Work

An arithmetic intersection for squares of elliptic curves with complex multiplication

Year: 2024

Type: preprint

Abstract: 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100$ and compare our results with an unpublished formula by the third author.

Source: HAL (Le Centre pour la Communication Scientifique Directe)

DOI: 10.48550/arxiv.2412.08738

Authors: García, Elisa Lorenzo, Ritzenthaler, Christophe, Villegas, Fernando Rodríguez

Institutions: Institut de recherche mathématique de Rennes, University of Neuchâtel, Centre National de la Recherche Scientifique, Université Côte d'Azur, Laboratoire Jean-Alexandre Dieudonné, The Abdus Salam International Centre for Theoretical Physics (ICTP)

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W4405354531

Cites:

Cited by: 1

Related to: 10