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Let $K$ be a local function field of characteristic $l$, $mathbb{F}$ be a finite field over $mathbb{F}_p$ where $l e p$, and $overline{rho}: G_K rightarrow text{GL}_n (mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $overline{text{cyc}}$, from mod $p$ representations of $text{GL}_n (mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{overline{rho}}^square$. This allows us to obtain a function field analog of the Breuil--Mezard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-Mezard conjecture for local number fields in the case of $l e p$, obtained by Shotton.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we pr
We propose a function-field analog of Pisots $d$-th root conjecture on linear recurrences, and prove it under some non-triviality assumption. Besides a recent result of Pasten-Wang on B{u}chis $d$-th power problem, our main tool, which is also develo
We establish the local Langlands conjecture for small rank general spin groups $GSpin_4$ and $GSpin_6$ as well as their inner forms. We construct appropriate $L$-packets and prove that these $L$-packets satisfy the properties expected of them to the
Sarnaks Density Conjecture is an explicit bound on the multiplicities of non-tempered representations in a sequence of cocompact congruence arithmetic lattices in a semisimple Lie group, which is motivated by the work of Sarnak and Xue. The goal of t
We consider natural variants of Lehmers unresolved conjecture that Ramanujans tau-function never vanishes. Namely, for $n>1$ we prove that $$tau(n) ot in {pm 1, pm 3, pm 5, pm 7, pm 691}.$$ This result is an example of general theorems for newforms w