Three projects in arithmetic geometry: torsion points and curves of low genus
Date
2019
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Abstract
This paper is an exposition of three different projects in arithmetic geometry. All of them consider problems related to smooth curves with low genus and the torsion points of their Jacobians. The first project studies curves over finite fields and two invariants of the p-torsion part of their Jacobians: the a-number (a) and p-rank (f). There are many open questions in the literature about the existence of curves with a certain genus g and given values of a and f. In particular, not much is known when g = 4 and the curve is non-hyperelliptic. This is the case that we focus on here; we collect and analyze statistical data of curves over Fp for p = 3, 5, 7, 11 and their invariants. Then, we study the existence of Cartier points, which are also related to the structure of J[p]. For curves with 0 ≤ a < g, the number of Cartier points is bounded, and it depends on a and f. The second project addresses the problem of computing the endomorphism ring of a supersingular elliptic curve. This question has gained recent interest as the basis of alternative cryptosystems that hope to be resistant to quantum attacks. Our strategy is to generate these endomorphism rings by finding cycles in the l-isogeny graph which correspond to generators of the ring. We were able to find a condition for cycles to be linearly independent and an obstruction for two of them to be generators. Finally, the last chapter considers the Galois representations associated to the n-torsion points of elliptic curves over Q. In concrete, we construct models for the modular curves associated to applicable subgroups of GL₂(Z/nZ) and find the rational points on all of those which result in genus 0 or 1 curves, or prove that they have infinitely many. We also analyze the curves with a hyperelliptic genus 2 model and provably find the rational points on all but seven of them.
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Cartier point
non-hyperelliptic curve
torsion points
elliptic curve
a-number
p-rank