nLab arithmetic gauge theory

Contents

Context

Arithmetic geometry

Chern-Weil theory

Contents

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 KK be a field of characteristic zero and let G KG_{K} be its absolute Galois group. A gauge group over KK is a topological group UU with a continuous action of G KG_{K}.

A UU-gauge field, or UU-principal bundle, is a topological space PP with compatible continuous left G KG_{K}-action and simply transitive continuous right UU action.

Examples

This section follows section 8 of Kim 2018.

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

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

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

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

the UU-torsor of pro-unipotent paths from bb to xx.

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

H f 1( S,U) H 1( v,U)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 pp the corresponding component is crystalline.

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

xP(x).x\mapsto P(x).

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

V() V( p) A A p H f 1( S,U) loc p H f 1( p,U) \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 VV is a smooth projective curve of genus g2g\geq 2. Then

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

Relation to L-functions

References

Last revised on July 3, 2022 at 14:45:41. See the history of this page for a list of all contributions to it.