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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.12466 |
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| _version_ | 1866909175564468224 |
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| author | Cojocaru, Alina Carmen Hinz, Auden Wang, Tian |
| author_facet | Cojocaru, Alina Carmen Hinz, Auden Wang, Tian |
| contents | For $E_1$ and $E_2$ elliptic curves defined over a number field $K$, without complex multiplication, we consider the function ${\mathcal{F}}_{E_1, E_2}(x)$ counting non-zero prime ideals $\mathfrak{p}$ of the ring of integers of $K$, of good reduction for $E_1$ and $E_2$, of norm at most $x$, and for which the Frobenius fields $\mathbb{Q}(π_{\mathfrak{p}}(E_1))$ and $\mathbb{Q}(π_{\mathfrak{p}}(E_2))$ are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that $E_1$ and $E_2$ are not potentially isogenous if and only if ${\mathcal{F}}_{E_1, E_2}(x) = \operatorname{o} \left(\frac{x}{\log x}\right)$, we investigate the growth in $x$ of ${\mathcal{F}}_{E_1, E_2}(x)$. We prove that if $E_1$ and $E_2$ are not potentially isogenous, then there exist positive constants $κ(E_1, E_2, K)$, $κ'(E_1, E_2, K)$, and $κ''(E_1, E_2, K)$ such that the following bounds hold: (i) ${\mathcal{F}}_{E_1, E_2}(x) < κ(E_1, E_2, K) \frac{ x (\log\log x)^{\frac{1}{9}}}{ (\log x)^{\frac{19}{18}}}$; (ii) ${\mathcal{F}}_{E_1, E_2}(x) < κ'(E_1, E_2, K) \frac{ x^{\frac{6}{7}}}{ (\log x)^{\frac{5}{7}}}$ under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) ${\mathcal{F}}_{E_1, E_2}(x) < κ''(E_1, E_2, K) x^{\frac{2}{3}} (\log x)^{\frac{1}{3}}$ under GRH, Artin's Holomorphy Conjecture for the Artin $L$-functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin $L$-functions of number field extensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12466 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative upper bounds related to an isogeny criterion for elliptic curves Cojocaru, Alina Carmen Hinz, Auden Wang, Tian Number Theory 11G05, 11G20, 11N05 For $E_1$ and $E_2$ elliptic curves defined over a number field $K$, without complex multiplication, we consider the function ${\mathcal{F}}_{E_1, E_2}(x)$ counting non-zero prime ideals $\mathfrak{p}$ of the ring of integers of $K$, of good reduction for $E_1$ and $E_2$, of norm at most $x$, and for which the Frobenius fields $\mathbb{Q}(π_{\mathfrak{p}}(E_1))$ and $\mathbb{Q}(π_{\mathfrak{p}}(E_2))$ are equal. Motivated by an isogeny criterion of Kulkarni, Patankar, and Rajan, which states that $E_1$ and $E_2$ are not potentially isogenous if and only if ${\mathcal{F}}_{E_1, E_2}(x) = \operatorname{o} \left(\frac{x}{\log x}\right)$, we investigate the growth in $x$ of ${\mathcal{F}}_{E_1, E_2}(x)$. We prove that if $E_1$ and $E_2$ are not potentially isogenous, then there exist positive constants $κ(E_1, E_2, K)$, $κ'(E_1, E_2, K)$, and $κ''(E_1, E_2, K)$ such that the following bounds hold: (i) ${\mathcal{F}}_{E_1, E_2}(x) < κ(E_1, E_2, K) \frac{ x (\log\log x)^{\frac{1}{9}}}{ (\log x)^{\frac{19}{18}}}$; (ii) ${\mathcal{F}}_{E_1, E_2}(x) < κ'(E_1, E_2, K) \frac{ x^{\frac{6}{7}}}{ (\log x)^{\frac{5}{7}}}$ under the Generalized Riemann Hypothesis for Dedekind zeta functions (GRH); (iii) ${\mathcal{F}}_{E_1, E_2}(x) < κ''(E_1, E_2, K) x^{\frac{2}{3}} (\log x)^{\frac{1}{3}}$ under GRH, Artin's Holomorphy Conjecture for the Artin $L$-functions of number field extensions, and a Pair Correlation Conjecture for the zeros of the Artin $L$-functions of number field extensions. |
| title | Quantitative upper bounds related to an isogeny criterion for elliptic curves |
| topic | Number Theory 11G05, 11G20, 11N05 |
| url | https://arxiv.org/abs/2404.12466 |