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SUMMARY:Seoyoung Kim (Queen's University)
DTSTART:20200806T173000Z
DTEND:20200806T183000Z
DTSTAMP:20260423T004758Z
UID:MAGIC/13
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MAGIC/13/">F
 rom the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture</a>\nby
  Seoyoung Kim (Queen's University) as part of MAGIC (Michigan - Arithmetic
  Geometry Initiative - Columbia)\n\n\nAbstract\nLet $E$ be an elliptic cur
 ve over $\\mathbb{Q}$ with discriminant $\\Delta_E$. For primes $p$ of goo
 d reduction\, let $N_p$ be the number of points modulo $p$ and write $N_p=
 p+1-a_p$. In 1965\, Birch and Swinnerton-Dyer formulated a conjecture whic
 h implies\n$$\\lim_{x\\to\\infty}\\frac{1}{\\log x}\\sum_{ {p\\leq x\,  p 
 \\nmid \\Delta_{E}}}\\frac{a_p\\log p}{p}=-r+\\frac{1}{2}\,$$\nwhere $r$ i
 s the order of the zero of the $L$-function $L_{E}(s)$ of $E$ at $s=1$\, w
 hich is predicted to be the Mordell-Weil rank of $E(\\mathbb{Q})$. We show
  that if the above limit exits\, then the limit equals $-r+1/2$. We also r
 elate this to Nagao's conjecture. This is a recent joint work with M. Ram 
 Murty.\n
LOCATION:https://researchseminars.org/talk/MAGIC/13/
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