Idempotents and Nilpotents Modulo n

A geometric interpretation of the nilpotent part of local Langlands correspondence modulo l

Let p and l be two distinct primes. We show how, under a certain congruence hypothesis, the mod l cohomology of the Lubin-Tate tower together with a certain Lefschetz operator provides a geometric interpretation of Vigneras' Langlands correspondence for a p-adic field with mod l coefficients.

On the modular Erdős-Burgess constant

Let $n$ be a positive integer. For any integer $a$, we say that $a$ is idempotent modulo $n$ if $a^2\equiv a\pmod n$. The $n$-modular Erdős-Burgess constant is the smallest positive integer $\ell$ such that any $\ell$ integers contain one or more integers whose product is idempotent modulo $n$. We gave a sharp lower bound of the $n$-modular Erdős-Burgess constant, in particular, we determined the $n$-modular Erdős-Burgess constant in the case when $n$ is a prime power or a product of pairwise distinct primes.

Formes modulaires modulo 2 : l'ordre de nilpotence des opérateurs de Hecke

The nilpotence order of the mod 2 Hecke operators. Let $Δ=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ be the reduction mod 2 of the $Δ$ series. A modular form f modulo 2 of level 1 is a polynomial in $Δ$. If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp(f) which is a polynomial in $Δ$ whose degree is smaller than the degree of f, so that Tp is nilpotent. The order of nilpotence of f is defined as the smallest integer g = g(f) such that, for every family of g odd primes p1, p2, ..., pg, the relation Tp1Tp2... Tpg (f) = 0 holds. We show how one can compute expl…

模 2 模形式：Hecke 算子的幂零阶（扩展版）

Formes modulaires modulo $2$ : L'ordre de nilpotence des opérateurs de Hecke (version développée)

本文研究模2模形式中Hecke算子的幂零性，给出显式计算幂零阶的方法，并证明对次数为d的多项式，其幂零阶小于3/2√d。