Local constancy of dimensions of Hecke eigenspaces of Automorphic forms

射影直线的 Hecke 特征空间

Hecke eigenspaces for the projective line

本文针对有限域上射影直线函数域与 GL_2，计算非分歧及分歧 Hecke 算子特征空间维数，并显式刻画其在自守形式上的作用。

Automorphic Forms on Feit's Hermitian Lattices

We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for $\mathbb{U}(2,2)$ automorphic forms.

Explicit calculations of automorphic forms for definite unitary groups

I give an algorithm for computing the full space of automorphic forms for definite unitary groups over Q, and apply this to calculate the automorphic forms of level $G(Z-hat)$ and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U_1 x U_1 x U_1 and U_1 x U_2, and to an example of a non-endoscopic form of weight (3,3) corresponding to a family of 3-dimensional irreducible l-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, …

Igusa簇上函数的完备Kirillov模型与局部-全局相容性

The completed Kirillov model and local-global compatibility for functions on Igusa varieties

本文将GL2的普通Igusa簇上的尖点函数描述为经典模形式Kirillov模型的完备化，建立弱局部-全局相容性定理，并猜想更一般情形下的Hida理论类似物。