Explicit division and torsion points on superelliptic Curves and jacobians

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 [68] 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 [15] and Grant-Shaulis [29] 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 [57] 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 [63].

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2020

Author dc:creator

• Arul, Vishal.

Advisor dc:contributor.advisor

• Bjorn Mikhail Poonen.

• Mathematics.

Statement dc:rights

• MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/127911

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/127911