15 research outputs found
An effective Chabauty-Kim theorem
Get PDFThe Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.https://arxiv.org/pdf/1803.10102.pdfFirst author draf
Topics in the theory of Selmer varieties
Get PDFThe Selmer varieties of a hyperbolic curve X over ℚ are refinements of the Selmer group arising from replacing the Tate module of the Jacobian with higher quotients of the unipotent étale fundamental group. It is hoped that these refinements carry extra arithmetic information. In particular the nonabelian Chabauty method developed by Kim uses the Selmer variety to give a new method to find the set X(ℚ). This thesis studies certain local and global properties of the Selmer varieties associated to finite dimensional quotients of the unipotent fundamental group of a curve over ℚ. We develop new methods to prove finiteness of the intersection of the Selmer varieties with the set of local points (and hence of the set of rational points) and new methods to implement this explicitly, giving the first examples of explicit nonabelian Chabauty theory for rational points on projective curves
2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II
Get PDFWe give refined methods for proving finiteness of the Chabauty--Coleman--Kim
set , when is a hyperelliptic curve with a rational
Weierstrass point. The main developments are methods for computing Selmer
conditions at and for the mod 2 Bloch--Kato Selmer group
associated to the higher Chow group . As a
result we show that most genus 2 curves in the LMFDB of Mordell--Weil rank 2
with exactly one rational Weierstrass point satsify . We also obtain a field-theoretic description of second descent on
the Jacobian of a hyperelliptic curve (under some conditions).Comment: 29 pages. Comments welcome
p-adic integrals and linearly dependent points on families of curves I
Get PDFWe prove that the set of `low rank' points on sufficiently large fibre powers
of families of curves are not Zariski dense. The recent work of
Dimitrov-Gao-Habegger and K\"uhne (and Yuan) imply the existence of a bound
which is exponential in the rank, and the Zilber-Pink conjecture implies a
bound which is linear in the rank. Our main result is a (slightly weaker)
linear bound for `low ranks'. We also prove analogous results for isotrivial
families (with relaxed conditions on the rank) and for solutions to the
-unit equation, where the bounds are now sub-exponential in the rank. Our
proof involves a notion of the Chabauty-Coleman(-Kim) method in families (or,
in some sense, for simply connected varieties). For Zariski non-density, we use
the recent work of Bl\`azquez-Sanz, Casale, Freitag and Nagloo on Ax-Schanuel
theorems for foliations on principal bundles.Comment: Comments welcom
Unlikely intersections and the Chabauty--Kim method over number fields
Get PDFThe Chabauty–Kim method is a tool for finding the integral or rational points on varieties over number fields via certain transcendental p-adic analytic functions arising from certain Selmer schemes associated to the unipotent fundamental group of the variety. In this paper we establish several foundational results on the Chabauty–Kim method for curves over number fields. The two main ingredients in the proof of these results are an unlikely intersection result for zeroes of iterated integrals, and a careful analysis of the intersection of the Selmer scheme of the original curve with the unipotent Albanese variety of certain Qp-subvarieties of the restriction of scalars of the curve. The main theorem also gives a partial answer to a question of Siksek on Chabauty’s method over number fields, and an explicit counterexample is given to the strong form of Siksek’s question
Quadratic Chabauty and rational points II: generalised height functions on Selmer varieties
Get PDFWe give new instances where Chabauty–Kim sets can be proved to be finite, by developing a notion of “generalised height functions” on Selmer varieties. We also explain how to compute these generalised heights in terms of iterated integrals and give the 1st explicit nonabelian Chabauty result for a curve X/Q whose Jacobian has Mordell–Weil rank larger than its genus
Quadratic Chabauty and rational points, I: p-adic heights
Get PDFWe give the first explicit examples beyond the Chabauty–Coleman method where Kim’s nonabelian Chabauty program determines the set of rational points of a curve defined over Q or a quadratic number field. We accomplish this by studying the role of p-adic heights in explicit non-Abelian Chabauty
Ogg\u27s Torsion conjecture: Fifty years later
Get PDFAndrew Ogg\u27s mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our subject, and his influence has been such a gift to all of us.
Ogg\u27s celebrated Torsion Conjecture -- as it relates to modular curves -- can be paraphrased as saying that rational points (on the modular curves that parametrize torsion points on elliptic curves) exist if and only if there is a good geometric reason for them to exist.
We give a survey of Ogg\u27s Torsion Conjecture and the subsequent developments in our understanding of rational points on modular curves over the last fifty years.This text is an expanded version of a 45-minute lecture that B.M. gave at the IAS, on the occasion of Talks Celebrating the Ogg Professorship in Mathematics - October 13, 2022. Appendix A, by Netan Dogra, gives a complete characterization of all quadratic points on Bring\u27s curv
