Titlebook: Arithmetic Geometry, Number Theory, and Computation; Jennifer S. Balakrishnan,Noam Elkies,John Voight Conference proceedings 2021 The Edit

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Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves, Chabauty–Coleman method to find the zero set of a certain system of .-adic integrals, which is known to be finite and include the set of rational points .. We implemented an algorithm in Sage to carry out the Chabauty–Coleman method on a database of 5870 curves.

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Curves with Sharp Chabauty-Coleman Bound,al points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman’s bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Cole

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Linear Dependence Among Hecke Eigenvalues,n cuspidal eigenform. Our motivation lies in its algorithmic application. For any fixed positive integer ., the bound established here yields an algorithm that computes cuspidal Hecke eigenforms with a given weight . whose Hecke eigenvalues generate a number field of degree .. The resulting algorith

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Curves with Sharp Chabauty-Coleman Bound,al points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman’s bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman’s theorem.

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Linear Dependence Among Hecke Eigenvalues,n cuspidal eigenform. Our motivation lies in its algorithmic application. For any fixed positive integer ., the bound established here yields an algorithm that computes cuspidal Hecke eigenforms with a given weight . whose Hecke eigenvalues generate a number field of degree .. The resulting algorithm reduces to Cremona’s when . = 1 and . = 2.