Extremal primes for elliptic curves

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• 作者：Kevin James^a ; ^{kevja@clemson.edu" class="auth_mail" title="E-mail the corresponding author} ; Brandon Tran^c ; ^{btran115@mit.edu" class="auth_mail" title="E-mail the corresponding author} ; Minh-Tam Trinh^d ; ^{mqt@uchicago.edu" class="auth_mail" title="E-mail the corresponding author} ; Phil Wertheimer^b ; ^{phil.wertheimer@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Dania Zantout^a ; ^{dzantou@g.clemson.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Frobenius distributions ; Trace of Frobenius ; Distribution of primes ; Elliptic curves ; Lang&ndash ; Trotter conjecture
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：164
• 期：Complete
• 页码：282-298
• 全文大小：378 K

文摘

For an elliptic curve 1600072X&_mathId=si1.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=c4f369ab71e24eb876269342b932aa75" title="Click to view the MathML source">E/Q$E/\mathbb{Q}$, we define an extremal prime for E to be a prime p of good reduction such that the trace of Frobenius of E at p   is 1600072X&_mathId=si2.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=6d796d49e9a86ed8bbaae660dca249fb">1600072X-si2.gif">$\pm \lfloor 2\sqrt{p}\rfloor$, i.e., maximal or minimal in the Hasse interval. Conditional on the Riemann Hypothesis for certain Hecke L  -functions, we prove that if 1600072X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=972058d844e84a7dc0b545d8a6a58ea7" title="Click to view the MathML source">End(E)=O_K$\mathrm{End}(E)={\mathcal{O}}_K$, where K   is an imaginary quadratic field of discriminant 1600072X&_mathId=si4.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=4c4849e726906cb52f630344f404b5d5" title="Click to view the MathML source">≠−3,−4$\ne -3,-4$, then the number of extremal primes ≤X for E   is asymptotic to 1600072X&_mathId=si5.gif&_user=111111111&_pii=S0022314X1600072X&_rdoc=1&_issn=0022314X&md5=d4b3fb7cf0aaedd57e609c8b60e65165" title="Click to view the MathML source">X^3/4/log⁡X$X^{3/4}/\log X$. We give heuristics for related conjectures.