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SUMMARY:Niven Achenjang (Harvard)
DTSTART:20261027T203000Z
DTEND:20261027T213000Z
DTSTAMP:20261023T093552Z
UID:MITNT/146
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/MITNT/146/">
 Two Arithmetic Applications of Additive Combinatorics</a>\nby Niven Achenj
 ang (Harvard) as part of MIT number theory seminar\n\nLecture held in Room
  2-449 in the Simons Building (building 2).\n\nAbstract\nDirichlet's theor
 em on primes in arithmetic progressions is a key input into the proof of t
 he local-global principle for quadratic forms over $\\mathbb Q$. It also m
 akes an appearance in Corollary X.6.2.1 of Silverman's "The Arithmetic of 
 Elliptic Curves" where it is used to prove that there are infinitely many 
 elliptic curves $E/\\mathbb Q$ of rank zero. Dirichlet's theorem can be un
 derstood as a characterization of which linear polynomials take on prime v
 alues infinitely often. More generally\, the multivariate Bateman-Horn con
 jecture makes precise predictions for how often systems of polynomials tak
 e on simultaneously prime values.\n\nIn ongoing joint work with Katy Woo\,
  we prove some new cases of the multivariate Bateman-Horn conjecture (util
 izing techniques of recent work of Green and Sawhney). We then apply our p
 rime producing theorem towards the study of (Brauer-Manin obstructions to)
  local-global principles on certain conic bundles and towards a constructi
 on of rank (exactly) 2 elliptic curves in quadratic twist families. Both o
 f these applications build on the work of too many people to list. \n\nThi
 s project is in progress and joint with Katy Woo.\n
LOCATION:https://researchseminars.org/talk/MITNT/146/
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