Rank-Favorable Bounds for Rational Points on Superelliptic Curves Open Access

Kantor, Noam (2017)

Permanent URL: https://etd.library.emory.edu/concern/etds/3j333311k?locale=en
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Abstract

Let C be a curve of genus at least two, and let r be the rank of the rational points on its Jacobian. Under mild hypotheses on r, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on C by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on r: when r is small, there should be fewer points on C. In a 2013 paper, Stoll established such a bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results to superelliptic curves. We also discuss a possible strategy for proving a rank-favorable bound for arbitrary curves based on the metrized complexes of Amini and Baker. Our results have stark implications for bounding the number of rational points on a curve, since r is expected to be small for "most" curves.

Table of Contents

1. Background and Main Theorems

1.1 Notation

2. Chabauty's Method and Stoll's Improvement

2.1 The Technique

3. Automorphisms of the Annulus

4. Annuli on Superelliptic Curves

5. Bounding Zeros of Differentials

6. The Final Count(down)

6.1 p-Adic Rolle's Theorem

6.2 Uniform Bounds

7. Discussion

7.1 Other Possible Attacks on Superelliptic Curves

8. Metrized Complexes

9. A New Chabauty Divisor

10. Bounding deg D_{chab}

10.1 Katz and Zureick-Brown's Analysis

10.2 deg D_{chab} on the Metrized Complex

10.3 Applying the Lemma to Chabauty

11. Evidence for the Main Assumption

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