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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si1.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=73b24f925854bb917ddc7b73b780f2f5" title="Click to view the MathML source">E/Qclass="mathContainer hidden">class="mathCode">. For each prime p   of good reduction, let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si2.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=a585410108c1cc0ddbcc08c169a00bc4" title="Click to view the MathML source">a_p=p+1−#E(F_p)class="mathContainer hidden">class="mathCode">. A well-known theorem of Hasse asserts that class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si3.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=75635932d918b360c0109aac2b311042">class="imgLazyJSB inlineImage" height="19" width="78" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302670-si3.gif">cript>cal-align:bottom" width="78" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16302670-si3.gif">cript>class="mathContainer hidden">class="mathCode">. We say that p is a champion prime for E   if class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si4.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=b0317567cba8877266612b1138030ab9">class="imgLazyJSB inlineImage" height="19" width="94" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302670-si4.gif">cript>cal-align:bottom" width="94" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16302670-si4.gif">cript>class="mathContainer hidden">class="mathCode">, that is, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si5.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=c8c364c001ebb6db299103a09a515017" title="Click to view the MathML source">#E(F_p)class="mathContainer hidden">class="mathCode"> is as large as allowed by the Hasse bound. Analogously, we call p   a trailing prime if class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si20.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=fe5d53bb3d71ec56d7f7f308e6d654a5">class="imgLazyJSB inlineImage" height="19" width="81" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302670-si20.gif">cript>cal-align:bottom" width="81" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16302670-si20.gif">cript>class="mathContainer hidden">class="mathCode">. 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 class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si138.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=d328a236520ca0653b4ebf1f59b1bc20">class="imgLazyJSB inlineImage" height="20" width="110" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16302670-si138.gif">cript>cal-align:bottom" width="110" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16302670-si138.gif">cript>class="mathContainer hidden">class="mathCode">, as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16302670&_mathId=si30.gif&_user=111111111&_pii=S0022314X16302670&_rdoc=1&_issn=0022314X&md5=eb17ce3140d1c8c1c0ec937f09191065" title="Click to view the MathML source">x→∞class="mathContainer hidden">class="mathCode">. This confirms (in corrected form) a recent conjecture of James–Tran–Trinh–Wertheimer–Zantout.