Remarks on normal bases

关于正规簇的同伴猜想的一些评注

Some remarks on the companions conjecture for normal varieties

本文研究有限域上正规簇的同伴猜想，通过引入态射的新性质并在若干情形下加以验证，从而将猜想推广到部分奇异正规簇。

Remarks on Automorphy of Residually Dihedral Representations

We prove automorphy lifting results for geometric representations $ρ:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, such that the residual representation $\barρ$ is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F(ζ_p)/F$. Such representations fail the Taylor-Wiles hypothesis and the patching techniques to prove automorphy do not work. We apply this to automorphy of elliptic curves $E$ over $F$, when $E$ has no $F$ ra…

A remark on the simple cuspidal representations of GL(n, F)

Let $F$ be a non-archimedean local field of residue characteristic $p$, $G$ be the group $GL(n, F)$. In this note, under the assumption $(n, p)=1$, we show a simple cuspidal representation $π$ (that with normalized level $\frac{1}{n}$) of $G$ is determined uniquely up to isomorphism by the local constants of $χ\circ \text{det}\otimes π$ for all characters $χ$ of $F^\times$.