambodh Ghat in New Delhi on Wednesday. (ANI Photo)
New Delhi, May 19 (ANI): India recorded the highest single-day COVID-19 fatalities with 4,529 new deaths in the last 24 hours, according to the Union Health Ministry's data on Wednesday.
Meanwhile, India's daily spike of coronavirus cases remained below the 3 lakh mark for the third consecutive day as only 2,67,334 fresh infections were recorded during a 24-hour period. At present, the cumulative caseload stands at 2,54,96,330, including 32,26,719 active cases, 2,19,86,363 recoveries and 2,83,248 deaths.
Karnataka, which has recently emerged as the new COVID-19 hotspot, continues to remain the worst affected state at present with 5,75,028 active infections followed by Maharashtra with 4,19,727 active cases.
As per the Indian Council of Medical Research (ICMR), a total of 32,03,01,177 samples have been tested for COVID-19 up to May 18, of these, 20,08,296 samples were tested on Tuesday.
A total of 18,58,09,302 doses of COVID-19 vaccines have been administered across the country so far.
Earlier on Tuesday, the country saw the highest ever recoveries with 4,22,436 fresh discharges in a single day for the first time.
On Monday (May 17), India's daily spike of coronavirus cases remained below the 3 lakh mark for the first time after 26 days as only 2,81,386 fresh infections were recorded in the last 24 hours. On April 22, the country achieved a grim milestone in its fight against the raging coronavirus as it reported over 3 lakh new cases in a 24-hour period for the first time since the pandemic began last year.
Previous articleCovid-19 update: Daily recoveries cross 4 lakh mark in India
Next article4,257 street children rescued across Andhra Pradesh under 'Operation Muskaan'


Morocco has well acclaimed Music Festivals all year round. Celebration is an important aspect of Moroccan culture. Morocco has become one of the leading world destinations for those wishing to attend a music festival. From the Fes in USESCO World Heritage Fes to the Gnaoua Festival in Seaside Essaouira, the Mawazine Festival in the capitol Rabat to the famous Timitar Festival in Agadir that highlights Berber Music and culture, attending a music festival in Morocco should be on the bucketlist for music lovers alike.
Featuring over 40 artists and 500,000 in attendance, Timitar Festival in Agadir, Morocco, is one of premiere African music festivals. Timitar provides its audience with an event well rooted in and actively working towards promoting Souss Massa Drâa culture. The district council of Agadir Under Massa Draâ initiated the Timitar Festival with the goal to set up a cultural project for the meeting between Amazigh artists and world musicians.
The Essaouira Gnaoua World Music Festival is one of Morocco's most popular festivals. It showcases Gnaoua music, which came to Morocco with sub-Saharan African slaves to the SultansThe favorite of many is when Maâlem (Master) Gnaoua and their bands begin to play, around the eleven pm. The Maâlems have venerable stringed-instrument traditions involving both bowed lutes like the gogo and plucked lutes like the gimbri, also called hajhuj a three-stringed bass instrument.
The Fes Festival of World Sacred Music is a celebration held in mid-summer (late May or early June) each year that takes place in the Imperial city of Fes. The festival was founded in 1994 by the Moroccan scholar and philanthropist Faouzi Skali and was created to showcase major musical traditions of sacred, spiritual music and world music. Each year the festival celebrates artists from Muslim, Jewish, Christian, Hindu and other faiths to perform together in a spirit of mutual respect and collaboration. The Fes Festival of World Sacred Music is one of the most respected Festivals in the World.
Imilchil located high up in the lake plateau of the Middle Atlas Mountains in Morocco. In this quaint village you will find primarily Berber tribal clans who have a strong sense of culture and tradition that has been preserved for decades. The Imilchil Marriage Festival is the prime attraction of this village and takes place annually in September. Each September, the surrounding tribes, Aït Sokham and Aït Bouguemmaz celebrate the Imilchil Marriage Festival, held in Souk Aam and Agdoud N'Oulmghenni.
The Marrakech Popular Arts Festival is a lively festival held annually each summer in Marrakech. Entertainers, artists and travelers from around the world come for this unique yet intimate show case of musicians, fortune-tellers, story-tellers, singers, snake charmers, dancers, acrobats, craftsmen and fantasia horse shows. There are theater performances also held in the city center of Marrakech, at El Badi Djemaa El Fna Square.
Held annually each spring, the M'hamid International Nomad Festival showcases desert musicians and performances in the heart of the M'hamid Sahara Desert. The festival takes place in the small village known as M'hamid El Ghizlane. M'hamid El Ghizlane is a small oasis town in Southern Morocco with over 7500 people and approximately 25 kilometers from the Algerian border. The M'hamid International Nomad Festival is a cultural event as well as a festival whereby the architectural heritage, fauna and agriculture of this small region are celebrated. Crafts, painting, weaving along with music combined with song, music and dance make it the ideal destination for a Morocco desert tour for the adventure traveler.
The M'hamid Targalte Festival of Desert Culture & Music isa three-day celebration of nomadic culture. It attracts both Moroccans and tourists alike. The festival includes music and debates on the desert region's patrimony and is one of Morocco's more intimate and serious festivals. Saharan music, Gnaoua and traditional music of the desert are part of this annual gathering.
Held each April the Printemps Musicals des Alizes is Essaouira's spring classical music festival that features a wide array of International and Moroccan artists within the classical music genre. Homages to Beethoven, Brahms and Schumann have been some of the highlights of past festivals. Classical piano and Opera are two important aspects of the festival that are widely celebrated annually. Many of the concerts are held in the 18th century merchant's house called Dar Souiri. As a UNESCO World Heritage site Essaouira is the place to be for classical lovers on a Morocco Tour in spring.
The Mawazine Festival is held in Rabat, Morocco each year and features musicians from over forty countries. The festival brings together a diverse group of artists and rhythms to pay tribute to the music of the world. The Mawazine Festival includes musical creations, and exhibition of emerging artists with a desire for cultural exchange.
The Sufi Cultural Festival is an 8-day celebration that takes place each April within the imperial city of Fes to honor Sufi music and spirituality. The Festival brings together religious leaders in Sufism and artists from around the world. Visitors come to enjoy ritual performances complemented by morning poetry readings, films and oriental art. The festival provides an opportunity for visitors to discover Morocco's spiritual heritage and to display a side of Islam that is not often understood. Scholars of various expertise have used the festival as an opportunity for debate and discussion, around such themes as "Sufism and human rights", "Sufism and Moroccan history", and "Sufism and cultural diversity".
Tanjazz is an International Jazz Festival held annually in Tangier. The line up is diverse including blues, rock and jazz along with klezmer and even gypsy jazz. A city with a long history known for writers, artists and the perfect look out post over Gibraltar jazz lovers are not to be disappointed when visiting Tangier for the TanJazz Festival. The festival splits its showcases between outdoor and indoor stage performances making for the perfect place to visit for a a Morocco traveler.
Held in Morocco each May, the Rose Festival in Kelaa M'Gouna is world famous for the Damask Rose first brought here in the 1938 by the French. At that time El Kelaa Des Mgouna's first rose water distillery was opened. Shortly after the first Rose Festival began and has been a tradition ever since. The three-day Rose Festival takes place in the Valley of the Roses and occupies the souk area of El Kelaa Des Mgouna. During this time, travelers come from all over to attend the festivities where a Rose Queen is elected to reign over the year's scented crop. The factories in El Kelaa Des Mgouna produce 3000-4000 petals a year. As with all festivals in Morocco, the annual Rose Festival boasts delicious food alongside traditional Berber local tribes singing, dancing, displaying sword maneuvers and playing traditional musical instruments; plus a parade of floats with the nominees for Ms. Roses who sit upon them.


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