Coherent State Transforms and Abelian Varieties

We extend the coherent state transform (CST) of Hall to the context of abelian varieties by considering them as quotients of the complexification of the abelian group K=U(1)^g. We show that this transform, applied to appropriate distributions on K, gives all classical theta functions, and that, by defining on this space of theta functions an inner product related to the K-averaged heat kernel, the unitarity of the CST transform is still preserved.     

Vanishing Results for Toric Varieties Associated To GL n and G2

Toric varieties associated with root systems appeared very naturally in the theory of group compactifications. Here they are considered in a very different context. We prove the vanishing of higher cohomology groups for certain line bundles on toric varieties associated to GL _n and G₂. This can be considered of general interest and it improves the previously known results for these varieties. We also show how these results give a simple proof of a converse to Mazur’s inequality for GL _n and G₂ respectively. It is known that the latter imply the nonemptiness of some affine Deligne–Lusztig varieties. Dedicated to Scarlett MccGwire and Dr. Christian Duhamel     

Divisors on Principally Polarized Abelian Varieties

The purpose of this paper is to show how generalizations of generic vanishing theorems to a -divisor setting can be used to study the geometric properties of pluritheta divisors on a principally polarized Abelian variety (PPAV for short).     

Projective Embeddings and Lagrangian Fibrations of Abelian Varieties

It is well known that every Abelian variety can be embedded into projective spaces by theta functions and the basis of theta functions are determined by choosing a Lagrangian fibration. In this paper, we prove that the restriction of natural Lagrangian fibrations (moment maps) of projective spaces converge to that of the Abelian variety in ``the Gromov-Hausdorff topology''. This is, in some sense, a Lagrangian fibration version of the convergence theorem of G. Tian [6] and S. Zelditch [7] for Kähler metrics.     

Visibility of Shafarevich-Tate Groups of Abelian Varieties

We investigate Mazur's notion of visibility of elements of Shafarevich-Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich-Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton-Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich-Tate groups.     

Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

Real Abelian Varieties with Many Line Bundles

Let X and Y be affine nonsingular real algebraic varieties.A general problem in real algebraic geometry is to try to decidewhen a continuous map f: X Y can be approximated by regularmaps in the space of c⁰ mappings from X to Y, equipped withthe c⁰ topology. This paper solves this problem when X is theconnected component containing the origin of the real part ofa complex Abelian variety and Y is the standard 2-dimensionalsphere.     

Varieties generated by countably compact Abelian groups

We prove under the assumption of Martin's Axiom that every precompact Abelian group of size belongs to the smallest class of groups that contains all Abelian countably compact groups and is closed under direct products, taking closed subgroups and continuous isomorphic images.

     

On the Specialization Theorem for Abelian Varieties

In this paper, we apply Moriwaki's arithmetic height functionsto obtain an analogue of Silverman's specialization theoremfor families of Abelian varieties over K, where K is any fieldfinitely generated over Q. 2000 Mathematics Subject Classification11G40, 14K15.     

The Tate Conjecture for Generic Abelian Varieties

Let A V be a Kuga fibre variety of Mumford's Hodge type, definedover a finitely generated subfield of C, and let be the genericpoint of V. We show that any element of which is invariant under , for some finite extension E of k(), is fixed bythe semisimple part of the Hodge group of A. If A V satisfiesthe H₂-condition, then the Hodge and Tate conjectures are equivalentfor A, and the Mumford–Tate conjecture is true.     

Varieties Generated by Wreath Products of Abelian Groups

This paper is a survey of our recent results concerning metabelian varieties, and more specifically, varieties generated by wreath products of Abelian groups. We give a full classification of cases where sets of wreath products of Abelian groups $ \mathfrak{X} $ Wr $ \mathfrak{Y} $ = { X Wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } and $ \mathfrak{X} $ wr $ \mathfrak{Y} $ = {X wr Y | X ∈ $ \mathfrak{X} $ , Y ∈ $ \mathfrak{Y} $ } generate the product variety $ \mathfrak{X} $ var ( $ \mathfrak{Y} $ ).     

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (\mathbbZ/l\mathbbZ)^m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.     

On Minimal Surfaces Incompressible in Homology and Abelian Varieties

In the years 1979–80 Sacks and Uhlenbeck [19] and Schoenand Yau [21] proved independently the following beautiful theorem.     

On Counting Certain Abelian Varieties Over Finite Fields

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers[Doc. Math., 21, 1607-1643 (2016)],[Taiwanese J. Math., 20(4), 723-741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author[Forum Math., 22(3), 565-582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.     

Measures of Simultaneous Approximation for Quasi-Periods of Abelian Varieties

We examine various extensions of a series of theorems proved by Chudnovsky in the 1980s on the algebraic independence (transcendence degree 2) of certain quantities involving integrals of the first and second kind on elliptic curves; these extensions include generalizations to abelian varieties of arbitrary dimensions, quantitative refinements in terms of measures of simultaneous approximation, as well as some attempt at unifying the aforementioned theorems. In the process we develop tools that might prove useful in other contexts, revolving around explicit “algebraic” theta functions on the one hand, and Eisenstein's theorem and G-functions on the other hand.     

The Rank of Abelian Varieties over Infinite Galois Extensions

G. Frey and M. Jarden (1974, Proc. London Math. Soc.28, 112-128) asked if every Abelian variety A defined over a number field k with dim A>0 has infinite rank over the maximal Abelian extension k^ab of k. We verify this for the Jacobians of cyclic covers of P¹, with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d -dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2).