scholarly journals An effective Chabauty–Kim theorem

2019 ◽  
Vol 155 (6) ◽  
pp. 1057-1075 ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Netan Dogra
Keyword(s):  
Rational Points ◽  
Higher Rank ◽  
Mordell Conjecture

The 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.

scholarly journals An Explicit Manin-Dem’janenko Theorem in Elliptic Curves

2018 ◽  
Vol 70 (5) ◽  
pp. 1173-1200 ◽  
Author(s):  
Evelina Viada
Keyword(s):  
Elliptic Curves ◽  
Rational Points ◽  
Sharp Bound ◽  
Explicit Bound ◽  
Minimal Dimension ◽  
Mordell Conjecture

AbstractLet be a curve of genus at least 2 embedded in E1 × … × EN, where the Ei are elliptic curves for i = 1, . . . , N. In this article we give an explicit sharp bound for the Néron–Tate height of the points of contained in the union of all algebraic subgroups of dimension < max(), where is the minimal dimension of a translate (resp. of a torsion variety) containing .As a corollary, we give an explicit bound for the height of the rational points of special curves, proving new cases of the explicit Mordell Conjecture and in particular making explicit (and slightly more general in the CM case) the Manin–Dem’janenko method for curves in products of elliptic curves.

scholarly journals THE EXPLICIT MORDELL CONJECTURE FOR FAMILIES OF CURVES

2019 ◽  
Vol 7 ◽  
Author(s):  
SARA CHECCOLI ◽  
FRANCESCO VENEZIANO ◽  
EVELINA VIADA
Keyword(s):  
Classical Method ◽  
Rational Points ◽  
Computer Search ◽  
General Shape ◽  
Sharp Estimates ◽  
Families Of Curves ◽  
Large Families ◽  
Mordell Conjecture

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin–Demjanenko and the analysis of our explicit examples is carried to conclusion.

scholarly journals Uniform Bound for the Number of Rational Points on a Pencil of Curves

2019 ◽  
Author(s):  
Vesselin Dimitrov ◽  
Ziyang Gao ◽  
Philipp Habegger
Keyword(s):  
Number Field ◽  
Rational Points ◽  
Parameter Family ◽  
Uniform Bound ◽  
Torsion Points ◽  
Uniform Bounds ◽  
The Family ◽  
Projective Curves ◽  
Mordell Conjecture

Abstract Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber’s Jacobian. Our proof uses Vojta’s approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.

scholarly journals Higher rank Clifford indices of curves on a K3 surface

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Soheyla Feyzbakhsh ◽  
Chunyi Li
Keyword(s):  
Vector Bundles ◽  
Plane Curve ◽  
Smooth Curve ◽  
Line Bundles ◽  
K3 Surface ◽  
Clifford Index ◽  
Smooth Plane ◽  
Higher Rank ◽  
Rank Vector ◽  
Semistable Vector Bundle

AbstractLet (X, H) be a polarized K3 surface with $$\mathrm {Pic}(X) = \mathbb {Z}H$$ Pic ( X ) = Z H , and let $$C\in |H|$$ C ∈ | H | be a smooth curve of genus g. We give an upper bound on the dimension of global sections of a semistable vector bundle on C. This allows us to compute the higher rank Clifford indices of C with high genus. In particular, when $$g\ge r^2\ge 4$$ g ≥ r 2 ≥ 4 , the rank r Clifford index of C can be computed by the restriction of Lazarsfeld–Mukai bundles on X corresponding to line bundles on the curve C. This is a generalization of the result by Green and Lazarsfeld for curves on K3 surfaces to higher rank vector bundles. We also apply the same method to the projective plane and show that the rank r Clifford index of a degree $$d(\ge 5)$$ d ( ≥ 5 ) smooth plane curve is $$d-4$$ d - 4 , which is the same as the Clifford index of the curve.

scholarly journals Processing of novel food reveals payoff and rank-biased social learning in a wild primate

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Charlotte Canteloup ◽  
Mabia B. Cera ◽  
Brendan J. Barrett ◽  
Erica van de Waal
Keyword(s):  
Social Learning ◽  
Learning Strategies ◽  
Individual Learning ◽  
Diffusion Experiment ◽  
Vervet Monkeys ◽  
Dynamic Learning ◽  
Novel Food ◽  
Bias Attention ◽  
Higher Rank ◽  
Learning From Others

AbstractSocial learning—learning from others—is the basis for behavioural traditions. Different social learning strategies (SLS), where individuals biasedly learn behaviours based on their content or who demonstrates them, may increase an individual’s fitness and generate behavioural traditions. While SLS have been mostly studied in isolation, their interaction and the interplay between individual and social learning is less understood. We performed a field-based open diffusion experiment in a wild primate. We provided two groups of vervet monkeys with a novel food, unshelled peanuts, and documented how three different peanut opening techniques spread within the groups. We analysed data using hierarchical Bayesian dynamic learning models that explore the integration of multiple SLS with individual learning. We (1) report evidence of social learning compared to strictly individual learning, (2) show that vervets preferentially socially learn the technique that yields the highest observed payoff and (3) also bias attention toward individuals of higher rank. This shows that behavioural preferences can arise when individuals integrate social information about the efficiency of a behaviour alongside cues related to the rank of a demonstrator. When these preferences converge to the same behaviour in a group, they may result in stable behavioural traditions.

scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

scholarly journals Higher rank FZZ-dualities

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Thomas Creutzig ◽  
Yasuaki Hikida
Keyword(s):  
Field Theory ◽  
Reduction Method ◽  
Operator Algebras ◽  
The Self ◽  
Liouville Theory ◽  
Conformal Field ◽  
Self Duality ◽  
Higher Rank ◽  
Corner Vertex ◽  
Liouville Field Theory

Abstract We examine strong/weak dualities in two dimensional conformal field theories by generalizing the Fateev-Zamolodchikov-Zamolodchikov (FZZ-)duality between Witten’s cigar model described by the $$ \mathfrak{sl}(2)/\mathfrak{u}(1) $$ sl 2 / u 1 coset and sine-Liouville theory. In a previous work, a proof of the FZZ-duality was provided by applying the reduction method from $$ \mathfrak{sl}(2) $$ sl 2 Wess-Zumino-Novikov-Witten model to Liouville field theory and the self-duality of Liouville field theory. In this paper, we work with the coset model of the type $$ \mathfrak{sl}\left(N+1\right)/\left(\mathfrak{sl}(N)\times \mathfrak{u}(1)\right) $$ sl N + 1 / sl N × u 1 and investigate the equivalence to a theory with an $$ \mathfrak{sl}\left(N+\left.1\right|N\right) $$ sl N + 1 N structure. We derive the duality explicitly for N = 2, 3 by applying recent works on the reduction method extended for $$ \mathfrak{sl}(N) $$ sl N and the self-duality of Toda field theory. Our results can be regarded as a conformal field theoretic derivation of the duality of the Gaiotto-Rapčák corner vertex operator algebras Y0,N,N+1[ψ] and YN,0,N+1[ψ−1].

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