An analogue of Serre’s conjecture for a ring of distributions

The set 𝒜 := 𝔺δ0+ 𝒟+′, obtained by attaching the identity δ0 to the set 𝒟+′ of all distributions on 𝕉 with support contained in (0, ∞), forms an algebra with the operations of addition, convolution, multiplication by complex scalars. It is shown that 𝒜 is a Hermite ring, that is, every finitely generated stably free 𝒜-module is free, or equivalently, every tall left-invertible matrix with entries from 𝒜 can be completed to a square matrix with entries from 𝒜, which is invertible.

Serre weights for U ⁢ ( n ) {U(n)}

We study the weight part of (a generalisation of) Serre’s conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when {n=3} and l splits completely in the underlying CM field.

On mod local-global compatibility for in the ordinary case

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

THE WEIGHT PART OF SERRE’S CONJECTURE FOR

Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.