In the doctrine of no arising does the past and future still exist? In the doctrine of no arising does the past and future still exist, and if so do they exist in the same way as the present does?

And if not, why?

From Dogen's Genjo-koan:


  Firewood becomes ash. Ash cannot turn back into firewood again. However, we should not view ash as after and firewood as before. We should know that firewood dwells in the dharma position of firewood and it has its own before and after. Although there is before and after, past and future are cut off. Ash stays at the position of ash and it has its own before and after. As firewood never becomes firewood again after it is burned and becomes ash, after person dies, there is no return to living. However, in buddha dharma, it is a never-changing tradition not to say that life becomes death. Therefore we call it no-arising. It is the laid-down way of buddha's turning the dharma wheel not to say that death becomes life. Therefore, we call it no-perishing. Life is a position at one time; death is also a position at one time. For instance, this is like winter and spring. We don't think that winter becomes spring, and we don't say that spring becomes summer. 


I wondered whether Dogen is saying that there is a before and after and these are the same as the past and future that are cut off -- from the present: so that anything that occurs only does so in the past or future.
Dogen is saying that from Buddhist perspective, a dharma does not "become" another dharma. Each dharma is a "presence" at its time, static in its own quality, followed by another "presence-quality" and so on.

So wood is wood and ash is ash, it's not the same "thing" changing while moving from point A in time to point B.

To this I can add that the temporal succession of dharmas is similar to a set of real numbers - each individual dharma is a generalization, and we can infinitely "zoom in" on a particular moment between "wood" and "ash" to see a some-level-of-burning-in-progress-dharma.

Wood and ash are connected as cause and effect. Ash exists due to wood (among other things).

Similarly, it's not like a living person dies. They exist in succession, first life, then death. Life and death are connected like cause and effect. Death exists due to life (among other things). Therefore, there is no perishing of the living.

Correspondingly, there is no arising. It's not like the newborn was dead before it was born. Nor is it like the mother turned into the newborn, nor mother's egg, nor the sperm. Instead, these dharmas exist in causal succession.

Now, if you think about it, it's not like any given dharma has exactly one dharma as its predecessor - not at all. Instead, most if not all dharmas have many, many "causing" or "conditioning" dharmas. Which one we consider "the main" cause is entirely up to us.

In this sense, someone who lives now is a "dharma" that has quadrillions of other dharmas as its "inputs" - going up the infinitely forking chains of physical ancestry, informational ancestry, circumstantial conditionality and who knows what else. 

This is why there is no arising, because it's an infinitely spanning network of static  non-discrete dharmas causing and conditioning each other. The whole thing is like a multidimensional vector space, without one global time dimension.



So, to answer your question, 

Past and future are something observed relative to a particular dharma. Past is the transitive closure of all its causes, and future is the superset of dharmas directly or indirectly conditioned by a given dharma.

Ontologically, there is no becoming, no objects moving through time, nor one global time dimension.