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Jonathan Pila

Publications and source records attributed to Jonathan Pila.

17 recordsLinked to original sources
arXiv2024

Periods in Families and Derivatives of Period Maps

Given a smooth proper family $\phi:X\rightarrow S$, we study the (quasi)-periods of the fibers of $\phi$ as (germs of) functions on $S$. We show that they field they generate has the same algebraic closure as that given by the flag variety co-ordinates parametrizing the corresponding Hodge filtration, together with their derivatives. Moreover, in the more general context of an arbitrary flat vector bundle, we determine the transcendence degree of the function field generated by the flat coordinates of algebraic sections. Our results are inspired by and generalize work of Bertrand--Zudilin.

math.AG↗
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.

math.AG↗
arXiv2021

Effective transcendental Zilber-Pink for variations of Hodge structures

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 $X \subsetneq S_K$, 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 $B(\operatorname{deg}(X),\dim(X))$. 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-$\aleph_0$-categorical induced structure.

math.AG↗
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--K\"uhne and Buium-Poonen.

math.NT↗
arXiv2016

Uniform parameterization of subanalytic sets and diophantine applications

We prove new parameterization theorems for sets definable in the structure $\mathbb{R}_{an}$ (i.e. for globally subanalytic sets) which are uniform for definable families of such sets. We treat both $C^r$-parameterization and (mild) analytic parameterization. In the former case we establish a polynomial (in $r$) bound (depending only on the given family) for the number of parameterizing functions. However, since uniformity is impossible in the latter case (as was shown by Yomdin via a very simple family of algebraic sets), we introduce a new notion, analytic quasi-parameterization (where many-valued complex analytic functions are used), which allows us to recover a uniform result. We then give some diophantine applications motivated by the question as to whether the $H^{o(1)}$ bound in the Pila-Wilkie counting theorem can be improved, at least for certain reducts of $\mathbb{R}_{an}$. Both parameterization results are shown to give uniform $(\log H)^{O(1)}$ bounds for the number of rational points of height at most $H$ on $\mathbb{R}_{an}$-definable Pfaffian surfaces. The quasi-parameterization technique produces the sharper result, but the uniform $C^r$-parametrization theorem has the advantage of also applying to $\mathbb{R}_{an}^{pow}$-definable families.

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arXiv2015

Rational points on Grassmannians and unlikely intersections in tori

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof uses a method introduced for the first time by Pila and Zannier to give an alternative proof of Manin-Mumford conjecture and a theorem to count points that satisfy a certain number of linear conditions with rational coefficients. This method has been largely used in many different problems in the context of "unlikely intersections".

math.NT↗
arXiv2015

On a modular Fermat equation

We consider some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of "unlikely intersections." We prove a special case of the Zilber-Pink conjecture for curves.

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arXiv2014

Multiplicative relations among singular moduli

We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)

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arXiv2014

Ax-Schanuel for the j-function

In this paper we prove a functional transcendence statement for the j-function which is an analogue of the Ax-Schanuel theorem for the exponential function. It asserts, roughly, that atypical algebraic relations among functions and their compositions with the j-function are governed by modular relations.

math.LO↗
arXiv2014

O-minimality and certain atypical intersections

We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and André-Oort. We verify the so-called Zilber-Pink Conjecture in a product of modular curves on assuming a lower bound for Galois orbits and a sufficiently strong modular Ax-Schanuel Conjecture. In the context of abelian varieties we obtain the Zilber-Pink Conjecture for curves unconditionally when everything is defined over a number field. For higher dimensional subvarieties of abelian varieties we obtain some weaker results and some conditional results.

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arXiv2013

Ax-Lindemann for \mathcal{A}_g

We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$, and affirm the André-Oort conjecture unconditionally for $\mathcal{A}_{g}$ for $g\le 6$.

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arXiv2008

Rational points in periodic analytic sets and the Manin-Mumford conjecture

We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation.

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arXiv2005

Note on the rational points of a pfaff curve

Let X be the graph in the plane of a pfaffian function f (in the sense of Khovanskii). Suppose X is not algebraic. This note gives an upper bound for the number of rational points on X of height up to X. The bound is uniform in the order and degre of f.

math.NT↗
arXiv2005

Counting points on curves over families in polynomial time

This note concerns the theoretical algorithmic problem of counting rational points on curves over finite fields. It explicates how the algorithmic scheme introduced by Schoof and generalized by the author yields an algorithm whose running time is uniformly polynomial time for curves in families.

math.NT↗