Extremal primes for elliptic curves with complex multiplication

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• 作者：Kevin James^a ; ^{kevja@clemson.edu" class="auth_mail" title="E-mail the corresponding author} ; Paul Pollack^b ; ^{pollack@uga.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11R11 ; 11R29
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：172
• 期：Complete
• 页码：383-391
• 全文大小：286 K

文摘

Fix an elliptic curve E/Q$E/\mathbb{Q}$. For each prime p   of good reduction, let a_p=p+1−#E(F_p)$a_p=p+1-\mathrm{\#}E({\mathbb{F}}_p)$. A well-known theorem of Hasse asserts that $|a_p|\le 2\sqrt{p}$. We say that p is a champion prime for E   if $a_p=-\lfloor 2\sqrt{p}\rfloor$, that is, #E(F_p)$\mathrm{\#}E({\mathbb{F}}_p)$ is as large as allowed by the Hasse bound. Analogously, we call p   a trailing prime if $a_p=\lfloor 2\sqrt{p}\rfloor$. In this note, we study the frequency of champion and trailing primes for CM elliptic curves. Our main theorem is that for CM curves, both the champion primes and trailing primes have counting functions c20">$\sim \frac{2}{3\pi }{x}^{3/4}/\log x$, as 17ce3140d1c8c1c0ec937f09191065" title="Click to view the MathML source">x→∞$x\to \infty$. This confirms (in corrected form) a recent conjecture of James–Tran–Trinh–Wertheimer–Zantout.