HodgeTate

The Hodge–Tate theorem is a fundamental result in the arithmetic theory of algebraic varieties and number theory, connecting algebraic geometry with representation theory. It was formulated by William Hodge in the context of complex manifolds and later generalized by John Tate to broader settings, particularly in the study of Galois representations.

The theorem provides a deep relationship between the étale cohomology of varieties over number fields and their associated Galois representations. Specifically, it describes how certain cohomology groups of a smooth projective variety over a number field, when viewed as representations of the absolute Galois group of the field, can be decomposed into direct sums of one-dimensional representations. These one-dimensional representations are often referred to as Hodge–Tate characters, and they are indexed by pairs of integers (r, s) corresponding to the Hodge numbers of the variety.

In the case of an abelian variety over a number field, the Hodge–Tate theorem asserts that the Galois representation on the Tate module of the abelian variety decomposes as a direct sum of characters of the form χ^r ⊗ χ^s, where χ is the cyclotomic character and r, s range over integers such that r + s = 1. This result is particularly significant because it bridges the geometric properties of the abelian variety, encoded in its Hodge structure, with its arithmetic properties, encoded in its Galois representation.

The theorem has profound implications for number theory, including the study of modular forms, elliptic curves, and the Birch and Swinnerton-Dyer conjecture. It also plays a crucial role in the Langlands program, which seeks to establish analogies between Galois representations and automorphic representations in various contexts. The Hodge–Tate theorem provides a key tool for understanding the arithmetic of algebraic varieties and their associated Galois representations.