Bounds on certain Higher-Dimensional Exponential Sums via the Self-Reducibility of the Weil Representation

Notes on the Self-Reducibility of the Weil Representation and Higher-Dimensional Quantum Chaos

In these notes we discuss the "self-reducibility property" of the Weil representation. We explain how to use this property to obtain sharp estimates of certain higher-dimensional exponential sums which originate from the theory of quantum chaos. As a result, we obtain the Hecke quantum unique ergodicity theorem for generic linear symplectomorphism $A$ of the torus $T^{2N}=R^{2N}/Z^{2N}.

Representation de Weil et beta-extensions

We study beta-extensions in a p-adic classical group and we produce a relation between some beta-extensions by means of a Weil representation. We apply this to the study of reducibility points of some parabolically induced representations.

Exponential sums and total Weil representations of finite symplectic and unitary groups

We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups $Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power of $p$, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, i.e., easy to remember, form. We also exhibit hypergeometric sheaves on $G_m$, whose geometric monodro…

Weil representations via abstract data and Heisenberg groups: a comparison

Let $B$ be a ring, not necessarily commutative, having an involution $*$ and let ${\mathrm U}_{2m}(B)$ be the unitary group of rank $2m$ associated to a hermitian or skew hermitian form relative to $*$. When $B$ is finite, we construct a Weil representation of ${\mathrm U}_{2m}(B)$ via Heisenberg groups and find its explicit matrix form on the Bruhat elements. As a consequence, we derive information on generalized Gauss sums. On the other hand, there is an axiomatic method to define a Weil representation of ${\mathrm U}_{2m}(B)$, and we compare the two Weil representations thus obtained under …

可约性点与特征$p$局部域 I——辛群的简单超尖点表示

Reducibility points and characteristic $p$ local fields I- Simple supercuspidal representations of symplectic groups

本文研究特征为奇素数p的非阿基米德局部域上辛群的简单超尖点表示，推广Moeglin可约性准则至任意正特征域，通过显式构造给出其到一般线性群表示的转移，推进了正特征下辛群泛型尖点表示的朗兰兹对应研究。

Quadratic forms in 8 prime variables

We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group $\operatorname{Sp}_8(\mathbf{Z}/q\mathbf{Z})$. Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convolution…