Hasse Principle Research Articles

Results on $\mathrm{K}_1$ of general quadratic groups

In the first part of this article we discuss the relative cases of Quillen-Suslin's local-global principle for the general quadratic (Bak's unitary) groups, and its applications for the (relative) stable and unstable $\mathrm{K}_1$-groups. The second part is dedicated to the graded version of the local-global principle for the general quadratic groups and its application to deduce a result for Bass' nil groups.

Distinction inside L-packets of SL(n)

If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker model with respect to a degenerate character of $N(E)/N(F)$. Then we establish a global analogue of this result. For this, let $E/F$ be a quadratic extension of number fields and let $\pi$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished square integrable automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Let $(\sigma,d)$ be the unique pair associated to $\pi$, where $\sigma$ is a cuspidal representation of $\mathrm{GL}_r(\mathbb{A}_E)$ with $n=dr$. Using an unfolding argument, we prove that an element of the L-packet of $\pi$ is distinguished with respect to $\mathrm{SL}_n(\mathbb{A}_F)$ if and only if it has a degenerate Whittaker model for a degenerate character $\psi$ of type $r^d:=(r,\dots,r)$ of $N_n(\mathbb{A}_E)$ which is trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular matrices of $\mathrm{SL}_n$. As a first application, under the assumptions that $E/F$ splits at infinity and $r$ is odd, we establish a local-global principle for $\mathrm{SL}_n(\mathbb{A}_F)$-distinction inside the L-packet of $\pi$. As a second application we construct examples of distinguished cuspidal automorphic representations $\pi$ of $\mathrm{SL}_n(\mathbb{A}_E)$ such that the period integral vanishes on some canonical copy of $\pi$, and of everywhere locally distinguished representations of $\mathrm{SL}_n(\mathbb{A}_E)$ such that their L-packets do not contain any distinguished representation.

Diagonal quartic surfaces with a Brauer–Manin obstruction

In this paper we investigate the quantity of diagonal quartic surfaces $a_0 X_0^4 + a_1 X_1^4 + a_2 X_2^4 +a_3 X_3^4 = 0$ which have a Brauer–Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

An O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}$$\\end{document}-acyclic variety of even index

We give the first examples of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}$$\\end{document}-acyclic smooth projective geometrically connected varieties over the function field of a complex curve, whose index is not equal to one. More precisely, we construct a family of Enriques surfaces over P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^{1}$$\\end{document} such that any multi-section has even degree over the base P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {P}}^{1}$$\\end{document} and show moreover that we can find such a family defined over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. This answers affirmatively a question of Colliot-Thélène and Voisin. Furthermore, our construction provides counterexamples to: the failure of the Hasse principle accounted for by the reciprocity obstruction; the integral Hodge conjecture; and universality of Abel–Jacobi maps.

Rigidity in elliptic curve local-global principles

Rigidity in elliptic curve local-global principles

On a local-global principle for quadratic twists of abelian varieties

Let A and A′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A'$$\\end{document} be abelian varieties defined over a number field k of dimension g≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\ge 1$$\\end{document}. For g≤3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\le 3$$\\end{document}, we show that the following local-global principle holds: A and A′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A'$$\\end{document} are quadratic twists of each other if and only if, for almost all primes p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document} of k of good reduction for A and A′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A'$$\\end{document}, the reductions Ap\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mathfrak {p}$$\\end{document} and Ap′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_\\mathfrak {p}'$$\\end{document} are quadratic twists of each other. This result is known when g=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=1$$\\end{document}, in which case it has appeared in works by Kings, Rajan, Ramakrishnan, and Serre. We provide an example that violates this local-global principle in dimension g=4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=4$$\\end{document}.

Failure of the Brauer–Manin principle for a simply connected fourfold over a global function field, via orbifold Mordell

Almost one decade ago, Poonen constructed the first examples of algebraic varieties over global fields for which Skorobogatov's etale Brauer-Manin obstruction does not explain the failure of the Hasse principle. By now, several constructions are known, but they all share common geometric features such as large fundamental groups. In this paper, we construct simply connected fourfolds over global fields of positive characteristic for which the Brauer-Manin machinery fails. Contrary to earlier work in this direction, our construction does not rely on major conjectures. Instead, we establish a new diophantine result of independent interest: a Mordell-type theorem for Campana's "geometric orbifolds" over function fields of positive characteristic. Along the way, we also construct the first example of simply connected surface of general type over a global field with a non-empty, but non-Zariski dense set of rational points.

Correction: Examples of abelian surfaces failing the local-global principle for isogenies

Correction: Examples of abelian surfaces failing the local-global principle for isogenies

Explicit realization of elements of the Tate–Shafarevich group constructed from Kolyvagin classes

We consider the Kolyvagin cohomology classes, associated to an elliptic curve E/{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 that represent non-trivial elements of , for p le 11, and hence are counterexamples to the Hasse principle.

On the Tits indices of absolutely almost simple algebraic groups over local and global fields

We present a new, detailed and uniform proof of the validity of those parts of the classification results regarding absolutely almost simple algebraic groups over p -adic and number fields, announced earlier by Tits, which also covers the case of local and global function fields. The main tools used in this paper are Prasad–Rapinchuk's reciprocity law for Tits indices over global fields and also some (new) Hasse principles for homogeneous spaces over global function fields and for forms in characteristic 2.

On genus one curves violating the local-global principle

For any number field not containing Q(i), we give an explicit construction to prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group is nontrivial.

Applications of the fibration method to the Brauer–Manin obstruction to the existence of zero-cycles on certain varieties

We study the Brauer–Manin obstruction to the existence of zero-cycles of degree [Formula: see text] on certain classes of varieties over number fields. We generalize existing results in the literature and prove some results about fibrations over the projective line, where the geometric Brauer group of the generic fiber is not assumed to be finite. The idea is to assume that the Brauer–Manin obstruction to the Hasse principle is the only one for certain fibers and then deduce analogous results for zero-cycles.

Explicit Tamagawa numbers for certain algebraic tori over number fields

Given a number field extension K / k K/k with an intermediate field K + K^+ fixed by a central element of Gal ⁡ ( K / k ) \operatorname {Gal}(K/k) of prime order p p , there exists an algebraic torus over k k whose rational points are elements of K × K^\times sent to k × k^\times by the norm map N K / K + N_{K/K^+} . The goal is to compute the Tamagawa number such a torus explicitly via Ono’s formula that expresses it as a ratio of cohomological invariants. A fairly complete and detailed description of the cohomology of the character lattice of such a torus is given when K / k K/k is Galois. Partial results including the numerator of Ono’s formula are given when the extension is not Galois, or more generally when the torus is defined by an étale algebra. We also present tools developed in SageMath for this purpose, allowing us to build and compute the cohomology and explore the local-global principles for such an algebraic torus. Particular attention is given to the case when [ K : K + ] = 2 [K:K^+]=2 and K K is a CM-field. This case corresponds to maximal tori in G S p 2 n \mathrm {GSp}_{2n} , and most examples will be in that setting. This is motivated by the application to abelian varieties over finite fields and the Hasse principle for bilinear forms.

Local-Global Principles for Constant Reductive Groups over Semi-Global Fields

We study local-global principles for torsors under reductive linear algebraic groups over semi-global fields; that is, over one-variable function fields over complete discretely valued fields. We provide conditions on the group and the semi-global field under which the local-global principle holds, and we compute the obstruction to the local-global principle in certain classes of examples. Using our description of the obstruction, we give the first example of a semisimple simply connected group over a semi-global field where the local-global principle fails. Our methods include patching and R-equivalence.

Exponential sums with multiplicative coefficients and applications

We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is “pretentious”. This leads to applications in the circle method, and a natural interpretation of the local-global principle.

Pairs Of Diagonal Quartic Forms: The Non-Singular Hasse Principle

Abstract We establish the non-singular Hasse principle for pairs of diagonal quartic equations in 22 or more variables.

Integral points on affine quadric surfaces

It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.

On a probabilistic local-global principle for torsion on elliptic curves

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid\#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability that $m$ divides the the order of the torsion subgroup of $E(\mathbb{Q})$: we find it is nonzero for all $m \in \{ 1, 2, \dots, 10, 12, 16\}$ and we compute it exactly when $m \in \{ 1,2,3,4,5,7 \}$. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve is torsion free of genus zero.

Local-global principle for classical groups over function fields of $p$-adic curves

Let K be a local field with residue field \kappa and F the function field of a curve over K . Let G be a connected linear algebraic group over F of classical type. Suppose \operatorname{char}(\kappa) is a good prime for G . Then we prove that projective homogeneous spaces under G over F satisfy a local-global principle for rational points with respect to discrete valuations of F . If G is a semisimple simply connected group over F , then we also prove that principal homogeneous spaces under G over F satisfy a local-global principle for rational points with respect to discrete valuations of F .

The Friedrichs angle and alternating projections in Hilbert C⁎-modules

Let B be a C⁎-algebra, X a Hilbert C⁎-module over B and M,N⊂X a pair of complemented submodules. We prove the C⁎-module version of von Neumann's alternating projections theorem: the sequence (PNPM)n is Cauchy in the ⁎-strong module topology if and only if M∩N is the complement of M⊥+N⊥‾. In this case, the ⁎-strong limit of (PMPN)n is the orthogonal projection onto M∩N. We use this result and the local-global principle to show that the cosine of the Friedrichs angle c(M,N) between any pair of complemented submodules M,N⊂X is well-defined and that c(M,N)<1 if and only if M∩N is complemented and M+N is closed.