The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions
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| Format: | Preprint |
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2024
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| _version_ | 1866913484532350976 |
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| author | Kane, Ben Pujahari, Sudhir Yang, Zichen |
| author_facet | Kane, Ben Pujahari, Sudhir Yang, Zichen |
| contents | In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in an arbitrary fixed arithmetic progression. Contrary to the classical setting of reductions of one-parameter families over the rationals, where it is conjectured by Steven J. Miller that the bias is always negative, we prove that in our setting the bias is positive for a positive density of arithmetic progressions and negative for a positive density of arithmetic progressions. Along the way, we obtain explicit formulas for moments of traces of Frobenius of elliptic curves over finite fields in arithmetic progressions and related moments of Hurwitz class numbers in arithmetic progressions, the distribution of which are of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14234 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions Kane, Ben Pujahari, Sudhir Yang, Zichen Number Theory 11E41, 11F11, 11F27, 11F37, 11G05, 11G07 In this paper, we consider a version of the bias conjecture for second moments in the setting of elliptic curves over finite fields whose trace of Frobenius lies in an arbitrary fixed arithmetic progression. Contrary to the classical setting of reductions of one-parameter families over the rationals, where it is conjectured by Steven J. Miller that the bias is always negative, we prove that in our setting the bias is positive for a positive density of arithmetic progressions and negative for a positive density of arithmetic progressions. Along the way, we obtain explicit formulas for moments of traces of Frobenius of elliptic curves over finite fields in arithmetic progressions and related moments of Hurwitz class numbers in arithmetic progressions, the distribution of which are of independent interest. |
| title | The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions |
| topic | Number Theory 11E41, 11F11, 11F27, 11F37, 11G05, 11G07 |
| url | https://arxiv.org/abs/2405.14234 |