High Rank Quadratic Twists of Elliptic Curves

Series: 
Algebra Seminar
Monday, March 24, 2014 - 15:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Department of Defense
Organizer: 
A notorious open problem in arithmetic geometry asks whether ranks ofelliptic curves are unbounded in families of quadratic twists.  A proof ineither direction seems well beyond the reach of current techniques, butcomputation can provide evidence one way or the other.  In this talk wedescribe two approaches for searching for high rank twists: the squarefreesieve, due to Gouvea and Mazur, and recursion on the prime factorization ofthe twist parameter, which uses 2-descents to trim the search tree.  Recentadvances in techniques for Selmer group computations have enabled analysisof a much larger search region; a large computation combining these ideas,conducted by Mark Watkins, has uncovered many new rank 7 twists of$X_0(32): y^2 = x^3 - x$, but no rank 8 examples.  We'll also describe aheuristic argument due to Andrew Granville that an elliptic curve hasfinitely many (and typically zero) quadratic twists of rank at least 8.