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arXiv · 2412.08738

An arithmetic intersection for squares of elliptic curves with complex multiplication

Elisa Lorenzo García·Christophe Ritzenthaler·Fernando Rodríguez Villegas
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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100$ and compare our results with an unpublished formula by the third author.

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Elisa Lorenzo García, Christophe Ritzenthaler, Fernando Rodríguez Villegas. 2024-12-11. An arithmetic intersection for squares of elliptic curves with complex multiplication. https://arxiv.org/abs/2412.08738

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