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Jacob Tsimerman

Publications and source records attributed to Jacob Tsimerman.

At least 37 records · Page 2Linked to original sources
arXiv2023

Finite Orbits in Surfaces with a Double Elliptic Fibration and Torsion Values of Sections

We consider surfaces with a double elliptic fibration, with two sections. We study the orbits under the induced translation automorphisms proving that, under natural conditions, the finite orbits are confined to a curve. This goes in a similar direction of (and is motivated by) recent work by Cantat-Dujardin, although we use very different methods and obtain related but different results. As a sample of application of similar arguments, we prove a new case of the Zilber-Pink conjecture, namely Theorem 1.5, for certain schemes over a 2-dimensional base, which was known to lead to substantial difficulties. Most results rely, among other things, on recent theorems by Bakker and the second author of `Ax-Schanuel Type'; we also relate a functional condition with a theorem of Shioda on ramified sections of the Legendre scheme. For one of our proofs, we also use recent height inequalities by Yuan-Zhang. Finally, in an appendix, we show that the Relative Manin-Mumford Conjecture over the complex number field is equivalent to its version over the field of algebraic numbers.

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arXiv2022

o-minimal GAGA and a conjecture of Griffiths

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample.

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arXiv2022

Ax-Schanuel and exceptional integrability

When can a primitive of a given algebraic function be con-structed by iteratively solving algebraic equations and composing withthe primitives of some other given algebraic functions or their inverses? We establish some results in this direction. Specifically, we establishdecision procedures for determining whether a given primitive can beexpressed in terms of finitely many others, or in terms of elliptic integrals.

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arXiv2021

Finiteness for self-dual classes in integral variations of Hodge structure

We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $\mathbb{R}_{\mathrm{an},\exp}$.

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arXiv2020

Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and Tsimerman

We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel-Venkatesh mixing conjecture about the equidistribution of CM points on a product of two modular curves.

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arXiv2020

Cohen-Lenstra heuristics and bilinear pairings in the presence of roots of unity

Let $L/K$ be a quadratic extension of global fields. We study Cohen-Lenstra heuristics for the $\ell$-part of the relative class group $G_{L/K} := \textrm{Cl}(L/K)$ when $K$ contains $\ell^n$th roots of unity. While the moments of a conjectural distribution in this case had previously been described, no method to calculate the distribution given the moments was known. We resolve this issue by introducing new invariants associated to the class group, $ψ_{L/K}$ and $ω_{L/K},$ and study the distribution of $(G_{L/K}, ψ_{L/K}, ω_{L/K})$ using a linear random matrix model. Using this linear model, we calculate the distribution (including our new invariants) in the function field case, and then make local adjustments at the primes lying over $\ell$ and $\infty$ to make a conjecture in the number field case, which agrees with some numerical experiments.

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arXiv2020

Quasiprojectivity of images of mixed period maps

We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasiprojective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions. Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating a $\mathbb{R}_{an,exp}$-definable structure to mixed period domains and admissible mixed period maps.

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arXiv2020

Tame topology of arithmetic quotients and algebraicity of Hodge loci

In this paper we prove the following results: $1)$ We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. $2)$ We prove that the period map associated to any pure polarized variation of integral Hodge structures $\mathbb{V}$ on a smooth complex quasi-projective variety $S$ is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. $3)$ As a corollary of $2)$ and of Peterzil-Starchenko's o-minimal Chow theorem we recover that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$, a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable $SL_2$-orbit theorem of Cattani-Kaplan-Schmid.

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arXiv2020

Definability of mixed period maps

We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}_{alg}$-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in $\mathbb{R}_{an,exp}$ with respect to this structure. As a consequence we reprove that the zero loci of admissible normal functions are algebraic.

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arXiv2020

Heuristics for the asymptotics of the number of $S_n$-number fields

We give a heuristic argument supporting conjectures of Bhargava on the asymptotics of the number of $S_n$-number fields having bounded discriminant. We then make our arguments rigorous in the case $n=3$ giving a new elementary proof of the Davenport-Heilbronn theorem. Our basic method is to count elements of small height in $S_n$-fields while carefully keeping track of the index of the monogenic ring that they generate.

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arXiv2020

High $\ell$-torsion rank for class groups over function fields

We prove that in the function field setting, $\ell$-torsion in the class groups of quadratic fields can be arbitrarily large. In fact, we explicitly produce a family whose $\ell$-rank growth matches the growth in the setting of genus theory, which might be best possible. We do this by specifically focusing on the Artin-Schreir curves $y^2=x^q-x$.

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arXiv2019

Densities of eigenspaces of Frobenius and distributions of R-modules

For any polynomial $p\left(x\right)$ over $\mathbb{F}_{l}$ we determine the asymptotic density of hyperelliptic curves over $\mathbb{F}_{q}$ of genus $g$ for which $p\left(x\right)$ divides the characteristic polynomial of Frobenius acting on the $l$-torsion of the Jacobian, and give an explicit formula for this density. We prove this result as a consequence of more general density theorems for quotients of Tate modules of such curves, viewed as modules over the Frobenius. The proof involves the study of measures on $R$-modules over arbitrary rings $R$ which are finite $\mathbb{Z}_{l}$-algebras. In particular we prove a result on the convergence of sequences of such measures, which can be applied to the moments computed in recent work of Lipnowski-Tsimerman to obtain the above results. We also extend the random model for finite $R$-modules proposed in that work to such rings $R$, and prove several of its properties. Notably the measure obtained is in general not inversely proportional to the size of the automorphism group.

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arXiv2019

Independence of CM points in Elliptic Curves

We prove a result which describes, for each $n\ge 1$, all linear dependencies among $n$ images in elliptic curves of special points in modular or Shimura curves under parameterizations (or correspondences). Our result unifies and improves in certain aspects previous work of Rosen-Silverman--Kühne and Buium-Poonen.

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arXiv2017

The Ax-Schanuel conjecture for variations of Hodge structures

We extend the Ax-Schanuel theorem recently proven for Shimura varieties by Mok-Pila-Tsimerman to all varieties supporting a pure polarized integral variation of Hodge structures. The essential new ingredient is a volume bound on Griffiths transverse subvarieties of period domains.

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arXiv2017

Constructing elliptic curves from Galois representations

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.

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arXiv2017

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_ε(|{\rm Disc}(K)|^{1/2+ε})$ by Brauer--Siegel). This yields corresponding improvements to: 1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves; 2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves; 3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves; and 4) bounds of Baily and Wong on the number of $A_4$-quartic fields of bounded discriminant.

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arXiv2016

Cohen Lenstra Heuristics for Étale Group Schemes and Symplectic Pairings

We generalize the Cohen-Lenstra heuristics over function fields to étale group schemes $G$ (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg-Venkatesh-Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of $\wedge^2G(1)$, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen-Lenstra heuristics for abelian $\ell$-groups in cases where $\ell\mid q-1$; the nature of this failure was suggested already in the works of Garton, EVW, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.

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