Abstract
We introduce a common generalization of essentially all known methods for explicit computation of Selmer groups, which are used to bound the ranks of abelian varieties over global fields. We also simplify and extend the proofs relating what is computed to the cohomologically defined Selmer groups. Selmer group computations have been practical for many Jacobians of curves over $\mathbb{Q}$ of genus up to 2 since the 1990s, but our approach is the first to be practical for general curves of genus 3. We show that our approach succeeds on some genus 3 examples defined by polynomials with small coefficients.
Highlights
We demonstrate its practicality by computing the rank of J (Q) for the Jacobians of several nonhyperelliptic genus 3 curves X, some of which have no special property beyond having a small discriminant; at least one of these curves can be handled unconditionally, without assuming the Generalized Riemann hypothesis
If we evaluate on any z ∈ Z0(X ) that is good for both f and f, the value of the homomorphism in L×/L×n is unchanged
If we evaluate on any z ∈ Z0(X ) that is good for both f and f, the value of the homomorphism in L×/L×nk× is unchanged
Summary
The main goal of this article is to develop a practical generalization of true and fake descent that contains essentially all previous instantiations of explicit descent. We demonstrate its practicality by computing the rank of J (Q) for the Jacobians of several nonhyperelliptic genus 3 curves X , some of which have no special property beyond having a small discriminant; at least one of these curves can be handled unconditionally, without assuming the Generalized Riemann hypothesis (see Section 12.9). This is the first time that Selmer group computations for ‘general’ genus 3 Jacobians have been possible.
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