On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$
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| Format: | Preprint |
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2017
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| _version_ | 1866914269921017856 |
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| author | Joshi, Kirti |
| author_facet | Joshi, Kirti |
| contents | For a given elliptic curve $E/\mathbb{Q}$, let $N_p(E)$ be the number of points on $E$ modulo $p$ for a prime of good reduction for $E$. Given integer $n$, let $G_k(E,n)$ be the number of $k$-tuples of $p_1<p_2<\ldots <p_k$ primes of good reduction for $E$, for which the equation in the title holds, then on assuming the Generalized Riemann Hypothesis for elliptic curves without CM (and unconditionally if the curves have complex multiplication), I show that $\varlimsup_{n\to\infty} G_k(E,n)=\infty$ for any integer $k\geq 3$. I conjecture that this result also holds for $k=1,2$ i.e. this conjecture says that there are arbitrarily long ``elliptic progressions of primes'' i.e. sequences of primes $p_1<p_2<\cdots <p_m$ of arbitrary lengths $m$ such that $N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1711_06283 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$ Joshi, Kirti Number Theory Algebraic Geometry 11G05, 11G20, 11A41, 11N05, 11B25, 11D45, 11G40 For a given elliptic curve $E/\mathbb{Q}$, let $N_p(E)$ be the number of points on $E$ modulo $p$ for a prime of good reduction for $E$. Given integer $n$, let $G_k(E,n)$ be the number of $k$-tuples of $p_1<p_2<\ldots <p_k$ primes of good reduction for $E$, for which the equation in the title holds, then on assuming the Generalized Riemann Hypothesis for elliptic curves without CM (and unconditionally if the curves have complex multiplication), I show that $\varlimsup_{n\to\infty} G_k(E,n)=\infty$ for any integer $k\geq 3$. I conjecture that this result also holds for $k=1,2$ i.e. this conjecture says that there are arbitrarily long ``elliptic progressions of primes'' i.e. sequences of primes $p_1<p_2<\cdots <p_m$ of arbitrary lengths $m$ such that $N_{p_1}(E)=N_{p_2}(E)=\cdots =N_{p_m}(E)$. |
| title | On the equation $N_{p_1}(E)\cdot N_{p_2}(E)\cdots N_{p_k}(E)=n$ |
| topic | Number Theory Algebraic Geometry 11G05, 11G20, 11A41, 11N05, 11B25, 11D45, 11G40 |
| url | https://arxiv.org/abs/1711.06283 |