Arithmetic Geometry and Number Theory RTG Seminar: Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture

Seminar | January 27 | 3:10-5 p.m. | 891 Evans Hall | Note change in location

 Daniel Kriz, MIT

 Department of Mathematics

In this talk, I formulate and prove a new Rubin-type Iwasawa main conjecture for imaginary quadratic fields in which $p$ is inert or ramified, as well as a Perrin-Riou type Heegner point main conjecture for certain supersingular CM elliptic curves. These main conjectures and their proofs are related to $p$-adic L-functions that I have previously constructed, and have applications to two classical problems of arithmetic. First, I prove the 1879 conjecture of Sylvester stating that if $p = 4,7,8 \mod 9$, then $x^3 + y^3 = p$ has a solution with $x,y$ rational numbers. Second, combined with previous Selmer distribution results, I show that $100\%$ of squarefree $d = 5,6,7 \mod 8$ are congruent numbers, thus establishing Goldfeld's conjecture for the family $y^2 = x^3 - d^2x$, and solving the congruent number problem in $100\%$ of cases.