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.

arXiv comment28 pages. Minor corrections

References in corpus (7)

• Ax-Schanuel for the j-function
• Exponential algebraicity in exponential fields
• The theory of the exponential differential equations of semiabelian varieties
• Pseudo-exponential maps, variants, and quasiminimality
• Polynomial exponential equations and Zilber's conjecture
• Blurrings of the $j$-function
• A Schanuel Property for $j$

Cited by in corpus (4)

• Generic Solutions of Equations Involving the Modular $j$-function
• Ax-Schanuel with derivatives for mixed period mappings
• Equations involving the modular $j$-function and its derivatives
• Hilbert polynomials for finitary matroids