An arithmetic intersection for squares of elliptic curves with complex multiplication
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910742202023936 |
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| author | García, Elisa Lorenzo Ritzenthaler, Christophe Villegas, Fernando Rodríguez |
| author_facet | García, Elisa Lorenzo Ritzenthaler, Christophe Villegas, Fernando Rodríguez |
| contents | 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100$ and compare our results with an unpublished formula by the third author. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08738 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An arithmetic intersection for squares of elliptic curves with complex multiplication García, Elisa Lorenzo Ritzenthaler, Christophe Villegas, Fernando Rodríguez Number Theory Algebraic Geometry 14G25, 14G40, 14H45, 14K22, 14K25, 14Q25, 11E20, 11F46, 11G05, 11G20, 11R52 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100$ and compare our results with an unpublished formula by the third author. |
| title | An arithmetic intersection for squares of elliptic curves with complex multiplication |
| topic | Number Theory Algebraic Geometry 14G25, 14G40, 14H45, 14K22, 14K25, 14Q25, 11E20, 11F46, 11G05, 11G20, 11R52 |
| url | https://arxiv.org/abs/2412.08738 |