  The ESA Euclid mission is scheduled to launch on July 1st 2023. This White
Paper discusses how Euclid observations of the Galactic Bulge Time Domain
Survey (GBTDS) area could dramatically enhance the exoplanet science output of
the Nancy Grace Roman Space Telescope (Roman). An early Euclid pre-imaging
survey of the Roman GBTDS fields, conducted soon after launch, can improve
proper motion determinations for Roman exoplanet microlenses that can yield a
factor of up to $\sim 5$ improvement in exoplanet mass measurements. An
extended Euclid mission would also enable the possibility of sustained
simultaneous observations of the GBTDS by Euclid and Roman that would achieve
large gains in several areas of Roman exoplanet science, including science that
is impossible to achieve with Roman alone. These include: a comprehensive
demographic survey for free-floating planets that includes precision mass
measurements to establish the true nature of individual candidates; detection,
confirmation and mass measurements of exomoons; direct exoplanet mass
measurements through parallax and finite source size effects for a large sample
of bound exoplanets detected jointly by Euclid and Roman; enhanced
false-positive discrimination for the large samples of transiting planets that
Roman will detect. Our main recommendation to NASA and ESA is to initiate a
Joint Study Group as early as possible that can examine how both missions could
best conduct a coordinated campaign. We also encourage flexibility in the GBTDS
scheduling.

Instrumentation and Methods for Astrophysics;Earth and Planetary Astrophysics