Hasse Principles for Higher-Dimensional Fields 3
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Selmer Groups in Degree $\Ell$ Twist Families
$\ell^{\infty}$-Selmer Groups in Degree $\ell$ Twist Families The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Smith, Alexander. 2020. $\ell^{\infty}$-Selmer Groups in Degree $\ell$ Twist Families. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37365902 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA `1-Selmer groups in degree ` twist families A dissertation presented by Alexander Smith to The Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics Harvard University Cambridge, Massachusetts April 2020 © 2020 – Alexander Smith All rights reserved. Dissertation Advisor: Professor Elkies and Professor Kisin Alexander Smith `1-Selmer groups in degree ` twist families Abstract Suppose E is an elliptic curve over Q with no nontrivial rational 2-torsion point. Given p a nonzero integer d, take Ed to be the quadratic twist of E coming from the field Q( d). For every nonnegative integer r, we will determine the natural density of d such that Ed has 2-Selmer rank r. We will also give a generalization of this result to abelian varieties defined over number fields. These results fit into the following general framework: take ` to be a rational prime, take F to be number field, take ζ to be a primitive `th root of unity, and take N to be an `- divisible Z`[ζ]-module with an action of the absolute Galois group GF of F . -
Conics Over Function Fields and the Artin-Tate Conjecture José Felipe
Conics over function fields and the Artin-Tate conjecture Jos´eFelipe Voloch Abstract: We prove that the Hasse principle for conics over function fields is a simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields. Hasse proved that a conic over a global field has a rational point if and only if it has points over all completions of the global field, an instance of the so-called local-global or Hasse principle. The case of the rational numbers is an old result of Legendre, who gave an elementary proof and a similar proof can be given in the case of rational functions over a finite field. Hasse’s proof, on the other hand, is a consequence of a more general result in Class Field Theory, the Hasse norm theorem. One purpose of this paper is to give a new proof of the local-global principle for conics over function fields, as a relatively simple consequence of a provable case of the Artin-Tate conjecture for surfaces over finite fields. This proof might be more complicated overall, once the work on the Artin-Tate conjecture is factored in, but it might be worthwhile recording since it suggests a new approach to local-global principles which might work in other contexts, such as number fields or higher dimensions. We also deduce from the Artin-Tate conjecture, together with a recent result [LLR] on Brauer groups, the Hilbert reciprocity law. We also do a careful study of conic bundles over curves over finite fields which might have independent interest. -
Counterexamples to the Hasse Principle
COUNTEREXAMPLES TO THE HASSE PRINCIPLE W. AITKEN AND F. LEMMERMEYER Abstract. This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexam- ples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate– Shafarevich groups. 1. Introduction In this article we develop counterexamplestotheHasseprincipleusingonlytech- niques from undergraduate number theory and algebra. By keeping the technical prerequisites to a minimum, we hope to provide a path for nonspecialists to this interesting area of number theory. The counterexamples considered here extend the classical counterexample of Lind and Reichardt. As discussed in an appendix, such counterexamples are important in the theory of elliptic curves, and today are interpreted as nontrivial elements in Tate–Shafarevich groups. 2. Background The problem of determining if the Diophantine equation aX2 + bY 2 + cZ2 =0 (1) has nontrivial solutions with values in Z has played a prominent role in the history of number theory. We assume that a, b, and c are nonzero integers and, using a simple argument, we reduce to the case where the product abc is square-free. Lagrange (1768) solved the problem by giving a descent procedure which determines in a finite number of steps whether or not (1) has a nontrivial Z-solution, but Legendre (1788) gave the definitive solution. Legendre proved that the following conditions, known by Euler to be necessary, are sufficient for the existence of a nontrivial Z- solution: (i) a, b,andc do not all have the same sign, and (ii) ab is a square modulo c , ca is a square modulo b ,and bc is a square modulo− a . -
Algebraic Tori — Thirty Years After
ALGEBRAIC TORI | THIRTY YEARS AFTER BORIS KUNYAVSKI˘I To my teacher Valentin Evgenyevich Voskresenski˘ı, with gratitude and admiration This article is an expanded version of my talk given at the Interna- tional Conference \Algebra and Number Theory" dedicated to the 80th anniversary of V. E. Voskresenski˘ı, which was held at the Samara State University in May 2007. The goal is to give an overview of results of V. E. Voskresenski˘ı on arithmetic and birational properties of algebraic tori which culminated in his monograph [Vo77] published 30 years ago. I shall try to put these results and ideas into somehow broader context and also to give a brief digest of the relevant activity related to the period after the English version of the monograph [Vo98] appeared. 1. Rationality and nonrationality problems A classical problem, going back to Pythagorean triples, of describing the set of solutions of a given system of polynomial equations by ratio- nal functions in a certain number of parameters (rationality problem) has been an attraction for many generations. Although a lot of various techniques have been used, one can notice that after all, to establish rationality, one usually has to exhibit some explicit parameterization such as that obtained by stereographic projection in the Pythagoras problem. The situation is drastically different if one wants to estab- lish non-existence of such a parameterization (nonrationality problem): here one usually has to use some known (or even invent some new) bi- rational invariant allowing one to detect nonrationality by comparing its value for the object under consideration with some \standard" one known to be zero; if the computation gives a nonzero value, we are done. -
Cohomological Hasse Principle and Motivic Cohomology for Arithmetic Schemes
COHOMOLOGICAL HASSE PRINCIPLE AND MOTIVIC COHOMOLOGY FOR ARITHMETIC SCHEMES by MORITZ KERZ and SHUJI SAITO ABSTRACT In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme X. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions. CONTENTS Introduction........................................................ 123 1.Homologytheory................................................... 129 2.Log-pairsandconfigurationcomplexes....................................... 135 3.Lefschetzcondition.................................................. 140 4.Pullbackmap(firstconstruction)........................................... 146 5.Pullbackmap(secondconstruction)......................................... 159 6.Maintheorem..................................................... 162 7.Resultwithfinitecoefficients............................................. 166 8.Kato’sconjectures................................................... 169 9.Applicationtocyclemaps............................................... 174 10.Applicationtospecialvaluesofzetafunctions.................................... 176 Acknowledgements..................................................... 178 Appendix A: Galois cohomology with compact support . .............................. 179 References........................................................ -
Counterexamples to the Local-Global Principle for Non-Singular Plane Curves and a Cubic Analogue of Ankeny-Artin-Chowla-Mordell Conjecture
COUNTEREXAMPLES TO THE LOCAL-GLOBAL PRINCIPLE FOR NON-SINGULAR PLANE CURVES AND A CUBIC ANALOGUE OF ANKENY-ARTIN-CHOWLA-MORDELL CONJECTURE YOSHINOSUKE HIRAKAWA AND YOSUKE SHIMIZU Abstract. In this article, we introduce a systematic and uniform construction of non- singular plane curves of odd degrees n ≥ 5 which violate the local-global principle. Our construction works unconditionally for n divisible by p2 for some odd prime number p. Moreover, our construction also works for n divisible by some p ≥ 5 which satisfies 1=3 1=3 a conjecture on p-adic properties of the fundamental units of Q(p ) and Q((2p) ). This conjecture is a natural cubic analogue of the classical Ankeny-Artin-Chowla-Mordell 1=2 conjecture for Q(p ) and easily verified numerically. 1. Introduction In the theory of Diophantine equations, the local-global principle for quadratic forms established by Minkowski and Hasse is one of the major culminations (cf. [31, Theorem 8, Ch. IV]). In contrast, there exist many homogeneous forms of higher degrees which violate the local-global principle (i.e., counterexamples to the local-global principle). For example, Selmer [30] found that a non-singular plane cubic curve defined by (1) 3X3 + 4Y 3 = 5Z3 has rational points over R and Qp for every prime number p but not over Q. From eq. (1), we can easily construct reducible (especially singular) counterexamples of higher degrees. After that, Fujiwara [13] found that a non-singular plane quintic curve defined by (2) (X3 + 5Z3)(X2 + XY + Y 2) = 17Z5 violates the local-global principle. More recently, Cohen [9, Corollary 6.4.11] gave several p p p counterexamples of the form x + by + cz = 0 of degree p = 3; 5; 7; 11 with b; c 2 Z, and Nguyen [23, 24] gave recipes for counterexamples of even degrees and more complicated forms. -
The Hasse Principle and the Brauer-Manin Obstruction for Curves
THE HASSE PRINCIPLE AND THE BRAUER-MANIN OBSTRUCTION FOR CURVES E.V. FLYNN Abstract. We discuss a range of ways, extending existing methods, to demonstrate violations of the Hasse principle on curves. Of particular interest are curves which contain a rational divisor class of degree 1, even though they contain no rational point. For such curves we construct new types of examples of violations of the Hasse principle which are due to the Brauer-Manin obstruction, subject to the conjecture that the Tate-Shafarevich group of the Jacobian is finite. 1. Introduction Let K be a number field and let AK denote the ad`eles of K. Suppose that X is a smooth projective variety over K violating the Hasse principle; that is, X (K) = ∅ even though X (AK ) 6= ∅. Global reciprocity Br applied to the Brauer-Grothendieck group Br(X ) defines a certain subset X (AK ) ⊂ X (AK ) which contains Br the diagonal image of X (K) (see [27], p.101). When X (AK ) = ∅ we say that the violation of the Hasse principle is explained by the Brauer-Manin obstruction. When X is a surface, examples have been constructed (see [27], §8) where X violates the Hasse principle in a way not explained by the Brauer-Manin obstruction. When X is a curve C it is an open question whether the Brauer-Manin obstruction is the only obstruction to the Hasse principle. When X is of genus 1, we have the following result of Manin (see [27], p.114). Lemma 1. Let C be a smooth proper curve of genus 1 defined over K, with Jacobian E. -
Episodes in the History of Modern Algebra (1800–1950)
HISTORY OF MATHEMATICS • VOLUME 32 Episodes in the History of Modern Algebra (1800–1950) Jeremy J. Gray Karen Hunger Parshall Editors American Mathematical Society • London Mathematical Society Episodes in the History of Modern Algebra (1800–1950) https://doi.org/10.1090/hmath/032 HISTORY OF MATHEMATICS v VOLUME 32 Episodes in the History of Modern Algebra (1800–1950) Jeremy J. Gray Karen Hunger Parshall Editors Editorial Board American Mathematical Society London Mathematical Society Joseph W. Dauben Alex D. D. Craik Peter Duren Jeremy J. Gray Karen Parshall, Chair Robin Wilson, Chair MichaelI.Rosen 2000 Mathematics Subject Classification. Primary 01A55, 01A60, 01A70, 01A72, 01A73, 01A74, 01A80. For additional information and updates on this book, visit www.ams.org/bookpages/hmath-32 Library of Congress Cataloging-in-Publication Data Episodes in the history of modern algebra (1800–1950) / Jeremy J. Gray and Karen Hunger Parshall, editors. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-8218-4343-7 (alk. paper) ISBN-10: 0-8218-4343-5 (alk. paper) 1. Algebra—History. I. Gray, Jeremy, 1947– II. Parshall, Karen Hunger, 1955– QA151.E65 2007 512.009—dc22 2007060683 AMS softcover ISBN: 978-0-8218-6904-8 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. -
L-Functions and Non-Abelian Class Field Theory, from Artin to Langlands
L-functions and non-abelian class field theory, from Artin to Langlands James W. Cogdell∗ Introduction Emil Artin spent the first 15 years of his career in Hamburg. Andr´eWeil charac- terized this period of Artin's career as a \love affair with the zeta function" [77]. Claude Chevalley, in his obituary of Artin [14], pointed out that Artin's use of zeta functions was to discover exact algebraic facts as opposed to estimates or approxi- mate evaluations. In particular, it seems clear to me that during this period Artin was quite interested in using the Artin L-functions as a tool for finding a non- abelian class field theory, expressed as the desire to extend results from relative abelian extensions to general extensions of number fields. Artin introduced his L-functions attached to characters of the Galois group in 1923 in hopes of developing a non-abelian class field theory. Instead, through them he was led to formulate and prove the Artin Reciprocity Law - the crowning achievement of abelian class field theory. But Artin never lost interest in pursuing a non-abelian class field theory. At the Princeton University Bicentennial Conference on the Problems of Mathematics held in 1946 \Artin stated that `My own belief is that we know it already, though no one will believe me { that whatever can be said about non-Abelian class field theory follows from what we know now, since it depends on the behavior of the broad field over the intermediate fields { and there are sufficiently many Abelian cases.' The critical thing is learning how to pass from a prime in an intermediate field to a prime in a large field. -
Local Galois Cohomology Computation) Let K Be a finite Extension of Qp Or Fp((T)) with Residue field Fq
Exercise 2 (due on October 26) Choose 4 out of 8 problems to submit. Problem 2.1. (Local Galois cohomology computation) Let K be a finite extension of Qp or Fp((t)) with residue field Fq. Let V be a representation of GK on an F`-vector space. 1 GK ∗ GK (1) Show that when ` 6= p, H (GK ;V ) = 0 unless V 6= 0 or V (1) 6= 0. When ` = p 1 and K = Qp, what is dim H (GQp ;V ) \usually"? (2) When ` 6= p, compute without using Euler characteristic formula, in an explicit way, i n dim H (GK ; F`(n)). Your answer will depend on congruences of q modulo `. Observe that the dimensions coincidence with the prediction of Tate local duality and Euler characteristic formula. i (3) When ` = p and K a finite extension of Qp, compute the dimension of dim H (GK ; Fp(n)). Problem 2.2. (Dimension of local Galois cohomology groups) Let K be a finite extension of Qp, and let V be a representation of GK over a finite dimensional F`-vector space. Suppose that V is irreducible as a representation of GK and dim V ≥ 2. 1 2 (1) When ` 6= p, show that H (GK ;V ) = 0. (Hint: compute H using local Tate duality and then use Euler characteristic.) 1 (2) When ` = p, what is dim H (GK ;V )? Problem 2.3. (An example of Poitou{Tate long exact sequence) Consider F = Q, and let S = fp; 1g for an odd prime p. Determine each term in the Poitou{Tate exact sequence for 2 the trivial representation M = Fp. -
Arxiv:1711.05231V2 [Math.NT]
RATIONAL POINTS IN FAMILIES OF VARIETIES MARTIN BRIGHT Contents 1. The Hasse principle 1 2. The Brauer–Manin obstruction 4 3. Understanding Brauer groups 7 4. Local solubility in families 11 5. The Brauer–Manin obstruction in families 16 References 21 These are notes from lectures given at the Autumn School on Algebraic and Arithmetic Geometry at the Johannes Gutenberg-Universit¨at Mainz in October 2017. I thank for organisers of the school, Dino Festi, Ariyan Javanpeykar and Davide Veniani, for their hospitality. 1. The Hasse principle Let X be a projective variety over the rational numbers Q. In order to study the rational points of X, we can embed Q in each of its completions, which are the real numbers R and the p-adic numbers Qp for every prime p. This gives an embedding of X(Q) into X(R) and into each of the X(Qp). It follows that a necessary condition for X(Q) to be non-empty is that X(R) and each X(Qp) be non-empty. This necessary condition is useful because it is a condition that can be checked algorithmically. 1.1. Testing local solubility. First consider testing, for a single prime p, whether r X has any Qp-points. For any fixed r, the finite set X(Z/p Z) can be enumerated. Hensel’s Lemma states that, as soon as we can find a point of X(Z/prZ) at which the derivatives of the defining polynomials of X satisfy certain conditions, then there exists a point of X(Qp). -
A Tate Duality Theorem for Local Galois Symbols II; the Semi-Abelian Case
A TATE DUALITY THEOREM FOR LOCAL GALOIS SYMBOLS II; THE SEMI-ABELIAN CASE EVANGELIA GAZAKI* Abstract. This paper is a continuation to [Gaz17]. For every integer n ≥ 1, we consider sn 2 the generalized Galois symbol K(k; G1, G2)/n −→ H (k, G1[n] ⊗ G2[n]), where k is a finite extension of Qp, G1, G2 are semi-abelian varieties over k and K(k; G1, G2) is the Somekawa K-group attached to G1, G2. Under some mild assumptions, we describe the exact anni- 2 hilator of the image of sn under the Tate duality perfect pairing, H (k, G1[n] ⊗ G2[n]) × 0 H (k, Hom(G1[n] ⊗ G2[n],µn)) → Z/n. An important special case is when both G1, G2 are abelian varieties with split semistable reduction. In this case we prove a finiteness result, which gives an application to zero-cycles on abelian varieties and products of curves. 1. Introduction In the early 90’s Somekawa ([Som90]), following a suggestion of K. Kato, defined a new K-group, K(k; G1, ··· ,Gr) attached to a finite family of semi-abelian varieties over a field M k. This group is a generalization of the Milnor K-group, Kr (k), of a field k. In fact, in the most basic case when G1 = ··· = Gr = Gm, Somekawa proved an isomorphism M Kr (k) ≃ K(k; Gm, ··· , Gm). Moreover, similarly to the case of the classical Milnor K-group, for every integer n ≥ 1 invertible in k, there is a well-defined homomorphism to Galois cohomology, r sn : K(k; G1, ··· ,Gr)/n → H (k,G1[n] ⊗···⊗ Gr[n]), known as the generalized Galois symbol.