##### Department of Mathematics,

University of California San Diego

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### Chabauty reading group

## David Corwin

#### UC Berkeley

## Explicit nonabelian Chabauty via motives

##### Abstract:

I will introduce a program, begun by Dan-Cohen and Wewers, to compute Minhyong Kim's Selmer varieties using mixed Tate motives. The idea is as follows. We care about the Galois action on the unipotent fundamental group of $\mathbf{P}^1 \setminus \{0,1,\infty\}$. This Galois action lives in a certain category of $p$-adic Galois representations known as mixed Tate representations. We will see that this category is Tannakian and has a fairly simple description, in terms of its Ext groups, which are just Bloch-Kato Selmer groups. The Bloch-Kato Selmer groups are $p$-adic vector spaces, but they also have a rational structure coming from algebraic K-theory. The category of mixed Tate motives gives us a $\mathbb{Q}$-linear Tannakian category that underlies the $\mathbb{Q}_p$-linear Tannakian category of Galois representations. This in turn allows us to define a Selmer variety over $\mathbb{Q}$ (rather than $\mathbb{Q}_p$), and a more explicit understanding of the category of mixed Tate motives allows us to compute this variety explicitly.

Host: Kiran Kedlaya

### February 28, 2020

### 8:30 AM

### AP&M 7218

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