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CATEGORIES:DPMMS Departmental Colloquia
SUMMARY:Equidistribution and reciprocity in number theory
- Jack Thorne (Cambridge)
DTSTART;TZID=Europe/London:20240118T160000
DTEND;TZID=Europe/London:20240118T170000
UID:TALK210031AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/210031
DESCRIPTION:A famous result in number theory is Dirichlet’s th
eorem that there exist infinitely many prime numbe
rs in any given arithmetic progression a\, a + N\,
a + 2 N\, … where a\, N are coprime. In fact\, a
stronger statement holds: the primes are equidistr
ibuted in the different residue classes modulo N.
In order to prove his theorem\, Dirichlet introduc
ed Dirichlet L-functions\, which are analogues of
the Riemann zeta function which depend on a choice
of character of the group of units modulo N. More
general L-functions appear throughout number theo
ry and are closely connected with equidistribution
questions\, such as the Sato—Tate conjecture (con
cerning the number of solutions to y2\n = x3 + a x
+ b in the finite field with p elements\, as the
prime p varies). L-functions also play a central r
ole in both the motivation for and the formulation
of the Langlands conjectures in the theory of the
automorphic forms. I will give a gentle introduct
ion to some of these ideas and discuss some recent
theorems in the area.\n\nA wine reception in the
central core will follow the lecture
LOCATION:CMS MR2
CONTACT:HoD Secretary\, DPMMS
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