A differential approach to the Ax-Schanuel, I

Blázquez-Sanz, David; Casale, Guy; Freitag, James; Nagloo, Joel

doi:10.48550/arxiv.2102.03384

Cited by 6 publications

(7 citation statements)

References 28 publications

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“…, h m at some point s 1 sufficiently close to s 0 do not satisfy a non-zero linear relation. After some setup, we will see that this is a consequence of the Ax-Schanuel theorem for periods, recently proved in [BT22] as an application of a general theorem of [BSCFN21].…”

Section: Proof Of 116mentioning

confidence: 81%

Geometric G-functions and Atypicality

Urbanik¹

2023

Preprint

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In a seminal research manuscript, André showed how the theory of G-functions could be used to give height bounds on the moduli of smooth projective algebraic varieties acquiring non-generic algebraic cycles. The method was limited by the lack of a suitable cohomological interpretation for these G-functions at finite places. In this paper we use recent developments in p-adic Hodge theory to remove this constraint.With respect to André's strategy for producing height bounds from algebraic relations on G-function values at special points, we give a general method for producing relations that hold at all finite places, and show that producing relations at the infinite places is the essential difficulty. This leads to new cases of the Zilber-Pink conjecture, as well as new height bounds on special moduli, including the first unconditional finiteness results for CM moduli in non-Shimura settings.As a more technical contribution, we introduce a refinement of the Pila-Zannier strategy capable of handling Zilber-Pink-type atypical intersection problems in arbitrary dimension and for arbitrary smooth projective families.

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Section: Proof Of 116mentioning

confidence: 81%

Geometric G-functions and Atypicality

Urbanik¹

2023

Preprint

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“…Much more general Ax-Schanuel theorems are announced by Blázquez Sanz, Casale, Freitag, and Nagloo in [8]. Analogous results on the model theoretic properties of the associated differential equations follow in each of the cases they consider.…”

Section: Trivial Minimal Types In Differentially Closed Fieldsmentioning

confidence: 82%

“…Analogous results on the model theoretic properties of the associated differential equations follow in each of the cases they consider. In [8], strong minimality and forking triviality for the differential equations associated to the covering maps of simple Shimura varieties are established and (non-)orthogonality is described geometrically in much the same way as is done here (which is not surprising as our methods and theirs follow the analysis of [12]). A subtlety here is that we consider as well the case where the underlying Shimura variety is not simple, observing that there can be a real distinction between minimality and strong minimality related to the notion of δ-Hodge genericity.…”

Section: Trivial Minimal Types In Differentially Closed Fieldsmentioning

confidence: 84%

Effective transcendental Zilber-Pink for variations of Hodge structures

Pila¹,

Scanlon²

2021

Preprint

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We prove function field versions of the Zilber-Pink conjectures for varieties supporting a variation of Hodge structures. A form of these results for Shimura varieties in the context of unlikely intersections is the following. Let S be a connected pure Shimura variety with a fixed quasiprojective embedding. We show that there is an explicitly computable function B of two natural number arguments so that for any field extension K of the complex numbers and Hodge generic irreducible proper subvariety X S K , the set of nonconstant points in the intersection of X with the union of all special subvarieties of X of dimension less than the codimension of X in S is contained in a proper subvariety of X of degree bounded by B(deg(X), dim(X)). Our techniques are differential algebraic and rely on Ax-Schanuel functional transcendence theorems. We use these results to show that the differential equations associated with Shimura varieties give new examples of minimal, and sometimes, strongly minimal, types with trivial forking geometry but non-ℵ 0categorical induced structure.Proof. If tp(a 1 /K) ⊥ tp(a 2 /K), then we can find some extension M of K and points c 1 and c 2 with tp(c i /M) being the nonforking extension of tp(a i /K) to M (for i = 1 and 2) and c 1 and c 2 are dependent over M. By Lemma 5.9, there is some proper special T ⊆ S 1 × S 2 with (c 1 , c 2 ) ∈ T and T → S i finite for i = 1 and 2. Set b := (c 1 , c 2 ).

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“…The proof of Bakker-Tsimerman's Ax-Schanuel crucially relies on the definability of period maps in the o-minimal structure R an,exp . It has been recently reproven in a more general setting using an o-minimal free approach ( [7]).…”

Section: 3mentioning

confidence: 99%

Existence and density of typical Hodge loci

Khelifa¹,

Urbanik²

2023

Preprint

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Motivated by a question of Baldi-Klingler-Ullmo, we provide a general sufficient criterion for the existence and analytic density of typical Hodge loci associated to a polarizable Z-variation of Hodge structures V. Our criterion reproves the existing results in the literature on density of Noether-Lefschetz loci. It also applies to understand Hodge loci of subvarieties of Ag . For instance, we prove that for g ě 4, if a subvariety S of Ag has dimension at least g then it has an analytically dense typical Hodge locus. This applies for example to the Torelli locus of Ag.

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A differential approach to the Ax-Schanuel, I

Cited by 6 publications

References 28 publications

Geometric G-functions and Atypicality

Geometric G-functions and Atypicality

Effective transcendental Zilber-Pink for variations of Hodge structures

Existence and density of typical Hodge loci

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