# Mock Heegner points and Sylvester's conjecture

## Algebra and Number Theory Seminar

## Meeting Details

For more information about this meeting, contact Kendra Stauffer, Robert Vaughan, Mihran Papikian, Ae Ja Yee, Kirsten Eisentraeger, Jack Huizenga.

**Speaker:** John Voight, Dartmouth College

**Abstract:** We consider the classical Diophantine problem of writing positive
integers $n$ as the sum of two rational cubes, i.e.\ $n=x^3+y^3$ for
$x,y \in {\mathbb Q}$. A conjecture attributed to Sylvester asserts
that a rational prime $p>3$ can be so expressed if $p \equiv 4,7,8
\pmod{9}$. The theory of mock Heegner points gives a method for
exhibiting such a pair $(x,y)$ in certain cases. We prove Sylvester's
conjecture in the case that $p \equiv 4,7 \pmod{9}$ and $3$ is not a
cube modulo $p$. This is joint work with Samit Dasgupta.

## Room Reservation Information

**Room Number:** 116 Osmond

**Date:** 02/01/2018

**Time:** 11:00am - 11:50am