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URL:https://uwaterloo.ca/pure-mathematics/events/number-theory-seminar-105
SUMMARY:Number Theory Seminar
CLASS:PUBLIC
DESCRIPTION:FÉLIX BARIL BOUDREAU\, UNIVERSITY OF LETHBRIDGE\n\n\"ARITHMETI
 C RANK BOUNDS FOR ABELIAN VARIETIES OVER FUNCTION FIELDS\"\n\nIt is known 
 since the works of Ogg (1962) and Shafarevich (1961)\n(under tameness assu
 mptions)\, followed by Grothendieck (1968)\, that\nthe rank of a given abe
 lian variety over the function field of a curve\nis bounded by a quantity 
 which depends on the genus of the base curve\nand on reduction data. This 
 bound is \"geometric\" in nature. In\nparticular\, it holds if we replace 
 the constant field by its algebraic\nclosure.\n\nUlmer asked in 2004 if\, 
 for an elliptic curve\, there was an arithmetic\nbound that could improve 
 on the geometric one. This question recently\nobtained a positive answer (
 Gillibert and Levin\, 2022).\n\nIn this talk\, we present a similar arithm
 etic refinement of the\ngeometric bound for higher-dimensional abelian var
 ieties. When\nspecialized to elliptic curves\, we improve on Gillibert-Lev
 in's bound.\nTime permitting\, we will discuss some consequences of our re
 sult.\n\nThis is joint work with Jean Gillibert and Aaron Levin.\n\nMC 550
 1
DTSTAMP:20260506T091634Z
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