Cohomological Mackey functors in number theory

We prove relations between the evaluations of cohomological Mackey functors over complete discrete valuation rings or fields and apply this to Mackey functors that arise naturally in number theory. This provides relations between λ- and μ-invariants in Iwasawa theory, between Mordell-Weil groups, Shafarevich-Tate groups, Selmer groups and zeta functions of elliptic curves, and between ideal class groups and regulators of number fields.     

Ranks of twists of elliptic curves and Hilbert’s tenth problem

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.     

Picard number of the generic fiber of an abelian fibered hyperkähler manifold

We shall show that the Picard number of the generic fiber of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O’Grady’s 10-dimensional manifold, we construct an abelian fibered hyperkähler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.     

Torsion in Mordell-Weil groups of Fermat Jacobians

We study the torsion in the Mordell-Weil group of the Jacobian of the Fermat curve of exponent p over the cyclotomic field obtained by adjoining a primitive p-th root of 1 to Q. We show that for all (except possibly one) proper subfields of this cyclotomic field, the torsion parts of the corresponding Mordell-Weil groups are elementary abelian p-groups.     

A note on the Mordell-Weil rank modulo n

Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is (conjecturally) the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.     

On Mordell-Weil Lattices of Type D5

We establish the analogue for D₅ of the theory of algebraic equation of type E_r (T. SHIODA: Construction of elliptic curves with high rank via the invariants of the Weyl groups, J. Math. Soc. Japan 43 , 1991, No. 4, 673-719), which is one of the results of the theory of Mordell-Weil lattices. As an application, we give a method of constructing an elliptic curve over Q(t) with rank 5, together with explicit generators of the Mordell-Weil group.     

Elliptic logarithms,diophantine approximation and the Birch and Swinnerton-Dyer conjecture

Most, if not all, unconditional results towards the abc-conjecture rely ultimately on classical Baker’s method. In this article, we turn our attention to its elliptic analogue. Using the elliptic Baker’s method, we have recently obtained a new upper bound for the height of the S-integral points on an elliptic curve. This bound depends on some parameters related to the Mordell-Weil group of the curve. We deduce here a bound relying on the conjecture of Birch and Swinnerton-Dyer, involving classical, more manageable quantities. We then study which abc-type inequality over number fields could be derived from this elliptic approach.     

On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields

If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional χ-eigenspace (with χ a complex ring class character) provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is non-zero. This represents the analogue in the function field setting of a theorem for elliptic curves over Q due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic.     

Globally irreducible representations of SL3(q) and SU3(q)

The notion ofglobally irreducible representations of finite groups was introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered by N. Elkies and T. Shioda using Mordell-Weil lattices of elliptic curves. In this paper we classify all globally irreducible representations coming from projective complex representations of the finite simple groups PSL₃(q) and PSU₃(q). The main result is that these representations are essentially those discovered by Gross.     

Right triangles with algebraic sides and elliptic curves over number fields

Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P _λ of infinite order in the Mordell-Weil group of the elliptic curve Y ² = X ³ − n ² X over ℚ(λ). Research of the rest of authors was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain).     

Integral points on elliptic curves and -torsion in class groups

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques and methods based on quasi-orthogonality in the Mordell-Weil lattice. We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the -torsion part of the class group of a quadratic field. The same ideas can be used to count rational points on curves of higher genus.

     

Fibred rational surfaces with extremal Mordell-Weil lattices

Slope inequalities are given for fibred rational surfaces according as the Clifford index of a general fibre. For fibred rational surfaces of Clifford index two, the Mordell-Weil lattices of maximal ranks are completely determined.Supported by The 21st Century COE Program named “Towards a new basic science: depth and synthesis”.     

The Lehmer inequality and the Mordell-Weil theorem for Drinfeld modules

In this paper we prove several Lehmer type inequalities for Drinfeld modules which will enable us to prove certain Mordell-Weil type structure theorems for Drinfeld modules.     

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

Given a large positive number x and a positive integer k, we denote by Qk(x) the set of congruent elliptic curves E(n): y2= z3- n2 z with positive square-free integers n x congruent to one modulo eight,having k prime factors and each prime factor congruent to one modulo four. We obtain the asymptotic formula for the number of congruent elliptic curves E(n)∈ Qk(x) with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)2. We also get a lower bound for the number of E(n)∈ Qk(x)with Mordell-Weil ranks zero and 2-primary part of Shafarevich-Tate groups isomorphic to(Z/2Z)4. The key ingredient of the proof of these results is an independence property of residue symbols. This property roughly says that the number of positive square-free integers n x with k prime factors and residue symbols(quadratic and quartic) among its prime factors being given compatible values does not depend on the actual values.     

Fibrations et conjecture de Tate

Résumé

Nous décrivons le comportement du rang du groupe de Mordell-Weil de la variété de Picard de la fibre générique d’une fibration en termes de contributions locales données par des moyennes de traces de Frobenius agissant sur les fibres. Les énoncés fournissent une réinterpretation de la conjecture de Tate (pour les diviseurs) et généralisent des résultats antérieurs de Nagao, Rosen-Silverman et des auteurs.

Abstract. Fibrations and Tate's conjecture

We describe the behaviour of the rank of the Mordell-Weil group of the Picard variety of the generic fibre of a fibration in terms of local contributions given by averaging traces of Frobenius acting on the fibres. The results give a reinterpretation of Tate's conjecture (for divisors) and generalises previous results of Nagao, Rosen-Silverman and the authors.     

Some examples of 5 and 7 descent for elliptic curves over Q

We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large. Received October 6, 2000 / final version received November 14, 2000?Published online February 15, 2001     

A refined conjecture of Mazur-Tate type for Heegner points

Summary In [MT1], Mazur and Tate present a refined conjecture of Birch and Swinnerton-Dyer type for a modular elliptic curveE. This conjecture relates congruences for certain integral homology cycles onE(C) (the modular symbols) to the arithmetic ofE overQ. In this paper we formulate an analogous conjecture forE over a suitable imaginary quadratic fieldK, in which the role of the modular symbols is played by Heegner points. A large part of this conjecture can be proved, thanks to the ideas of Kolyvagin on the Euler system of Heegner points. In effect the main result of this paper can be viewed as a generalization of Kolyvagin's result relating the structure of the Selmer group ofE overK to the Heegner points defined in the Mordell-Weil groups ofE over ring class fields ofK. An explicit application of our method to the Galois module structure of Heegner points is given in Sect. 2.2.Oblatum 19-XII-1991, & 25-II-1992     

Automorphism groups of rational elliptic surfaces with section and constant J-map

In this paper, the automorphism groups of relatively minimal rational elliptic surfaces with section which have constant J-maps are classified. The ground field is ?. The automorphism group of such a surface β: B → ?¹, denoted by Au t(B), consists of all biholomorphic maps on the complex manifold B. The group Au t(B) is isomorphic to the semi-direct product MW(B) ? Aut _σ (B) of the Mordell-Weil groupMW(B) (the group of sections of B), and the subgroup Aut _σ (B) of the automorphisms preserving a fixed section σ of B which is called the zero section on B. The Mordell-Weil group MW(B) is determined by the configuration of singular fibers on the elliptic surface B due to Oguiso and Shioda [9]. In this work, the subgroup Aut _σ (B) is determined with respect to the configuration of singular fibers of B. Together with a previous paper [4] where the case with non-constant J-maps was considered, this completes the classification of automorphism groups of relatively minimal rational elliptic surfaces with section.     

Dyson's theorem for curves

Let K be a number field and X₁ and X₂ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X₁×X₂. If Xi=P₁ we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell-Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.     

Tables of octic fields with a quartic subfield

We describe the computation of extended tables of degree 8 fields with a quartic subfield, using class field theory. In particular we find the minimum discriminants for all signatures and for all the possible Galois groups. We also discuss some phenomena and statistics discovered while making the tables, such as the occurrence of 11 non-isomorphic number fields having the same discriminant, or several pairs of non-isomorphic number fields having the same Dedekind zeta function.