CM点的p-adic分布与Hecke轨道. II: 超奇异轨迹上的Linnik等分布

Simultaneous supersingular reductions of CM elliptic curves

We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show -- under additional congruence assumptions on the CM order -- that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut-Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.

关于Shimura曲线乘积上的等分布与André-Oort的一个注记

A note on equidistribution on a product of Shimura curves and André--Oort

应用Aka-Luethi-Michel-Wieser的adelic环面包等分布定理，证明在r≥2个非同构Shimura曲线乘积上CM点的Galois轨道等分布，并推导出这些曲线乘积的André-Oort猜想，用Linnik型分裂条件替代GRH。

定志村曲线皮卡群上赫克轨道的等分布

Equidistribution of Hecke Orbits on the Picard group of definite Shimura curves

本文利用自守方法及Drinfeld型尖点自守形式的系数界，证明了函数域上定志村曲线皮卡群中赫克轨道的等分布，特别是秩2超奇异Drinfeld模的赫克轨道等分布。

Integer points and their orthogonal lattices

Linnik proved in the late 1950's the equidistribution of integer points on large spheres under a congruence condition. The congruence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition.

Equidistribution of Gross points over rational function fields

In this paper we prove a sparse equidistribution theorem for Gross points over the rational function field $\mathbb{F}_q(t)$. We apply this result to study the reduction map from CM Drinfeld modules to supersingular Drinfeld modules. Our proofs rely crucially on a period formula due to M. Papikian and F.-T. Wei/J. Yu, and a Lindelöf-type bound for central values of Rankin-Selberg $L$-functions associated to twists of automorphic forms of Drinfeld-type by ideal class group characters.

Equidistribution of Heegner Points and Ternary Quadratic Forms

We prove new equidistribution results for Galois orbits of Heegner points with respect to reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize one of the equidistribution theorems established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from …

模曲线乘积中特殊轨迹的p进等分布性

$p$-adic Equidistribution of Special Loci in a Product of Modular Curves

该论文研究定义在ℂₚ上的光滑曲线C与模子簇Y₀(n)交轨迹的p进等分布性，给出等分布成立的条件，证明对应交集合非离散，并给出等分布不成立的典型例子。

CM点和RM曲线的等分布

Equidistribution of CM points and RM curves

研究CM点和RM曲线沿固定测地线和固定点周围的等分布问题，通过解决任意二元二次型的聚合Linnik问题来证明。

Large Deviations and Effective Equidistribution

We study large deviations for measurable averaging operators on state spaces of dynamical systems. Our main motivation is the Hecke operators on the modular curve Y_0(p^n) and their generalization to higher rank S-arithmetic quotients. We prove a relatively sharp large deviations result in terms of the norm of the averaging operator restricted to the orthogonal complement of the constant functions in L2. In the self-adjoint case this norm is expressible by the spectral gap. Developing ideas of Linnik and Ellenberg, Michel and Venkatesh, we use this large deviation result to prove an effective …

Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a "microsquare", a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice…

Atkin--Lehner 子空间上的有效 Hecke 特征值等分布

Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces

本文证明固定素数下 Atkin--Lehner 子空间上 Hecke 特征值的显式有效等分布，并给出五个应用，包括推广 Kim 猜想、新形式计数、模雅可比簇点数的显式渐近、下界推广及超奇异模雅可比簇的有限性。

更高海根点非 torsion 性的马祖尔猜想加强

Strengthenings of Mazur's Conjecture for Higher Heegner Points

本文通过伽罗瓦轨道与海根轨道的相互作用，定量加强了马祖尔猜想关于更高海根点非 torsion 性的结论，避免了对水平结构的严格限制。

高斯周期的渐近Mahler测度

The asymptotic Mahler measure of Gaussian periods

构造具有特别小Mahler测度的分圆整数（高斯周期）序列，研究其Mahler测度作为导子函数的渐近行为，发现增长率是维数递增的log Calabi-Yau簇族的多元Mahler测度。

某些阿贝尔四簇的基本约化在无穷多个素数处成立

Infinitely many primes of basic reduction for some abelian fourfolds

本文将 Elkies 关于椭圆曲线超奇异约化无穷多素数的结果推广到某些亏格 4 曲线，通过 Deligne–Mostow Shimura 簇的一致化、测地线与二次型表示及 CM 点的等分布，证明其雅可比簇在无穷多个素数处具有基本约化。