A closure operator respecting the modular j-function

Abstract

We prove some unconditional cases of the Existential Closedness problem for the modular j-function. For this, we show that for any finitely generated field we can find a "convenient" set of generators. This is done by showing that in any field equipped with functions replicating the algebraic behaviour of the modular j-function and its derivatives, one can define a natural closure operator in three equivalent different ways.

Schanuel type conjectures and disjointness

“…However, obtaining a similar result about the j function and π is not expected. As shown in [1,Remark 6.21], the generalized period conjecture of Grothendieck-André which implies MSCD, also implies that π / ∈ C j . Choosing F = Q will give that J , K ⊆ C j (see [1, Proposition 6.6]), so this prevents π from being in either J or K .…”

“…5] for the case of j). One can find more explicit descriptions of the sets C exp and C j by using Khovanskii systems of equations (see [1,Sect. 6] for j and [10, Sect.…”

“…The methods used to prove all of our main results rely mostly on the work of [1] on convenient generators, which in turn rely heavily on the so-called Ax-Schanuel theorems: [2] in the case of exp and [16] in the case of j. We expect that similar constructions can be performed whenever an Ax-Schanuel theorem is available in differential form, and so the methods presented here can be expected to extend to other situations, such as the exponential maps of semi-abelian varieties (using [3] or [9]) or the uniformization maps of Shimura varieties (using [14]).…”