Local conductor bounds for modular abelian varieties
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910915700457472 |
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| author | Martin, Kimball |
| author_facet | Martin, Kimball |
| contents | Brumer and Kramer gave bounds on local conductor exponents for an abelian variety $A/\mathbb Q$ in terms of the dimension of $A$ and the localization prime $p$. Here we give improved bounds in the case that $A$ has maximal real multiplication, i.e., $A$ is isogenous to a factor of the Jacobian of a modular curve $X_0(N)$. In many cases, these bounds are sharp. The proof relies on showing that the rationality field of a newform for $Γ_0(N)$, and thus the endomorphism algebra of $A$, contains $\mathbb Q(ζ_{p^r})^+$ when $p$ divides $N$ to a sufficiently high power. We also deduce that certain divisibility conditions on $N$ determine the endomorphism algebra when $A$ is simple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_13127 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local conductor bounds for modular abelian varieties Martin, Kimball Number Theory Brumer and Kramer gave bounds on local conductor exponents for an abelian variety $A/\mathbb Q$ in terms of the dimension of $A$ and the localization prime $p$. Here we give improved bounds in the case that $A$ has maximal real multiplication, i.e., $A$ is isogenous to a factor of the Jacobian of a modular curve $X_0(N)$. In many cases, these bounds are sharp. The proof relies on showing that the rationality field of a newform for $Γ_0(N)$, and thus the endomorphism algebra of $A$, contains $\mathbb Q(ζ_{p^r})^+$ when $p$ divides $N$ to a sufficiently high power. We also deduce that certain divisibility conditions on $N$ determine the endomorphism algebra when $A$ is simple. |
| title | Local conductor bounds for modular abelian varieties |
| topic | Number Theory |
| url | https://arxiv.org/abs/2302.13127 |