In a series of articles published in the C.R. Paris more than a century ago, T. Pépin announced a list of ``theorems'' concerning the solvability of diophantine equations of the type $ax^4 + by^4 = z^2$. In this article, we show how to prove these claims using the structure of 2-class groups of imaginary quadratic number fields. We will also look at a few related results (including FLT for $n = 7$) from a modern point of view.
In this article we develop counterexamples to the Hasse principle using only techniques 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, this type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in the Tate--Shafarevich group.
After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by Pépin in a series of articles on unsolvable quartic diophantine equations, and for constructing families of counterexamples to the Hasse Principle for curves of genus 1 defined by equations of the form $ax^4 + by^4 = z^2$.
In this paper we inspect from closer the local and global points of the twists of the Klein quartic. For the local ones we use geometric arguments, while for the global ones we strongly use the modular interpretation of the twists. The main result is providing families with (conjecturally infinitely many) twists of the Klein quartic that at counter-examples to the Hasse principle.
Let $p$ be an odd prime number. In this paper, we are concerned with the behaviour of Fermat curves defined over ${\bf Q}$ given by equations $ax^p+by^p+cz^p=0$, with respect to the local-global Hasse principle. It is conjectured that there exist infinitely many Fermat curves of exponent $p$ which are counterexamples to the Hasse principle. It is a consequence of the abc-conjecture if $p\geq 5$. Using a cyclotomic approach due to H. Cohen and Chebotarev's density theorem, we obtain a partial result towards this conjecture, by proving it for $p\leq 19$.
We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby, we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.
Let k be a global field of characteristic not 2. The classical Hasse-Minkowski theorem states that if two quadratic forms become isomorphic over all the completions of k, then they are isomorphic over k as well. It is natural to ask whether this is also true for G-quadratic forms, where G is a finite group. In the case of number fields the Hasse principle for G-quadratic forms does not hold in general, as shown by Jorge Morales. The aim of this paper is to study this question when k is a global field of positive characteristic. We give a sufficient criterion for the Hasse principle to hold, an…
We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin-Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin-Lehner involutions. A particular example is the quotient of the Shimura curve X attached to the indefinite rational quaternion algebra of discriminant 23*107 by the Atkin-Lehner involution w107. The quadratic twist of X by Q(sqrt{-23}) with respect to this involution is also a counter-example to the Hasse principle over Q.
Châtelet surfaces provide a rich source of geometrically rational surfaces which do not always satisfy the Hasse principle. Restricting attention to a special class of Châtelet surfaces, we investigate the frequency that such counter-examples arise over the rationals.
Cubic surfaces failing the integral Hasse principle
University of Bath（巴斯大学）；Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
In $2000$, Colliot-Thélène and Poonen showed how to construct algebraic families of genus one curves violating the Hasse principle. Poonen explicitly constructed an algebraic family of genus one cubic curves violating the Hasse principle using the general method developed by Colliot-Thélène and himself. The main result in this paper generalizes the result of Colliot-Thélène and Poonen to arbitrarily high genus hyperelliptic curves. More precisely, for $n > 5$ and $n \not\equiv 0 \pmod{4}$, we show that there is an algebraic family of hyperelliptic curves of genus $n$ that is counterexamples to…
Given a number field $k$ and a positive integer $d$, in this paper we consider the following question: does there exist a smooth diagonal surface of degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every completion of $k$, yet no line over $k$? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erdős.
A generalized Mordell curve of degree $n \ge 3$ over $\bQ$ is the smooth projective model of the affine curve of the form $Az^2 = Bx^n + C$, where $A, B, C$ are nonzero integers. A generalized Fermat curve of signature $(n, n, n)$ with $n \ge 3$ over $\bQ$ is the smooth projective curve of the form $Ax^n + By^n + Cz^n = 0$ for some nonzero integers $A, B, C$. In this paper, we show that for each prime $p$ with $p \equiv 1 \pmod{8}$ and $p \equiv 2 \pmod{3}$, there exists a threefold $\cX_p \subseteq \bP^6$ such that certain rational points on $\cX_p$ produce infinite families of non-isomorphic…
In this note, we are interested in local-global principles for multinorm equations of the form $\prod_{i=1}^n N_{L_i /k}(z_i) = a$ where $k$ is a global field, $L_i/k$ are finite separable field extensions and $a \in k^*$. In particular, we prove a result relating weak approximation for this equation to weak approximation for some classical norm equation $N_{F/k}(w) = a$ where $F := \bigcap_{i=1}^n L_i$. It provides a proof of a "weak approximation" analogue of a recent conjecture by Pollio and Rapinchuk about multinorm principle. We also provide a counterexample to the original conjecture con…
We give large families of Shimura curves defined by congruence conditions, all of whose twists lack $p$-adic points for some $p$. For each such curve we give analytically large families of counterexamples to the Hasse principle via the descent (or equivalently étale Brauer-Manin) obstruction to rational points applied to étale coverings coming from the level structure. More precisely, we find infinitely many quadratic fields defined using congruence conditions such that a twist of a related Shimura curve by each of those fields violates the Hasse principle. As a minimal example, we find the tw…
Skorobogatov constructed a bielliptic surface which is a counterexample to the Hasse principle not explained by the Brauer-Manin obstruction. We show that this surface has a $0$-cycle of degree 1, as predicted by a conjecture of Colliot-Thélène.
Quaternion extensions are often the smallest extensions to exhibit special properties. In the setting of the Hasse-Arf Theorem, for instance, quaternion extensions are used to illustrate the fact that upper ramification numbers need not be integers. These extensions play a similar role in Galois module structure. To better understand these examples, we catalog the ramification filtrations that are possible in totally ramified extensions of dyadic number fields. Interestingly, we find that the catalog depends, for sharp lower bounds, upon the refined ramification filtration, which is associated…
Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(\mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=\mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper mod…
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops further the technique initiated by Swinnerton-Dyer in 1993, and later generalised by Colliot-Thélène, Skorobogatov and Swinnerton-Dyer, for studying rational points on pencils of curves of genus 1. The second half, which builds upon the first, is devoted to quartic del Pezzo surfaces and to higher-dimensional intersections of two quadrics. Conditionally on two wel…
In this paper, we study the properties of weak approximation with Brauer-Manin obstruction and the Hasse principle with Brauer-Manin obstruction for surfaces with respect to field extensions of number fields. We assume a conjecture of M. Stoll. For any nontrivial extension of number fields $L/K,$ we construct two kinds of smooth, projective, and geometrically connected surfaces defined over $K.$ For the surface of the first kind, it has a $K$-rational point, and satisfies weak approximation with Brauer-Manin obstruction off $\infty_K,$ while its base change by $L$ does not so off $\infty_L.$ F…
We consider the Kolyvagin cohomology classes associated to an elliptic curve $E$ defined over $\mathbb{Q}$ from a computational point of view. We explain how to go from a model of a class as an element of $(E(L)/pE(L))^{\mathrm{Gal}(L/\mathbb{Q})}$, where $p$ is prime and $L$ is a dihedral extension of $\mathbb{Q}$ of degree $2p$, to a geometric model as a genus one curve embedded in $\mathbb{P}^{p-1}$. We adapt the existing methods to compute Heegner points to our situation, and explicitly compute them as elements of $E(L)$. Finally, we compute explicit equations for several genus one curves …
