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SUMMARY:Lance Miller (University of Arkansas)
DTSTART:20210415T210000Z
DTEND:20210415T220000Z
DTSTAMP:20260423T024647Z
UID:UCSD_NTS/33
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UCSD_NTS/33/
 ">Finiteness of quasi-canonical lifts of elliptic curves</a>\nby Lance Mil
 ler (University of Arkansas) as part of UCSD number theory seminar\n\nLect
 ure held in normally APM 7321\, currently online.\n\nAbstract\nFix a prime
  integer $p$. Set $R$ the completed valuation ring of the maximal unramifi
 ed extension of $\\mathbb{Q}_p$. For  $X := X_1(N)$ the modular curve with
  $N$ at least 4 and coprime to $p$\, Buium-Poonen in 2009 showed that the 
 locus of canonical lifts enjoys finite intersection with preimages of fini
 te rank subgroups of $E(R)$ when $E$ is an elliptic curve with a surjectio
 n from $X$. This is done using Buium's theory of arithmetic ODEs\, in part
 icular interesting homomorphisms $E(R) \\to R$ which are arithmetic analog
 ues of Manin maps. \n\nIn this talk\, I will review the general idea behin
 d this result and other applications of arithmetic jet spaces to Diophanti
 ne geometry and discuss a recent analogous result for quasi-canonical lift
 s\, i.e.\, those curves with Serre-Tate parameter a root of unity. Here th
 e ODE Manin maps are insufficient\, so we introduce a PDE version of Buium
 's theory to provide the needed maps. All of this is joint work with A. Bu
 ium.\n\npre-talk at 1:30\n
LOCATION:https://researchseminars.org/talk/UCSD_NTS/33/
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