Event Detail

Event Type: 
Number Theory Seminar
Tuesday, October 12, 2021 - 10:00 to 10:50
ALS 0006

Speaker Info

University of California, Berkeley

We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over the rationals with fixed degree n, discriminant bounded by X, and Galois closure S_n. For C a fixed curve given by an affine equation y^m = f(x) where m is at least 2 and deg f(x) = d is at least m, we find that for all degrees n divisible by gcd(m, d) and sufficiently large, the number of such fields is asymptotically bounded below by X^{c_n} , where c_n tends to 1/m^2 as n tends to infinity. This bound is determined explicitly by parameterizing x and y by rational functions, counting specializations, and accounting for multiplicity. We then give geometric heuristics suggesting that for n not divisible by gcd(m, d), degree n points may be less abundant than those for which n is divisible by gcd(m, d). Namely, we discuss the obvious geometric sources from which we expect to find points on C and discuss the relationship between these sources and our parametrization. When one a priori has a point on C of degree not divisible by gcd(m, d), we argue that a similar counting argument applies. As a proof of concept we show in the case that C has a rational point that our methods can be extended to bound the number of fields generated by a degree n point of C, regardless of divisibility of n by gcd(m, d). This talk is based on joint work with Christopher Keyes.