Supercongruences and complex multiplication

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• 作者：Jonas Kibelbek^a ; ^{jckibelbek@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Ling Long^b ; ^{llong@math.lsu.edu" class="auth_mail" title="E-mail the corresponding author} ; Kevin Moss^a ; ^{kmoss@iastate.edu" class="auth_mail" title="E-mail the corresponding author} ; Benjamin Sheller^a ; ^{bsheller@iastate.edu" class="auth_mail" title="E-mail the corresponding author} ; Hao Yuan^a ; ^{hyuan@iastate.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：33C20 ; 11G07 ; 11G15 ; 44A20
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：164
• 期：Complete
• 页码：166-178
• 全文大小：364 K

文摘

We study congruences involving truncated hypergeometric series of the form where p   is a prime and 16000445&_mathId=si2.gif&_user=111111111&_pii=S0022314X16000445&_rdoc=1&_issn=0022314X&md5=53aae8e18b8c7e4ea95c3a6257e595b7" title="Click to view the MathML source">m,s$m,s$ are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of K3 surfaces. For special values of λ  , with 16000445&_mathId=si3.gif&_user=111111111&_pii=S0022314X16000445&_rdoc=1&_issn=0022314X&md5=07db47f26ae37d9e902e7cdd0e34e2f4" title="Click to view the MathML source">s=1$s=1$, our congruences are stronger than those predicted by the theory of formal groups, because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Stienstra and Beukers for the 16000445&_mathId=si4.gif&_user=111111111&_pii=S0022314X16000445&_rdoc=1&_issn=0022314X&md5=78d5258c05da441b420edd765ad8d608" title="Click to view the MathML source">λ=1$\lambda =1$ case and confirm some other supercongruence conjectures at special values of λ.