An arithmetic intersection formula for denominators of Igusa class polynomials

PIMS CRG in Explicit Methods for Abelian Varieties

授课人:Bianca Viray

机构:Pacific Institute for the Mathematical Sciences(PIMS)

关键词: Scientific;    Mathematics;    Number Theory;    polynomials;   

加拿大|英语

【 摘 要 】

Igusa class polynomials are the genus 2 analogue of Hilbert class polynomials; their roots are invariants of genus 2 curves that have complex multiplication by a fixed order. The coefficients of Igusa class polynomials are rational, but, unlike in genus 1, are not integral. An exact formula, or tight upper bound, for these denominators is needed to compute Igusa class polynomials and has applications to cryptography. In this talk, we explain how to obtain a formula for the arithmetic intersection number G1.CM(K) and how this results in a bound for denominators of Igusa class polynomials. We also explain how the formula for G1.CM(K) leads us to a generalization of Gross and Zagier's formula for differences of CM j-invariants. This is joint work with Kristin Lauter.

【 授权许可】

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