[Liste-CICMA] Seminar tomorrow
Farzad Aryan
aryan at CRM.Umontreal.ca
Wed Aug 10 11:17:12 EDT 2016
Dear all,
Tomorrow at 11:00 am in room 5448 André Aisenstadt, Chantal David
(Concordia University) will give a talk.
Title: One-parameter families of elliptic curves with non-zero average
root number.
Abstract: We investigate in this talk the average root number (i.e. sign
of the functional equation) of one-parameter families of elliptic curves
(i.e elliptic curves over Q(t), or elliptic surfaces over Q). For most
one-parameter families of elliptic curves, the average root number is
predicted to be 0. Helfgott showed that under Chowla’s conjecture and the
square-free conjecture, the average root number is 0 unless the curve has
no place of multiplicative reduction over Q(t). We then build families of
elliptic curves with no place of multiplicative reduction, and compute the
average root number of the families. Some families have periodic root
number, giving a rational average, and some other families have an average
root number which is expressed as an infinite Euler product. We also show
several density results for the average root number of families of
elliptic curves, and exhibit some surprising examples, for example,
non-isotrivial families of elliptic curves with rank r over Q(t) and
average root number −(−1)^r , which were not found in previous
literature.
See you there,
Farzad
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