On a strange invariant bilinear form on the space of automorphic forms

On an invariant bilinear form on the space of automorphic forms via asymptotics

This article concerns the study of a new invariant bilinear form $\mathcal B$ on the space of automorphic forms of a split reductive group $G$ over a function field. We define $\mathcal B$ using the asymptotics maps from Bezrukavnikov-Kazhdan and Sakellaridis-Venkatesh, which involve the geometry of the wonderful compactification of $G$. We show that $\mathcal B$ is naturally related to miraculous duality in the geometric Langlands program through the functions-sheaves dictionary. In the proof, we highlight the connection between the classical non-Archimedean Gindikin-Karpelevich formula and c…

Sato-Tate theorem for families and low-lying zeros of automorphic $L$-functions

We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let $G$ be a reductive group over a number field $F$ which admits discrete series representations at infinity. Let $^{L}G=\hat G \rtimes \mathrm{Gal}(\bar F/F)$ be the associated $L$-group and $r:{}^L G\to \mathrm{GL}(d,\mathbb{C})$ a continuous homomorphism which is irreducible and does not factor through $\mathrm{Gal}(\bar F/F)$. The families under consideration consist of discrete automorphic representations of $G(\mathbb{A}_F)$ of given weight and level a…

Bilinear forms in Weyl sums for modular square roots and applications

Let $q$ be a prime, $P \geq 1$ and let $N_q(P)$ denote the number of rational primes $p \leq P$ that split in the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$. The first part of this paper establishes various unconditional and conditional (under existence of a Siegel zero) lower bounds for $N_q(P)$ in the range $q^{1/4+\varepsilon} \leq P \leq q$, for any fixed $\varepsilon>0$. This improves upon what is implied by work of Pollack and Benli-Pollack. The second part of this paper is dedicated to proving an estimate for a bilinear form involving Weyl sums for modular square roots (equivalen…

GL(3)系数与线性及非线性相位加法特征之间的正交关系

Orthogonality relation between $GL(3)$ coefficients and additive characters with linear and non-linear phases

该研究针对SL(3,Z)的Hecke-Maass形式的素数支撑傅里叶系数，通过构造Vaughan恒等式自守变体、改进双线性型等方法，突破了Jiang与吕的2/3障碍，拓宽了非平凡抵消的容许范围。

关于Brun-Titchmarsh定理。I

On the Brun--Titchmarsh theorem. I

本文改进了Brun-Titchmarsh定理的上界，通过双线性或三线性结构优化了不同范围内的素数计数估计。