Invariants de classes : exemples de non-annulation en dimension supérieure

Consider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences generalize a theorem of Olson [9] ( K = Q) and give divisibility conditions for the discriminant and the coefficients of E related with the paper of Zimmer [13] as well as diophantine equations related with Fermat's equation that are discussed for K Q and K a function field. We are interested in elliptic curves over global fields K (i.e. : K is a finite number field or K is a function field of one variable over a finite field) and especially in the torsion group of E(K), where E(K) is the group of K-rational points of E. It is well known that E(K) is finitely generated, it is conjectured that if K is a number field then the order of the torsion group of E(K) is bounded by some number depending only on K (cf. Demjanenko [1]). In any case in order to handle with E(K) the first step is to determine the torsion group. In principle this is not so difficult; if one uses the results of Lutz [6] and Zimmer [13], one sees immediately that for every E there exist points of q-power-order only for a finite number of primes q, as the equations for points of order q are known (in principle) one has only t ~ test what orders really occur. But as the computational work grows very rapidly with q it is usefull to look for sharper necessary conditions, and this shall be done in this paper.

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck’s conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.

This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R . The focus is on lattices, namely, finitely generated G -modules that are projective as R -modules, and on the full subcategory of all G -modules projective over R generated by the lattices. The stable category of such G -modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G . Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

Let [Formula: see text] be a number field, [Formula: see text] a finite set of places. For [Formula: see text] or an elliptic curve [Formula: see text] defined over [Formula: see text], we establish uniformity results on the number of [Formula: see text]-integral torsion points relative to a non-torsion point [Formula: see text], as [Formula: see text] varies over number fields of bounded degree. In particular for [Formula: see text], if [Formula: see text] is a positive integer, we prove a uniform bound on the degree of a torsion point [Formula: see text] that is [Formula: see text]-integral relative to a non-torsion point [Formula: see text] with degree [Formula: see text].

The main purpose of this paper is to give an overview over the theory of abelian varieties, with main focus on Jacobian varieties of curves reaching from well-known results till to latest developments and their usage in cryptography. In the first part we provide the necessary mathematical background on abelian varieties, their torsion points, Honda-Tate theory, Galois representations, with emphasis on Jacobian varieties and hyperelliptic Jacobians. In the second part we focus on applications of abelian varieties on cryptography and treating separately, elliptic curve cryptography, genus 2 and 3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard groups, isogenies of Jacobians via correspondences and applications to discrete logarithms. Several open problems and new directions are suggested.

On a weak variant of the geometric torsion conjecture

Let E/ℚ be an elliptic curve with good ordinary reduction at a prime p > 2. It has a welldefined Iwasawa μ-invariant μ(E)p which encodes part of the information about the growth of the Selmer group ) as Kn ranges over the subfields of the cyclotomic Zp-extension K∞/ℚ. Ralph Greenberg has conjectured that any such E is isogenous to a curve E′ with μ(E′)p = 0. In this paper we prove Greenberg's conjecture for infinitely many curves E with a rational p-torsion point, p = 3 or 5, no two of our examples having isomorphic p-torsion. The core of our strategy is a partial explicit evaluation of the global duality pairing for finite flat group schemes over rings of integers.

Let A_K be an abelian variety over a discrete valuation field K. Let A be the Neron model of A_K over the ring of integers O_K of K and A_k its special fibre. We study the set of rational points of the group of components \phi_A of A_k. In particular, we look at the cases where A_K is a Jacobian or where A_K is an abelian variety having semi-stable reduction. We also consider algebraic tori over K instead of abelian varieties.

Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $\lambda $-invariants of these signed Selmer groups for tw

A be an abelian variety defined over an algebraically closed field.We give an elementary proof of the following result (Theorem 1): If 7 is a O-cycle of degree 0 on -A, and c € A is a point of finite order, then 7 is rationally equivalent to 7 C , the translate of 7 under c.From this follows Theorem 2: Given any effective 0-cycle rç = (ai) + • • • + (a r ) on A, and any points of finite order ci,... c r 6 A satisfying ci + -• • + c r = o -the identity of A, we have that 77 is rationally equivalent to the 0-cycle (a\ + c{) H h (a r + c r ).Consequently, for r > 2, the set E[rj] of effective 0-cycles rationally equivalent to r\ is always at least a countably infinite set (Corollary 1).Further corollaries of Theorem 2 are given, including a generalization of Theorem 1 to higher dimensional cycles (Corollary 4).

Suppose that $O=\Bbb F_q[\pi ]$ is a polynomial ring and $R$ is a commutative unitary $O$-algebra. The category of finite group schemes over $R$ with a strict action of $O$ was recently introduced by Faltings and appears as an equal characteristic analogue of the classical category of finite flat group schemes in the equal characteristic case. In this paper we obtain a classification of these modules and apply it to prove analogues of properties that were known earlier for classical group schemes.

We introduce the space Π(G) of equivalence classes of π-points of a finite group scheme G and associate a subspace Π(G)M to any G-module M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large G-modules, the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite-dimensional) G-module can be detected by its restriction along π-points of G. Unlike the cohomological support variety of a G-module M, the invariant M↦Π(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite-dimensional G-modules. Finally, using the stable module category of G, we provide Π(G) with the structure of a ringed space which we show to be isomorphic to the scheme ProjH•(G,k)

Let${\mathbb M}$be an affine variety equipped with a foliation, both defined over a number field${\mathbb K}$. For an algebraic$V\subset {\mathbb M}$over${\mathbb K}$, write$\delta _{V}$for the maximum of the degree and log-height ofV. Write$\Sigma _{V}$for the points where the leaves intersectVimproperly. Fix a compact subset${\mathcal B}$of a leaf${\mathcal L}$. We prove effective bounds on the geometry of the intersection${\mathcal B}\cap V$. In particular, when$\operatorname {codim} V=\dim {\mathcal L}$we prove that$\#({\mathcal B}\cap V)$is bounded by a polynomial in$\delta _{V}$and$\log \operatorname {dist}^{-1}({\mathcal B},\Sigma _{V})$. Using these bounds we prove a result on the interpolation of algebraic points in images of${\mathcal B}\cap V$by an algebraic map$\Phi $. For instance, under suitable conditions we show that$\Phi ({\mathcal B}\cap V)$contains at most$\operatorname {poly}(g,h)$algebraic points of log-heighthand degreeg.We deduce several results in Diophantine geometry. Following Masser and Zannier, we prove that given a pair of sections$P,Q$of a nonisotrivial family of squares of elliptic curves that do not satisfy a constant relation, whenever$P,Q$are simultaneously torsion their order of torsion is bounded effectively by a polynomial in$\delta _{P},\delta _{Q}$; in particular, the set of such simultaneous torsion points is effectively computable in polynomial time. Following Pila, we prove that given$V\subset {\mathbb C}^{n}$, there is an (ineffective) upper bound, polynomial in$\delta _{V}$, for the degrees and discriminants of maximal special subvarieties; in particular, it follows that the André–Oort conjecture for powers of the modular curve is decidable in polynomial time (by an algorithm depending on a universal, ineffective Siegel constant). Following Schmidt, we show that our counting result implies a Galois-orbit lower bound for torsion points on elliptic curves of the type previously obtained using transcendence methods by David.