Related papers: Quaternions, polarizations and class numbers

Let $k$ be a number field and $B$ be a central simple algebra over $k$ of dimension $p^2$ where $p$ is prime. In the case that $p=2$ we assume that $B$ is not totally definite. In this paper we study sets of pairwise nonisomorphic maximal…

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal $(x_1,x_2,\cdots,x_m)^n$ of a polynomial ring in $m$ variables. We also give a combinatorial description of the Alexander…

We study the loci of principally polarized abelian varieties with points of high multiplicity on the theta divisor. Using the heat equation and degeneration techniques, we relate these loci and their closures to each other, as well as to…

We construct spaces of coinvariants at principally polarized abelian varieties with respect to the action of an infinite-dimensional Lie algebra. We show how these spaces globalize to twisted $\mathcal{D}$-modules on moduli of principally…

The u-invariant of a field is the supremum of the dimensions of anisotropic quadratic forms over the field. We define corresponding u-invariants for hermitian and generalised quadratic forms over a division algebra with involution in…

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…

Let $F$ be a totally real number field and $A/F$ a principally polarized abelian variety with real multiplication by the ring of integers $\mathcal{O}$ of a totally real field. Assuming $A$ admits an $\mathcal{O}$-linear 3-isogeny over $F$,…

Let $D$ be a quaternion algebra over a number field $F$, and $\mathscr{G}$ be an arbitrary genus of $O_F$-orders of full rank in $D$. Let $K$ be a quadratic field extension of $F$ that embeds into $D$, and $B$ be an $O_F$-order in $K$ that…

Let G(A) be an AF-algebra given by periodic Bratteli diagram with the incidence matrix A in GL(n, Z). For a given polynomial p(x) in Z[x] we assign to G(A) a finite abelian group Z^n/p(A) Z^n. It is shown that if p(0)=1 or p(0)=-1 and…

Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence…

We prove that the automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields are cyclic, and give a complete list of finite groups that can be realized as such automorphism groups.

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…

We study the groups of rational points of abelian varieties defined over a finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or, equivalently, whose isogeny classes are determined by squarefree characteristic…

Let $A$ be a $g$-dimensional abelian variety over $\mathbb{Q}$ whose adelic Galois representation has open image in $\text{GSp}_{2g} \widehat{\mathbb{Z}}$. We investigate the endomorphism algebras $\text{End}(A_p) \otimes \mathbb{Q} =…

In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture…

Let $R$ be the maximal order in a quadratic imaginary field $K$. We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over $\mathbb{C}$ with complex…

Using the Polya Enumeration Theorem, we count with particular attention to C^3/Gamma up to C^6/Gamma, abelian orbifolds in various dimensions which are invariant under cycles of the permutation group S_D. This produces a collection of…

Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B)=[B^\times: A^\times]>1$. We also calculate numerical invariants…

G. Prasad and A. Rapinchuk asked if two quaternion division F -algebras that have the same subfields are necessarily isomorphic. The answer is known to be "no" for some very large fields. We prove that the answer is "yes" if F is an…

We show Laplacian algebras are maximal, and give applications to the Classical Invariant Theory of real orthogonal representations of compact groups, including: The solution of the Inverse Invariant Theory problem for finite groups. An…