Number Theory Seminar: Computing Galois images of Picard curves

SEMINARS, NUMBER THEORY

Speaker:

Shiva Chidambaram - MIT

Let C be a genus 3 curve whose Jacobian is geometrically simple and has geometric endomorphism algebra equal to an imaginary quadratic field. In particular, consider Picard curves y^3 = f_4(x) where the geometric endomorphism algebra is Q (zeta3). We study the associated mod-l Galois representations and their images. I will discuss an algorithm, developed in ongoing joint work with Pip Goodman, to compute the set of primes mod-l for which the images are not maximal. By running it on several datasets of Picard curves, the largest non-maximal prime we obtain is 13. This may be compared with genus 1, where Serre's uniformity question asks if the mod-l Galois image of non-CM elliptic curves over Q is maximal for all primes l > 37

For more info, see https://ashvin-swaminathan.github.io/home/NTSeminar.html