On the non-Archimedean metric Mahler measure

Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted M_∞, and prove that M_∞(α)=1 if and only if α is a root of unity. We further show that M_∞ defines a projective height on as a vector space over Q. Finally, we demonstrate how to compute M_∞(α) when α is a surd.     

The structure of analytic τ-sheaves

Let R be a complete discrete valuation -algebra whose residue field is algebraic over , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T_?(M) of M.For a given τ-sheaf M on , we prove the existence of a maximal model for M on , an R-model of , and, over a finite separable extension R′ of R, of nondegenerate models for M.We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂C, and a filtration

     

Application of covering techniques to families of curves

Much success in finding rational points on curves has been obtained by using Chabauty's Theorem, which applies when the genus of a curve is greater than the rank of the Mordell-Weil group of the Jacobian. When Chabauty's Theorem does not directly apply to a curve , a recent modification has been to cover the rational points on by those on a covering collection of curves , obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty's Theorem applies to each . So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty's Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with non-trivial elements of the Tate-Shafarevich group killed by a bielliptic isogeny.     

Infinite multiplicity of roots of unity of the Galois group in the representation on elliptic curves

Let K be a number field, an algebraic closure of K and E/K an elliptic curve defined over K. Let G_K be the absolute Galois group of over K. This paper proves that there is a subset Σ⊆G_K of Haar measure 1 such that for every σ∈Σ, the spectrum of σ in the natural representation of G_K consists of all roots of unity, each of infinite multiplicity. Also, this paper proves that any complex conjugation automorphism in G_K has the eigenvalue -1 with infinite multiplicity in the representation space of G_K.     

Non-archimedean analytic curves in the complements of hypersurface divisors

We study the degeneration dimension of non-archimedean analytic maps into the complement of hypersurface divisors of smooth projective varieties. We also show that there exist no non-archimedean analytic maps into where Di, 1?i?n, are hypersurfaces of degree at least 2 in general position and intersecting transversally. Moreover, we prove that there exist no non-archimedean analytic maps into when D₁, D₂ are generic plane curves with degD₁+degD₂?4.     

Height difference bounds for elliptic curves over number fields

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψ_υ:E(K_υ)→R over the set of places υ of K. A standard method for bounding , (due to Lang, and previously employed by Silverman) is to bound each function Ψ_υ and sum these local ‘contributions’.In this paper, we give simple formulae for the extreme values of Ψ_υ for non-archimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ.For real archimedean υ a method for sharply bounding Ψ_υ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψ_υ for complex archimedean υ.     

The vanishing order of certain Hecke L-functions of imaginary quadratic fields

Let −D<−4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of exists. Let d be a fundamental discriminant prime to D. Let 2k−1 be an odd natural number prime to the class number of . Let χ be the twist of the (2k−1)th power of a canonical Hecke character of by the Kronecker's symbol . It is proved that the vanishing order of the Hecke L-function L(s,χ) at its central point s=k is determined by its root number when , where the constant implied in the symbol ? depends only on k and ?, and is effective for L-functions with root number −1.     

The support problem for abelian varieties

Let A be an abelian variety over a number field K. If P and Q are K-rational points of A such that the order of the reduction of Q divides the order of the ) reduction of P for almost all prime ideals , then there exists a K-endomorphism φ of A and a positive integer k such that φ(P)=kQ.     

Incomplete quadratic exponential sums in several variables

We consider incomplete exponential sums in several variables of the form

     

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

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In a previous paper Najman (in press) [9], the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and . Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and , we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and .

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VPhCkJTGB_o.     

On elliptic units and p-adic Galois representations attached to elliptic curves

Let K be a quadratic imaginary number field with discriminant D_K≠-3,-4 and class number one. Fix a prime p?7 which is not ramified in K and write h_p for the class number of the ray class field of K of conductor p. Given an elliptic curve A/K with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that if then the image of a certain deformation of is “as big as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Z_p). The proof rests on the theory of Siegel functions and elliptic units as developed by Kubert, Lang and Robert.     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.     

Openness of the Galois image for τ-modules of dimension 1

Let C be a smooth projective absolutely irreducible curve over a finite field , F its function field and A the subring of F of functions which are regular outside a fixed point ∞ of C. For every place ? of A, we denote the completion of A at ? by .In [Pi2], Pink proved the Mumford-Tate conjecture for Drinfeld modules. Let φ be a Drinfeld module of rank r defined over a finitely generated field K containing F. For every place ? of A, we denote by Γ_? the image of the representation

     

Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group of a field F is whether or not is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that ? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how is determined by for an equicharacteristic Henselian field with arbitrary residue field k.     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .