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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
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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