LMS Durham Symposium on Computational Number Theory
24 July - 3 August 2000

Fermat Quartics and a Challenge Curve of Serre

Victor FLYNN (Liverpool)

Abstract: A study of rational points on Fermat quartics of the form X^4 + Y^4 = c immediately reveals many values of c which can be dismissed by congruence considerations. Many other values of c can be dismissed if one of two associated elliptic curves has rank 0. What remains are the stubborn values of c which cannot be trivially dismissed: c = 17, 82, 97 and 257 being the only such less than 300. A solution of the case c = 17 (posed as a challenge by Serre) is presented, representing the first success with a nontrivial value of c, and we discuss the extent to which the method might hope to solve other difficult values of c. The talk is based on joint work with Joe Wetherell.

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