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Sum–product estimates for rational functions
We establish several sum–product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)+|AA| is substantially larger than |A| for an arbitrary polynomial f
Ax-Lindemann for Ag
We prove the Ax-Lindemann theorem for the coarse moduli space Ag of principally polarized abelian varieties of dimension g ≥ 1. We affirm the André-Oort conjecture unconditionally for Ag for g ≤ 6,
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 polarizable integral variation of Hodge structures. In fact, Hodge
Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points
In \cite{S}, Shyr derived an analogue of Dirichlet's class number formula for arithmetic Tori. We use this formula to derive a Brauer-Siegel formula for Tori, relating the Discriminant of a torus to
The André–Oort conjecture for the moduli space of abelian surfaces
Abstract We provide an unconditional proof of the André–Oort conjecture for the coarse moduli space 𝒜2,1 of principally polarized abelian surfaces, following the strategy outlined by Pila–Zannier.
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
On the Davenport–Heilbronn theorems and second order terms
We give simple proofs of the Davenport–Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion
Ax-Schanuel for Shimura varieties
We prove the Ax-Schanuel theorem for a general (pure) Shimura variety.
Non-split sums of coefficients of GL(2)-automorphic forms
Given a cuspidal automorphic form π on GL2, we study smoothed sums of the form $$\sum\nolimits_n {{a_\pi }({n^2} + d)V({n \over x})} $$. The error term we get is sharp in that it is uniform in both d
Local spectral equidistribution for Siegel modular forms and applications
Abstract We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to