Bounds for the distribution of the Frobenius traces associated to a generic abelian variety
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2022
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| _version_ | 1866914486800089088 |
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| author | Cojocaru, Alina Carmen Wang, Tian |
| author_facet | Cojocaru, Alina Carmen Wang, Tian |
| contents | Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, $A$ has a surjective residual modulo $\ell$ Galois representation. For $t\in \mathbb{Z}$ and $x>0$, denote by $π_A(x, t)$ the number of primes $p \leq x$ for which the Frobenius trace $a_{1, p}(A)$ associated to $A \pmod p$ equals $t$. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), we obtain that $π_A(x, 0) \ll_A x^{1 - \frac{1}{2g^2+g+1}}/(\log x)^{1 - \frac{2}{2g^2+g+1}}$ and $π_A(x, t) \ll_A x^{1 - \frac{1}{2g^2+g+2}}/(\log x)^{1 - \frac{2}{2g^2+g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)| > p^{\frac{1}{2 g^2 + g + 1}}/ (\log p)^{\frac{2}{2g^2+g+1}+\varepsilon}$ for any $\varepsilon>0$. Assuming, in addition to GRH, Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we obtain that $π_A(x, 0) \ll_A x^{1 - \frac{1}{g+1}}/(\log x)^{1 - \frac{4}{g+1}}$ and $π_A(x, t) \ll_A x^{1 - \frac{1}{g+2}}/(\log x)^{1 - \frac{4}{g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)|> p^{\frac{1}{g + 2} - \varepsilon }$ for any $\varepsilon>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_02913 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Bounds for the distribution of the Frobenius traces associated to a generic abelian variety Cojocaru, Alina Carmen Wang, Tian Number Theory 11G05, 11G20, 11N05 (Primary), 11N36, 11N37, 11N56 (Secondary) Let $A$ be an abelian variety defined over $\mathbb{Q}$ and of dimension $g$. Assume that, for each sufficiently large prime $\ell$, $A$ has a surjective residual modulo $\ell$ Galois representation. For $t\in \mathbb{Z}$ and $x>0$, denote by $π_A(x, t)$ the number of primes $p \leq x$ for which the Frobenius trace $a_{1, p}(A)$ associated to $A \pmod p$ equals $t$. Assuming the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH), we obtain that $π_A(x, 0) \ll_A x^{1 - \frac{1}{2g^2+g+1}}/(\log x)^{1 - \frac{2}{2g^2+g+1}}$ and $π_A(x, t) \ll_A x^{1 - \frac{1}{2g^2+g+2}}/(\log x)^{1 - \frac{2}{2g^2+g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)| > p^{\frac{1}{2 g^2 + g + 1}}/ (\log p)^{\frac{2}{2g^2+g+1}+\varepsilon}$ for any $\varepsilon>0$. Assuming, in addition to GRH, Artin's Holomorphy Conjecture and a Pair Correlation Conjecture for Artin L-functions, we obtain that $π_A(x, 0) \ll_A x^{1 - \frac{1}{g+1}}/(\log x)^{1 - \frac{4}{g+1}}$ and $π_A(x, t) \ll_A x^{1 - \frac{1}{g+2}}/(\log x)^{1 - \frac{4}{g+2}}$ if $t \neq 0$, and deduce that almost all primes $p$ satisfy $|a_{1, p}(A)|> p^{\frac{1}{g + 2} - \varepsilon }$ for any $\varepsilon>0$. |
| title | Bounds for the distribution of the Frobenius traces associated to a generic abelian variety |
| topic | Number Theory 11G05, 11G20, 11N05 (Primary), 11N36, 11N37, 11N56 (Secondary) |
| url | https://arxiv.org/abs/2207.02913 |