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SUMMARY:Niven Achenjang (Harvard University)
DTSTART:20260922T200000Z
DTEND:20260922T210000Z
DTSTAMP:20261004T233531Z
UID:FCNTS/37
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/FCNTS/37/">T
 wo Arithmetic Applications of Additive Combinatorics</a>\nby Niven Achenja
 ng (Harvard University) as part of Five College Number Theory Seminar\n\nL
 ecture held in Seeley Mudd 207 @Amherst College.\n\nAbstract\nDirichlet's 
 theorem on primes in arithmetic progressions is a key input into the proof
  of the local-global principle for quadratic forms over Q. It also makes a
 n appearance in Corollary X.6.2.1 of Silverman's "The Arithmetic of Ellipt
 ic Curves" where it is used to prove that there are infinitely many ellipt
 ic curves E/Q of rank zero. Dirichlet's theorem can be understood as a cha
 racterization of which linear polynomials take on prime values infinitely 
 often. More generally\, the multivariate Bateman-Horn conjecture makes pre
 cise predictions for how often systems of polynomials take on simultaneous
 ly prime values.\n\nIn joint work with Katy Woo\, we prove some new cases 
 of the multivariate Bateman-Horn conjecture (utilizing techniques of recen
 t work of Green and Sawhney). We then apply our prime producing theorem to
 wards the study of (Brauer-Manin obstructions to) local-global principles 
 on certain conic bundles and towards a construction of rank 2 elliptic cur
 ves in quadratic twist families. Both of these applications build on the w
 ork of too many people to list. \n\nThis project is in progress and joint 
 with Katy Woo.\n
LOCATION:https://researchseminars.org/talk/FCNTS/37/
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