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#### Serre weight conjectures for p-adic unitary groups of rank 2

##### External Resource

https://doi.org/10.2140/ant.2022.16.2005

(Publisher version)

https://doi.org/10.48550/arXiv.1810.03827

(Preprint)

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##### Fulltext (public)

Koziol-Morra_Serre weight conjectures for p-adic unitary groups of rank 2_2022.pdf

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##### Supplementary Material (public)

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##### Citation

Kozioł, K., & Morra, S. (2022). Serre weight conjectures for p-adic unitary groups
of rank 2.* Algebra & Number Theory,* *16*(9), 2005-2097.
doi:10.2140/ant.2022.16.2005.

Cite as: https://hdl.handle.net/21.11116/0000-000A-21E4-F

##### Abstract

We prove a version of the weight part of Serre's conjecture for mod $p$

Galois representations attached to automorphic forms on rank 2 unitary groups

which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension

of a totally real field such that every place of $F^+$ above $p$ is unramified

and inert in $F$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+)

\longrightarrow {}^C\mathbf{U}_2(\overline{\mathbb{F}}_p)$ be a Galois

parameter valued in the $C$-group of a rank 2 unitary group attached to

$F/F^+$. We assume that $\overline{r}$ is semisimple and sufficiently generic

at all places above $p$. Using base change techniques and (a strengthened

version of) the Taylor-Wiles-Kisin conditions, we prove that the set of Serre

weights in which $\overline{r}$ is modular agrees with the set of Serre weights

predicted by Gee-Herzig-Savitt.

Galois representations attached to automorphic forms on rank 2 unitary groups

which are non-split at $p$. More precisely, let $F/F^+$ denote a CM extension

of a totally real field such that every place of $F^+$ above $p$ is unramified

and inert in $F$, and let $\overline{r}: \textrm{Gal}(\overline{F^+}/F^+)

\longrightarrow {}^C\mathbf{U}_2(\overline{\mathbb{F}}_p)$ be a Galois

parameter valued in the $C$-group of a rank 2 unitary group attached to

$F/F^+$. We assume that $\overline{r}$ is semisimple and sufficiently generic

at all places above $p$. Using base change techniques and (a strengthened

version of) the Taylor-Wiles-Kisin conditions, we prove that the set of Serre

weights in which $\overline{r}$ is modular agrees with the set of Serre weights

predicted by Gee-Herzig-Savitt.