A Schanuel property for j

Eterović, Sebastian

doi:10.1112/blms.12135

Cited by 8 publications

(16 citation statements)

References 22 publications

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“…In particular, Theorem 6.9 proves that kcl is a pregeometry. In [Ete18], the fact that C = jcl(∅) ⊂ C is countable was proven using o-minimality arguments (which cannot be extended to other j-fields), but this theorem now gives that jcl(∅) = kcl(∅), and so the j-closure of any countable set is countable in every j-field. 6.2.…”

Section: J-polynomialsmentioning

confidence: 99%

“…The definitions of these operators will be given later on. It was shown in [Ete18] that jcl is a pregeometry; Theorem 4.11 gives a new proof of this.…”

Section: Introductionmentioning

confidence: 97%

“…The methods there are based on Ax's proof of the Ax-Schanuel theorem ([Ax71, Theorem 3]). In the case of the modular j-function, the Ax-Schanuel theorem was established in [PT16], and a corresponding study of j-fields was given in [Ete18]. However, it was left as an open question whether proving Theorem 1.1 could be achieved.…”

Section: Introductionmentioning

confidence: 99%

“…In[Ete18] this was called "geodesic closure", and was denoted as gcl. We have opted to maintain the notation used in[EH19].3 This definition of j-fields is slightly simpler than the one presented in[Ete18], but it defines the same structures.…”

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confidence: 99%

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A closure operator respecting the modular $j$-function

Aslanyan¹,

Eterović²,

Kirby³

2020

Preprint

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We give three equivalent characterisations of the natural closure operator on a field equipped with functions replicating the algebraic behaviour of the modular j-function and its derivatives, following similar work on exponential fields. As an application of these results, we give unconditional cases of a theorem of Eterović-Herrero on the Existential Closedness problem for the complex j-function which previously assumed the modular version of Schanuel's conjecture.

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Section: J-polynomialsmentioning

confidence: 99%

“…The definitions of these operators will be given later on. It was shown in [Ete18] that jcl is a pregeometry; Theorem 4.11 gives a new proof of this.…”

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

A closure operator respecting the modular $j$-function

Aslanyan¹,

Eterović²,

Kirby³

2020

Preprint

Self Cite

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“…Indeed, Ax-Schanuel is one of the main ingredients in proofs of many Zilber-Pink type statements (see [Pil15, Zil02, Kir09, PT16, HP16, Zan12, DR18, Asl18b, Asl19]). Ax-Schanuel theorems also contribute to our understanding of the corresponding number theoretic conjectures like Schanuel's conjecture (see [BKW10,Kir10,Ete18]).…”

Section: Introductionmentioning

confidence: 98%

Adequate Predimension Inequalities in Differential Fields

Aslanyan¹

2018

Preprint

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In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation and its analogue for the differential equation of the jfunction (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which, in its turn, is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences.

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Schanuel type conjectures and disjointness

Broudy

Eterović

2023

Ramanujan J

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Given a subfield F of $${\mathbb {C}}$$ C , we study the linear disjointess of the field E generated by iterated exponentials of elements of $${\overline{F}}$$ F ¯ , and the field L generated by iterated logarithms, in the presence of Schanuel’s conjecture. We also obtain similar results replacing $$\exp $$ exp by the modular j-function, under an appropriate version of Schanuel’s conjecture, where linear disjointness is replaced by a notion coming from the action of $$\textrm{GL}_2$$ GL 2 on $${\mathbb {C}}$$ C . We also show that for certain choices of F we obtain unconditional versions of these statements.

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A Schanuel property for j

Cited by 8 publications

References 22 publications

A closure operator respecting the modular $j$-function

A closure operator respecting the modular $j$-function

Adequate Predimension Inequalities in Differential Fields

Schanuel type conjectures and disjointness

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