nLab arithmetic gauge theory

Idea

Arithmetic gauge theory is a theory proposed by Kim 2018 to look at Galois representations with an action of another group (the gauge group) as being analogous to gauge fields in gauge theory. It is an approach to Diophantine problems such as the effective Mordell conjecture and is influenced by the theory of the Selmer group and Chabauty’s method.

Definitions

Let $K$ be a field of characteristic zero and let $G_{K}$ be its absolute Galois group. A gauge group over $K$ is a topological group $U$ with a continuous action of $G_{K}$.

A $U$-gauge field, or $U$-principal bundle, is a topological space $P$ with compatible continuous left $G_{K}$-action and simply transitive continuous right $U$ action.

Examples

This section follows section 8 of Kim 2018.

Let $V$ be a variety over $\mathbb{Q}$ equipped with a base point $b\in V(\mathbb{Q})$. Our motivation is determining the set $V(\mathbb{Q})$ of rational points of $V$. For our gauge group $U$ we let

$$U=\pi_{1}(\overline{V},b)_{\mathbb{Q}},$$

the $\mathbb{Q}_{p}$-pro-unipotent fundamental group of $V$. For our gauge field $P$ we let

$$P(x)=\pi_{1}(\overline{V},b,x),$$

the $U$-torsor of pro-unipotent paths from $b$ to $x$.

Let $S$ be a finite set of primes. Let $H_{f}^{1}(\mathbb{Z}_{S},U)$ be the set of $U$-torsors over $\mathbb{Q}$ which are unramified outside $S$ and crystalline at $p$ (see p-adic Hodge theory for the meaning of crystalline). Similarly let $H^{1}(\mathbb{Q}_{p},U)$ be the set of $U$-torsors over $\mathbb{Q}_{p}$. We have a localization map

$$H_{f}^{1}(\mathbb{Z}_{S},U)\to\prod^{'}H^{1}(\mathbb{Q}_{v},U)$$

where the restricted product on the right means that all but finitely many of the components are unramified, and at $p$ the corresponding component is crystalline.

Recall that we are interested in $V(\mathbb{Q})$, the set of rational points of $V$. Now we have a map $A:V(\mathbb{Q})\to H_{f}^{1}(\mathbb{Z}_{S},U)$ given by

$$x\mapsto P(x).$$

where $P(x)=\pi_{1}(\overline{V},b,x)$ as above. This map fits into the diagram

$$\array{& V(\mathbb{Q}) & \rightarrow & V(\mathbb{Q}_{p}) & \\ A & \downarrow &&\downarrow & A_{p} \\ & H_{f}^{1}(\mathbb{Z}_{S},U) & \underset{loc_{p}}\rightarrow& H_{f}^{1}(\mathbb{Q}_{p},U) & }$$

Kim conjectures the following:

Conjecture

Suppose $V$ is a smooth projective curve of genus $g\geq 2$. Then

$$V(\mathbb{Q})=A_{p}^{-1}(Im(loc_{p})).$$

Relation to L-functions

Related ideas

• gauge theory

• Mordell conjecture

• arithmetic Chern-Simons theory

References

• Minhyong Kim, Arithmetic Gauge Theory: A Brief Introduction, Modern Physics Letters A 33 29 (2018) 1830012 [arxiv:1712.07602, doi:10.1142/S0217732318300124]

• Minhyong Kim, Recent Progress on the Effective Mordell Problem, lecture at the Sydney Mathematical Research Institute YouTube

• Minhyong Kim, Foundations of nonabelian Chabauty, lectures at the Arizona Winter School 2020 Slides1, Slides2 Slides3 Slides4