Sato-Tate groups of genus 2 curves
Abstract
We describe the analogue of the Sato-Tate conjecture for an abelian variety over a number field; this predicts that the zeta functions of the reductions over various finite fields, when properly normalized, have a limiting distribution predicted by a certain group-theoretic construction related to Hodge theory, Galois images, and endomorphisms. After making precise the definition of the "Sato-Tate group" appearing in this conjecture, we describe the classification of Sato-Tate groups of abelian surfaces due to Fite-Kedlaya-Rotger-Sutherland. (These are notes from a three-lecture series presented at the NATO Advanced Study Institute "Arithmetic of Hyperelliptic Curves" held in Ohrid (Macedonia) August 25-September 5, 2014, and are expected to appear in a proceedings volume.)
Keywords
Cite
@article{arxiv.1408.6968,
title = {Sato-Tate groups of genus 2 curves},
author = {Kiran S. Kedlaya},
journal= {arXiv preprint arXiv:1408.6968},
year = {2014}
}
Comments
20 pages; includes custom class file; v2: formula of Birch corrected