227 research outputs found
Heights and measures on analytic spaces. A survey of recent results, and some remarks
This paper has two goals. The first is to present the construction, due to
the author, of measures on non-archimedean analytic varieties associated to
metrized line bundles and some of its applications. We take this opportunity to
add remarks, examples and mention related results.Comment: 41 pages, final version. To appear in: Motivic Integration and its
Interactions with Model Theory and Non-Archimedean Geometry, edited by Raf
Cluckers, Johannes Nicaise, Julien Seba
On the leading terms of Zeta isomorphisms and p-adic L-functions in non-commutative Iwasawa theory
We discuss the formalism of Iwasawa theory descent in the setting of the
localized K_1-groups of Fukaya and Kato. We then prove interpolation formulas
for the `leading terms' of the global Zeta isomorphisms that are associated to
certain Tate motives and of the p-adic L-functions that are associated to
certain critical motives.Comment: 38 pages; added references and corrected typo
Computational tools for quadratic Chabauty
http://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfhttp://math.bu.edu/people/jbala/2020BalakrishnanMuellerNotes.pdfFirst author draf
p-adic Integration on Hyperelliptic Curves of Bad Reduction: Algorithms and Applications
For curves over the field of p-adic numbers, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally, and Vologodsky integrals with desirable number-theoretic properties. These integrals have the advantage of being insensitive to the reduction type at p, but are known to coincide with Coleman integrals in the case of good reduction. Moreover, there are practical algorithms available to compute Coleman integrals.Berkovich-Coleman and Vologodsky integrals, however, differ in general. In this thesis, we give a formula for passing between them. To do so, we use combinatorial ideas informed by tropical geometry. We also introduce algorithms for computing Berkovich-Coleman and Vologodsky integrals on hyperelliptic curves of bad reduction. By covering such a curve by basic wide open spaces, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on hyperelliptic curves of good reduction. We then convert the Berkovich-Coleman integrals into Vologodsky integrals using our formula.As an application, we provide an algorithm for computing Coleman-Gross p-adic heights on Jacobians of bad reduction hyperelliptic curves, whose definition relies on Vologodsky integration. This algorithm, for instance, can be used in the quadratic Chabauty method to find rational points on hyperelliptic curves of genus at least two
A heuristic for boundedness of ranks of elliptic curves
We present a heuristic that suggests that ranks of elliptic curves over the
rationals are bounded. In fact, it suggests that there are only finitely many
elliptic curves of rank greater than 21. Our heuristic is based on modeling the
ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies
on a theorem counting alternating integer matrices of specified rank. We also
discuss analogues for elliptic curves over other global fields.Comment: 41 pages, typos fixed in torsion table in section
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