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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2402.12218 |
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| _version_ | 1866916833231110144 |
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| author | Wang, Tian |
| author_facet | Wang, Tian |
| contents | Let $A/K$ be an absolutely simple abelian surface defined over a number field $K$. We give unconditional upper bounds for the number of prime ideals $\mathfrak{p}$ of $K$ with norm up to $x$ such that $A$ has supersingular reduction at $\mathfrak{p}$. These bounds are obtained in three distinct settings, depending on the endomorphism algebra of $A$, namely, the case of trivial endomorphisms, real multiplication (RM), and quaternion multiplication (QM). In the RM case and when $K=\mathbb{Q}$, our results further implies an unconditional upper bound on the distribution of Frobenius traces of $A$. Furthermore, in the RM setting, we study the distribution of the middle coefficients of Frobenius polynomials of $A$ at primes where the reduction of $A$ splits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12218 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of supersingular primes for abelian surfaces Wang, Tian Number Theory 11G10, 11N45, 11G18, 11N13, 11N36 Let $A/K$ be an absolutely simple abelian surface defined over a number field $K$. We give unconditional upper bounds for the number of prime ideals $\mathfrak{p}$ of $K$ with norm up to $x$ such that $A$ has supersingular reduction at $\mathfrak{p}$. These bounds are obtained in three distinct settings, depending on the endomorphism algebra of $A$, namely, the case of trivial endomorphisms, real multiplication (RM), and quaternion multiplication (QM). In the RM case and when $K=\mathbb{Q}$, our results further implies an unconditional upper bound on the distribution of Frobenius traces of $A$. Furthermore, in the RM setting, we study the distribution of the middle coefficients of Frobenius polynomials of $A$ at primes where the reduction of $A$ splits. |
| title | Distribution of supersingular primes for abelian surfaces |
| topic | Number Theory 11G10, 11N45, 11G18, 11N13, 11N36 |
| url | https://arxiv.org/abs/2402.12218 |