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SUMMARY:Equidistribution and reciprocity in number theory  - Jack Thorne (
 Cambridge) 
DTSTART:20240118T160000Z
DTEND:20240118T170000Z
UID:TALK210031@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:A famous result in number theory is Dirichlet’s theorem that
  there exist infinitely many prime numbers in any given arithmetic progres
 sion a\, a + N\, a + 2 N\, … where a\, N are coprime. In fact\, a strong
 er statement holds: the primes are equidistributed in the different residu
 e classes modulo N. In order to prove his theorem\, Dirichlet introduced D
 irichlet L-functions\, which are analogues of the Riemann zeta function wh
 ich depend on a choice of character of the group of units modulo N. More g
 eneral L-functions appear throughout number theory and are closely connect
 ed with equidistribution questions\, such as the Sato—Tate conjecture (c
 oncerning the number of solutions to y2\n = x3 + a x + b in the finite fie
 ld with p elements\, as the prime p varies). L-functions also play a centr
 al role in both the motivation for and the formulation of the Langlands co
 njectures in the theory of the automorphic forms. I will give a gentle int
 roduction to some of these ideas and discuss some recent theorems in the a
 rea.\n\nA wine reception in the central core will follow the lecture
LOCATION:CMS MR2
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