2026-09-20T23:05:09Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1279112026-06-17T14:45:48Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Bjorn Mikhail Poonen.
Arul, Vishal.
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2020-10-08T21:30:01Z
2020-10-08T21:30:01Z
2020
2020
https://hdl.handle.net/1721.1/127911
1197636351
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020
Cataloged from the official PDF of thesis.
Includes bibliographical references (pages 113-116).
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].
by Vishal Arul.
Ph.D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
116 pages
eng
Massachusetts Institute of Technology
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.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Explicit division and torsion points on superelliptic Curves and jacobians
Thesis
application/pdf
2020-10-08T21:30:00Z
Publication
Doctoral
Math
unknown
unknown
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<Abstract>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].</Abstract>
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