An arithmetic intersection for squares of elliptic curves with complex multiplication

Autor:	García, Elisa Lorenzo, Ritzenthaler, Christophe, Villegas, Fernando Rodríguez
Rok vydání:	2024
Předmět:	Mathematics - Number Theory Mathematics - Algebraic Geometry 14G25 14G40 14H45 14K22 14K25 14Q25 11E20 11F46 11G05 11G20 11R52
Druh dokumentu:	Working Paper
Popis:	Let $C$ be a genus $2$ curve with Jacobian isomorphic to the square of an elliptic curve with complex multiplication by a maximal order in an imaginary quadratic field of discriminant $-d<0$. We show that if the stable model of $C$ has bad reduction over a prime $p$ then $p \leq d/4$. We give an algorithm to compute the set of such $p$ using the so-called refined Humbert invariant introduced by Kani. Using results from Kudla-Rapoport and the formula of Gross-Keating, we compute for each of these primes $p$ its exponent in the discriminant of the stable model of $C$. We conclude with some explicit computations for $d<100$ and compare our results with an unpublished formula by the third author. Comment: 31 pages, 3 tables, programs are available as ancillary files
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2412.08738 Zobrazit plný text záznamu View this record from Arxiv