Hasse Principle Research Articles

ARBITRARILY LARGE p-TORSION IN TATE-SHAFAREVICH GROUPS

Abstract We show that, for any prime p, there exist absolutely simple abelian varieties over $\mathbb {Q}$ with arbitrarily large p-torsion in their Tate-Shafarevich groups. To prove this, we construct explicit $\mu _p$ -covers of Jacobians of curves of the form $y^p = x(x-1)(x-a)$ which violate the Hasse principle. In the appendix, Tom Fisher explains how to interpret our proof in terms of a Cassels-Tate pairing.

A local-global principle for similarities over function fields of p-adic curves

Let p∈N be a prime with p≠2, and let K be a p-adic field. Let F be the function field of a curve over K. Let ΩF be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups.

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

We study an infinite family of j-invariant zero elliptic curves ED:y2=x3+16D and their λ-isogenous curves ED′:y2=x3−27⋅16D, where D and D′=−3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field KD=Q(D) always has non-trivial 3-class group. We prove a series of results related to the set of rational points ED′(Q)∖λ(ED(Q)), and the SL(2,Z)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of ED and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

An analytic viewpoint on the Hasse principle

An analytic viewpoint on the Hasse principle

Locally conjugate Galois sections

Abstract We consider sections of the étale homotopy exact sequence of a hyperbolic curve over a number field. We prove that two sections whose restrictions to decomposition groups are conjugate on a set of valuations of density one are globally conjugate, which establishes the local-global principle for the conjugacy classes of sections. In fact, we obtain this result as a corollary of a more general property concerning sections of the étale homotopy exact sequence, the finite covering property, which we prove as our main result.

Nonabelian descent types

We present the notion of nonabelian descent type, which classifies torsors up to twisting by a Galois cocycle. This relies on the previous construction of kernels and nonabelian Galois 2-cohomology due to Springer and Borovoi. The necessity of descent types arises in the context of the descent theory where no torsors are given a priori, for example, when we wish to study the arithmetic properties such as the Brauer–Manin obstruction to the Hasse principle on homogeneous spaces without rational points. This new definition also unifies the types by Colliot-Thélène–Sansuc, the extended types by Harari–Skorobogatov, and the finite descent types by Harpaz–Wittenberg.

The local-global principle for the artinianness dimensions

Let R be a commutative noetherian ring and a an ideal of R. The goal of this paper is to establish the local-global principle for the artinianness dimension r a ( M ) , where r a ( M ) is the smallest integer such that the local homology module of M is not artinian. For an artinian R-module M with the set Coass R H r a ( M ) a ( M ) finite, we show that r a ( M ) = inf { r a R p ( Hom R ( R p , M ) ) | p ∈ Spec R } . And the class of all modules N such that Coass R N is finite is studied.

Class field theory, Hasse principles and Picard–Brauer duality for two-dimensional local rings

Class field theory, Hasse principles and Picard–Brauer duality for two-dimensional local rings

Brauer–Manin obstruction for Wehler K3 surfaces of Markoff type

Following a recent work of Fuchs, Litman, Silverman, and Tran, we study the Brauer–Manin obstruction for integral points on Wehler K3 surfaces of Markoff type. In particular, we construct some families which fail the integral Hasse principle via the Brauer–Manin obstruction with some counting results of similar nature to those in recent works of Ghosh and Sarnak; Loughran and Mitankin; and Colliot-Thélène, Wei, and Xu. We also give some counterexamples to strong approximation which can be explained by the Brauer–Manin obstruction, and study a few aspects of rational points on affine surfaces.

Decision Support for Managing Common Musculoskeletal Pain Disorders: Development of a Case-Based Reasoning Application.

Common interventions for musculoskeletal pain disorders either lack evidence to support their use or have small to modest or short-term effects. Given the heterogeneity of patients with musculoskeletal pain disorders, treatment guidelines and systematic reviews have limited transferability to clinical practice. A problem-solving method in artificial intelligence, case-based reasoning (CBR), where new problems are solved based on experiences from past similar problems, might offer guidance in such situations. This study aims to use CBR to build a decision support system for patients with musculoskeletal pain disorders seeking physiotherapy care. This study describes the development of the CBR system SupportPrim PT and demonstrates its ability to identify similar patients. Data from physiotherapy patients in primary care in Norway were collected to build a case base for SupportPrim PT. We used the local-global principle in CBR to identify similar patients. The global similarity measures are attributes used to identify similar patients and consisted of prognostic attributes. They were weighted in terms of prognostic importance and choice of treatment, where the weighting represents the relevance of the different attributes. For the local similarity measures, the degree of similarity within each attribute was based on minimal clinically important differences and expert knowledge. The SupportPrim PT's ability to identify similar patients was assessed by comparing the similarity scores of all patients in the case base with the scores on an established screening tool (the short form Örebro Musculoskeletal Pain Screening Questionnaire [ÖMSPQ]) and an outcome measure (the Musculoskeletal Health Questionnaire [MSK-HQ]) used in musculoskeletal pain. We also assessed the same in a more extensive case base. The original case base contained 105 patients with musculoskeletal pain (mean age 46, SD 15 years; 77/105, 73.3% women). The SupportPrim PT consisted of 29 weighted attributes with local similarities. When comparing the similarity scores for all patients in the case base, one at a time, with the ÖMSPQ and MSK-HQ, the most similar patients had a mean absolute difference from the query patient of 9.3 (95% CI 8.0-10.6) points on the ÖMSPQ and a mean absolute difference of 5.6 (95% CI 4.6-6.6) points on the MSK-HQ. For both ÖMSPQ and MSK-HQ, the absolute score difference increased as the rank of most similar patients decreased. Patients retrieved from a more extensive case base (N=486) had a higher mean similarity score and were slightly more similar to the query patients in ÖMSPQ and MSK-HQ compared with the original smaller case base. This study describes the development of a CBR system, SupportPrim PT, for musculoskeletal pain in primary care. The SupportPrim PT identified similar patients according to an established screening tool and an outcome measure for patients with musculoskeletal pain.

A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

Primes of the form $X^{3} + NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

Primes of the form $X^{3} + NY^{3}$ and a family of non-singular plane curves which violate the local-global principle

Reflect-Push Methods Part I: Two Dimensional Techniques

We determine all maximum weight downsets in the product of two chains, where the weight function is a strictly increasing function of the rank. Many discrete isoperimetric problems can be reduced to the maximum weight downset problem. Our results generalize Lindsay's edge-isoperimetric theorem in two dimensions in several directions. They also imply and strengthen (in several directions) a result of Ahlswede and Katona concerning graphs with maximal number of adjacent pairs of edges. We find all optimal shifted graphs in the Ahlswede-Katona problem. Furthermore, the results of Ahlswede-Katona are extended to posets with a rank increasing and rank constant weight function. Our results also strengthen a special case of a recent result by Keough and Radcliffe concerning graphs with the fewest matchings. All of these results are achieved by applications of a key lemma that we call the reflect-push method. This method is geometric and combinatorial. Most of the literature on edge-isoperimetric inequalities focuses on finding a solution, and there are no general methods for finding all possible solutions. Our results give a general approach for finding all compressed solutions for the above edge-isoperimetric problems.
 By using the Ahlswede-Cai local-global principle, one can conclude that lexicographic solutions are optimal for many cases of higher dimensional isoperimetric problems. With this and our two dimensional results we can prove Lindsay's edge-isoperimetric inequality in any dimension. Furthermore, our results show that lexicographic solutions are the unique solutions for which compression techniques can be applied in this general setting.

On the Hasse principle for complete intersections

We prove the Hasse principle for a smooth projective variety $X\subset \mathbb {P}^{n-1}_\mathbb {Q}$ defined by a system of two cubic forms $F,G$ as long as $n\geq 39$. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defined over $\mathbb {Q}$.

Most odd-degree binary forms fail to primitively represent a square

Let $F$ be a separable integral binary form of odd degree $N \geq 5$ . A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree- $N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree- $N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$ , ordered by height. For every sufficiently large $N$ , we prove that among equations in the family $\mathscr {F}_N(f_0)$ , more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$ , respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points.

On the local-global principle for isogenies of abelian surfaces

Let ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} be a prime number. We classify the subgroups G of Sp4(Fℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Sp}}_4({\\mathbb {F}}_\\ell )$$\\end{document} and GSp4(Fℓ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {GSp}}_4({\\mathbb {F}}_\\ell )$$\\end{document} that act irreducibly on Fℓ4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_\\ell ^4$$\\end{document}, but such that every element of G fixes an Fℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_\\ell $$\\end{document}-vector subspace of dimension 1. We use this classification to prove that a local-global principle for isogenies of degree ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} between abelian surfaces over number fields holds in many cases—in particular, whenever the abelian surface has non-trivial endomorphisms and ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} is large enough with respect to the field of definition. Finally, we prove that there exist arbitrarily large primes ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document} for which some abelian surface A/Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A/{\\mathbb {Q}}$$\\end{document} fails the local-global principle for isogenies of degree ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}.

There are genus one curves violating Hasse principle over every number field

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.

A quantitative Hasse principle for weighted quartic forms

AbstractWe derive, via the Hardy–Littlewood method, an asymptotic formula for the number of integral zeros of a particular class of weighted quartic forms under the assumption of nonsingular local solubility. Our polynomials satisfy the condition that . Our conclusions improve on those that would follow from a direct application of the methods of Birch. For example, we show that in many circumstances the expected asymptotic formula holds when and .

Weak approximation versus the Hasse principle for subvarieties of abelian varieties

Weak approximation versus the Hasse principle for subvarieties of abelian varieties

On the analytic theory of isotropic ternary quadratic forms

A new local-global result about the primitive representations of zero by integral ternary quadratic forms is proven. By an extension of a result of Kneser (given in the Appendix), it yields a quantitative supplement to the Hasse principle on the number of automorphic orbits of primitive zeros of a genus of forms. One ingredient in its proof is an asymptotic formula for a count of the zeros of a given form in such an orbit.