Given an elliptic curve E1 over a number field and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2-Selmer groups of E1, E2 and A, we can obtain information about Sha(E1/K)[2] and we give examples where we use this to obtain a sharp bound on the Mordell-Weil rank of an elliptic curve.
As a tool, we give a precise description of the m-Selmer group of an Abelian surface A that is m-isogenous to a product of elliptic curves E1 x E2. One of the constructions can be applied iteratively to obtain information about Sha(E1/K)[2^n]. We give an example where we use this iterated application to exhibit an element of order 4 in Sha(E1/Q).
Two abelian varieties $A_1, A_2$ over a number field $K$ are called strongly iso-Kummerian if the torsion fields $K(A_1[d])$ and $K(A_2[d])$ coincide for all $d \geq 1$. For all $g \geq 4$ we construct geometrically simple, strongly iso-Kummerian $g$-dimensional abelian varieties over number fields that are not geometrically isogenous. We also discuss related examples and put significant constraints on any further iso-Kummerian pair.
We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the $p$-part. Along the way, we generalize previous results on the Tate-Shafarevich group in this situation.
This note aims at providing evidence for the section conjecture of anabelian geometry by establishing its behaviour under Weil restriction of scalars. In particular, the etale fundamental group of the Weil restriction is determined by means of a Shapiro Lemma for non-abelian group cohomology.
We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in $\mathbb{Q}_p$-unipotent fundamental groups. For $\mathbb{P}^1\setminus \{0,1,\infty\}$ and the complement of the origin in semi-stable elliptic curves of rank 0, we compute the local image of global Selmer schemes, which then allows us to numerically confirm our conjecture in a wide range of cases.
We show that the p-torsion in the Tate-Shafarevich group of any principally polarized abelian variety over a number field is unbounded as one ranges over extensions of degree O(p), the implied constant depending only on the dimension of the abelian variety.
For each prime $p$, we show that there exist geometrically simple abelian varieties $A/\mathbb Q$ with non-trivial $p$-torsion in their Tate-Shafarevich groups. Specifically, for any prime $N\equiv 1 \pmod{p}$, let $A_f$ be an optimal quotient of $J_0(N)$ with a rational point $P$ of order $p$, and let $B = A_f/\langle P \rangle$. Then the number of positive integers $d \leq X$, such that the Tate-Shafarevich group of $\hat B_d$ has non-trivial $p$-torsion, is $\gg X/\log X$, where $\hat B_d$ is the dual of the $d$-th quadratic twist of $B$. We prove this more generally for abelian varieties o…
We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and `discrete Shafarevich-Tate groups', and prove that they are finitely generated $\Bbb Z$-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich-Tate group vanishes and the discrete Selmer group coincides with the Mordell-Weil group. One of the key ingredients to prove these results is a new specialisation theorem à la Néron for first Galo…
Given an abelian variety J and an abelian subvariety A of J over a number field K, we study the visible elements of the Shafarevich-Tate group of A with respect to J over certain number field extension M of K. The notion of visible elements in Shafarevich-Tate group of an abelian variety was introduced by Mazur. In this article, we study the image of Visible elements of A with respect to J under the natural restriction map of the Galois cohomology of A over K to the Galois cohomology of A over M. In particular, for a fixed odd prime p, we investigate the conditions under which visible elements…
Let A be an abelian surface over a fixed number field. If A is principally polarised, then it is known that the order of the Tate-Shafarevich group of A must, if finite, be a square or twice a square. The situation for A not principally polarised remains unclear. For each k in {1,2,3,5,6,7,10,13} we construct a non-simple non-principally polarised abelian surface over the rationals whose Tate-Shafarevich group has order k times a square. To obtain this result, we explore the invariance under isogeny of the Birch and Swinnerton-Dyer conjecture.
Let (A,λ) be a principally polarized abelian variety defined over a global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd \rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternatin…
The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of an elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non-trivial elements of the Shafarevich-Tate group of an elliptic curve by means of the Mordell-Weil group of an ambient curve. In this paper, we generalize Cremona and Mazur's work and give precise conditions under which such a construction can be made for the Shafarevich-Tate gro…
An abelian surface A over a field K has potential quaternionic multiplication if the ring End_\bar K (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is sharp.
Let $K/F$ be a finite Galois extension of number fields with Galois group $G$, let $A$ be an abelian variety defined over $F$, and let ${\cyr W}(A_{^{/ K}})$ and ${\cyr W}(A_{^{/ F}})$ denote, respectively, the Tate-Shafarevich groups of $A$ over $K$ and of $A$ over $F$. Assuming that these groups are finite, we derive, under certain restrictions on $A$ and $K/F$, a formula for the order of the subgroup of ${\cyr W}(A_{^{/ K}})$ of $G$-invariant elements. As a corollary, we obtain a simple formula relating the orders of ${\cyr W}(A_{^{/ K}})$, ${\cyr W}(A_{^{/ F}})$ and ${\cyr W}(A_{^{/ F}}^χ)…
We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share isomorphic unpolarized modular Jacobian varieties over \Q ; we also show a method to obtain genus two curves over \Q whose Jacobian varieties are isomorphic to Weil's restriction of quadratic \Q-curves, and present examples.
Modular abelian surfaces of small conductor with nontrivial Tate--Shafarevich groups
Inria and Laboratoire d’Informatique de l’École polytechnique, CNRS, Institut Polytechnique de Paris（法国国家信息与自动化研究所、巴黎综合理工学院计算机实验室、法国国家科学研究中心、巴黎理工学院）；École Normale Supérieure de Lyon（里昂高等师范学院）
Let $A$ be an abelian variety over a number field $F$ and let $p$ be a prime. Cohen-Lenstra-Delaunay-style heuristics predict that the Tate-Shafarevich group of $A_s$ should contain an element of order $p$ for a positive proportion of quadratic twists $A_s$ of $A$. We give a general method to prove instances of this conjecture by exploiting independent isogenies of $A$. For each prime $p$, there is a large class of elliptic curves for which our method shows that a positive proportion of quadratic twists have nontrivial $p$-torsion in their Tate-Shafarevich groups. In particular, when the modul…
Let $K$ be a field which is complete for a discrete valuation. We prove a logarithmic version of the Néron-Ogg-Shafarevich criterion: if $A$ is an abelian variety over $K$ which is cohomologically tame, then $A$ has good reduction in the logarithmic setting, i.e. there exists a projective, log smooth model of $A$ over $\mathcal{O}_K$. This implies in particular the existence of a projective, regular model of $A$, generalizing a result of Künnemann. The proof combines a deep theorem of Gabber with the theory of degenerations of abelian varieties developed by Mumford, Faltings-Chai et al.
We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic $p$. We prove the prime-to-$p$ part conditionally on the finiteness of the $p$-primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to-$p$ part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-$p$ part for (1) relative elliptic curve…
Let $k$ be a higher-dimensional local field and $X$ be a smooth projective geometrically integral curve over $k$. Let $K$ be the function field of $X$. We define Tate-Shafarevich groups of an abelian variety via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove local duality theorems for abelian varieties over $k$, as well as global duality theorems for Tate-Shafarevich groups of abelian varieties over $K$. Soient $k$ un corps local supérieur et $X$ une courbe projective lisse géométriquement intègre de corps de fonctions $K$. On définit l…
Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety $A$ over $K$, we proved a similar result for the Tate-Shafarevich group of $A$, generalizing a theorem of Schneider for abelian varieties over global function fields.
Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $μ$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-extension of $K$. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of $A$ and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the $μ$-invariant) in terms of other quantities including the Faltings h…
Let $a(n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivić proved an asymptotic formula of the sum $\sum_{n\leq x} a(n+ a(n)).$ In this paper, we will prove a sharper asymptotic formula for this sum.
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Néron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators…
We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato-Tate group of the abelian variety, which had been proved for abelian surfaces by Fite-Kedlaya-Rotger-Sutherland. The proof uses Minkowski's reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.
Hindry has proposed an analogue of the classical Brauer-Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell-Weil group and the order of the Tate-Shafarevich group should have size similar to the exponential differential height. Hindry-Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate-Shafarevich group and the regulator. We r…
We study the relationship between the period and the index of a principal homogeneous space over an abelian variety, obtaining results which generalize work of Cassels and Lichtenbaum on curves of genus one. In addition, we show that the p-torsion in the Shafarevich-Tate group of a fixed abelian variety over a number field k grows arbitrarily large when considered over field extensions l/k of bounded degree. Essential use is made of an abelian variety version of O'Neil's period-index obstruction.
We develop a theory of `non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of $p$-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate-Shafarevich and Selmer groups of abelian varieties.
Let $F$ be a global function field of characteristic $p>0$ and $A/F$ an abelian variety. Let $K/F$ be an $ł$-adic Lie extension ($ł\neq p$) unramified outside a finite set of primes $S$ and such that $\Gal(K/F)$ has no elements of order $ł$. We shall prove that, under certain conditions, $Sel_A(K)_ł^\vee$ has no nontrivial pseudo-null submodule.
A well-known result by Kemperman describes the structure of those pairs (A,B) of finite subsets of an abelian group satisfying |A+B|\le|A|+|B|-1. We establish a description which is, in a sense, dual to Kemperman's, and as an application sharpen several results due to Deshouillers, Hamidoune, Hennecart, and Plagne.
Let A be an abelian variety of positive dimension defined over a number field K and let Kbar be a fixed algebraic closure of K. For each element sigma of the absolute Galois group Gal(Kbar/K), let Kbar(sigma) be the fixed field of sigma in Kbar. We shall prove that the torsion subgroup of A(Kbar(sigma)) is infinite for all sigma in Gal(Kbar/K) outside of some set of Haar measure zero. This proves the number field case of a conjecture of Geyer and Jarden from 1978.
Let $L/K$ be a finite Galois extension whose Galois group $G$ is non-abelian and characteristically simple. Using tools from graph theory, we shall give a closed formula for the number of Hopf-Galois structures on $L/K$ with associated group isomorphic to $G$.
Let E be an elliptic curve defined over a number field K. Let h be an element of order 4 in the Shafarevich-Tate group of E. We prove that h is visible in infinitely many abelian surfaces up to isomorphism. This is to say that there are infinitely many abelian surfaces J such that E\hookrightarrow J and h lies in the kernel of the natural map H^1(K,E)\rightarrow H^1(K,J).
We study Néron models of pseudo-Abelian varieties over excellent discrete valuation rings of equal characteristic $p>0$ and generalize the notions of good reduction and semiabelian reduction to such algebraic groups. We prove that the well-known representation-theoretic criteria for good and semiabelian reduction due to Néron-Ogg-Shafarevich and Grothendieck carry over to the pseudo-Abelian case, and give examples to show that our results are the best possible in most cases. Finally, we study the order of the group scheme of connected components of the Néron model in the pseudo-Abelian case. O…
Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. In this paper we shall characterise the largest possible sum-free subsets of G in case the order of G is only divisible by primes which are congruent to 1 modulo 3. The result is based on a recent result of Ben Green and Imre Ruzsa.
Suppose $X$ is a torsor under an abelian variety $A$ over a number field. We show that any adelic point of $X$ that is orthogonal to the algebraic Brauer group of $X$ is orthogonal to the whole Brauer group of $X$. We also show that if there is a Brauer-Manin obstruction to the existence of rational points on $X$, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that $A$ has finite Tate-Shafarevich group. Our results are unconditional.
Let A be a subset of a finite abelian group G. We say that A is sum-free if there is no solution of the equation x + y = z, with x, y, z belonging to the set A. Let SF(G) denotes the set of all sum-free subets of $G$ and $σ(G)$ denotes the number $ n^{-1}(\log_2 |SF(G)|) $. In this article we shall improve the error term in the asymptotic formula of $σ(G)$ which was obtained recently by Ben Green and Ruzsa. The methods used are a slight refinement of methods developed by Ben Green and Ruzsa.
The main aim of this article is to compute all the moments of the number of $p^\ell$-torsion elements in some type of nite abelian groups. The averages involved in these moments are those de ned for the Cohen-Lenstra heuristics for class groups and their adaptation for Tate-Shafarevich groups. In particular, we prove that the heuristic model for Tate-Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of $p$-Selmer groups of elliptic curves. For our purpose, we are led to de ne certain polynomials indexed by integer partitions and to …
We study the parity of 2-Selmer ranks in the family of quadratic twists of a fixed principally polarised abelian variety over a number field. Specifically, we determine the proportion of twists having odd (resp. even) 2-Selmer rank. This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for Jacobians of hyperelliptic curves. Several differences in the statistics arise due to the possibility that the Shafarevich--Tate group (if finite) may have order twice a square. In particular, the statistics for parities of 2-Selmer ranks and 2-infinity Selmer ranks need no longer agree…
This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In case the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by Min Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism gro…
