Explicit division and torsion points on superelliptic Curves and jacobians
Massachusetts Institute of Technology. Department of Mathematics.
Bjorn Mikhail Poonen.
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In this thesis, I study two problems in the arithmetic of superelliptic curves. By a superelliptic curve, I mean the smooth projective model of the affine plane curve y[superscript n] = f(x) where f(x) is separable, n is coprime to deg(f), and the characteristic of the ground field does not divide n. When n = 2, this is commonly referred to as a hyperelliptic curve. I first generalize Zarhin's formula for division by 2  on hyperelliptic curves to the superelliptic case. Rather than divide by n, I invert the 1[zeta] endomorphism on the jacobian. My formula reduces to Zarhin's when n = 2. Next, I study torsion points on superelliptic curves. Work of Coleman  and Grant-Shaulis  together classifies all torsion points on the hyperelliptic curve y² = x[superscript d] + 1, where d >/= 5 is prime. I extend their results to the superelliptic curve y[superscript n] = x[superscript d] + 1, where n, d >/= 2 are coprime. Using a specialization argument, I also classify torsion points on a generic superelliptic curve, extending Theorem 7.1 of Poonen-Stoll  to the hyperelliptic case. In order to classify torsion points, I prove a result about Galois action on the p-torsion of the jacobian of y[superscript p] = x[superscript q]+1, where p and q are distinct primes. This problem is equivalent to a new p-adic congruence for Jacobi sums, which I state and prove. This congruence is related to (but does not follow from) a congruence of Uehara .
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020Cataloged from the official PDF of thesis.Includes bibliographical references (pages 113-116).
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology