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\n Let\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be an elliptic curve over\n \n \n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with discriminant\n \n \n \n \n \n \u0394\n \n <\/mml:mi>\n E<\/mml:mi>\n <\/mml:msub>\n \\Delta _E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . For primes\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of good reduction, let\n \n \n \n \n N<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n N_p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be the number of points modulo\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and write\n \n \n \n \n \n N<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n p<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n a<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n N_p=p+1-a_p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In 1965, Birch and Swinnerton-Dyer formulated a conjecture which implies\n \n \n \n \n \n lim<\/mml:mo>\n \n x<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:munder>\n \n 1<\/mml:mn>\n \n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n x<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mfrac>\n \n \n \u2211\n \n <\/mml:mo>\n \n \n \n \n \n p<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n x<\/mml:mi>\n <\/mml:mtd>\n <\/mml:mtr>\n \n \n p<\/mml:mi>\n \n \u2224\n \n <\/mml:mo>\n \n \n \u0394\n \n <\/mml:mi>\n \n E<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mstyle>\n <\/mml:mrow>\n <\/mml:munder>\n \n \n \n a<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n log<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n p<\/mml:mi>\n <\/mml:mrow>\n p<\/mml:mi>\n <\/mml:mfrac>\n =<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n r<\/mml:mi>\n +<\/mml:mo>\n \n 1<\/mml:mn>\n 2<\/mml:mn>\n <\/mml:mfrac>\n ,<\/mml:mo>\n <\/mml:mrow>\n \\begin{equation*} \\lim _{x\\to \\infty }\\frac {1}{\\log x}\\sum _{\\substack {p\\leq x\\\\ p\\nmid \\Delta _{E}}}\\frac {a_p\\log p}{p}=-r+\\frac {1}{2}, \\end{equation*}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/disp-formula>\n where\n \n \n \n r<\/mml:mi>\n r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the order of the zero of the\n \n \n \n L<\/mml:mi>\n L<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -function\n \n \n \n \n \n L<\/mml:mi>\n \n E<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n L_{E}(s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n at\n \n \n \n \n s<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n s=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , which is predicted to be the Mordell-Weil rank of\n \n \n \n \n E<\/mml:mi>\n (<\/mml:mo>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n E(\\mathbb {Q})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We show that if the above limit exits, then the limit equals\n \n \n \n \n \n \u2212\n \n <\/mml:mo>\n r<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n -r+1\/2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We also relate this to Nagao\u2019s conjecture. This paper also includes an appendix by Andrew V. Sutherland which gives evidence for the convergence of the above-mentioned limit.\n <\/p>","DOI":"10.1090\/mcom\/3773","type":"journal-article","created":{"date-parts":[[2022,9,12]],"date-time":"2022-09-12T14:14:09Z","timestamp":1662992049000},"page":"385-408","source":"Crossref","is-referenced-by-count":6,"title":["From the Birch and Swinnerton-Dyer conjecture to Nagao\u2019s conjecture"],"prefix":"10.1090","volume":"92","author":[{"given":"Seoyoung","family":"Kim","sequence":"first","affiliation":[]},{"given":"M.","family":"Murty","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2022,9,12]]},"reference":[{"issue":"4","key":"1","doi-asserted-by":"publisher","first-page":"843","DOI":"10.1090\/S0894-0347-01-00370-8","article-title":"On the modularity of elliptic curves over \ud835\udc10: wild 3-adic exercises","volume":"14","author":"Breuil, Christophe","year":"2001","journal-title":"J. Amer. Math. 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