Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions
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| Format: | Preprint |
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2021
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| _version_ | 1866917698962718720 |
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| author | Bringmann, Kathrin Kane, Ben Pujahari, Sudhir |
| author_facet | Bringmann, Kathrin Kane, Ben Pujahari, Sudhir |
| contents | In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the $(2k+1)$-th moment to the zeroeth moment as the size of the finite field $\mathbb{F}_{p^r}$ goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato--Tate measure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2112_08205 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions Bringmann, Kathrin Kane, Ben Pujahari, Sudhir Number Theory 11E41, 11F27, 11F37, 11G05 In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the $(2k+1)$-th moment to the zeroeth moment as the size of the finite field $\mathbb{F}_{p^r}$ goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato--Tate measure. |
| title | Odd moments for the trace of Frobenius and the Sato--Tate conjecture in arithmetic progressions |
| topic | Number Theory 11E41, 11F27, 11F37, 11G05 |
| url | https://arxiv.org/abs/2112.08205 |