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SUMMARY:Dino Lorenzini
DTSTART:20211108T183000Z
DTEND:20211108T193000Z
DTSTAMP:20260423T035711Z
UID:UNYONTC/21
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UNYONTC/21/"
 >Torsion and Tamagawa numbers</a>\nby Dino Lorenzini as part of Upstate Ne
 w York Online Number Theory Colloquium\n\n\nAbstract\nAssociated with an a
 belian variety $A/K$ over a number field $K$ is a finite\nset of integers 
 greater than $1$ called the local Tamagawa numbers of $A/K$. Assuming\ntha
 t the abelian variety $A/K$ has a $K$-rational torsion point of prime orde
 r $N$\, we can\nask whether it is possible for none of the local Tamagawa 
 numbers to be divisible by\n$N$. The ratio $\\textrm{(product of the Tamag
 awa numbers)}/ |\\textrm{Torsion in }E(K) |$ appears in the\nconjectural l
 eading term of the L-function of $A$ in the Birch and Swinnerton-Dyer\ncon
 jecture\, and we are thus interested in understanding whether there are of
 ten\ncancellation in this ratio.\n\nWe will present some finiteness result
 s on this question in the case of elliptic\ncurves. More precisely\, let $
 d > 0$ be an integer\, and assume that there exist\ninfinitely many fields
  $K/\\mathbb{Q}$ of degree $d$ with an elliptic curve $E/K$ having a $K$-r
 ational\npoint of order $N$. We will show that for certain such pairs $(d\
 ,N)$\, there are only\nfinitely many fields $K/\\mathbb{Q}$ of degree $d$ 
 such that there exists an elliptic curve $E/K$\nhaving a $K$-rational poin
 t of order $N$ and none of the local Tamagawa numbers are\ndivisible by $N
 $. The lists of known exceptions are surprisingly small when $d$ is at\nmo
 st $7$.\n
LOCATION:https://researchseminars.org/talk/UNYONTC/21/
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