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Let $n>1$ be an integer such that $X_{0}\!\left( n\right) $ has genus $0$, and let $K$ be a field of characteristic $0$ or relatively prime to $6n$. In this article, we explicitly classify the isogeny graphs of all rational elliptic curves…

We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…

Let ${\bf X}$ and ${\bf X}$ be two $n$-dimensional elliptical random vectors, we establish an identity for $E[f({\bf Y})]-E[f({\bf X})]$, where $f: \Bbb{R}^n \rightarrow \Bbb{R}$ fulfilling some regularity conditions. Using this identity we…

We consider generalized $\Lambda$-structures on algebras and schemes over the ring of integers $\mathit{O}_K$ of a number field $K$. When $K=\mathbb{Q}$, these agree with the $\lambda$-ring structures of algebraic K-theory. We then study…

A probabilistic representation for a class of weighted $p$-radial distributions, based on mixtures of a weighted cone probability measure and a weighted uniform distribution on the Euclidean $\ell_p^n$-ball, is derived. Large deviation…

This paper proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector…

We construct new families of elliptic curves over \(\FF_{p^2}\) with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant-Lambert-Vanstone (GLV) and…

Let $\mathbf{k}$ be an algebraically closed field. Recently, K. Erdmann classified the symmetric $\mathbf{k}$-algebras $\Lambda$ of finite representation type such that every non-projective module $M$ has period dividing four. The goal of…

Let $E/\mathbb{Q}$ be a non-CM elliptic curve. Assuming GRH, we prove that, for a set of primes $p$ of density $1$, the absolute discriminant of the $\mathbb{F}_p$-endomorphism ring of the reduction of $E$ modulo $p$ is close to maximal.

Given a finite abelian group $\Gamma$, we study the distribution of the $p$-part of the class group $\operatorname{Cl}(K)$ as $K$ varies over Galois extensions of $\mathbb{Q}$ or $\mathbb{F}_q(t)$ with Galois group isomorphic to $\Gamma$.…

In his previous papers (J. reine angew. Math. 544 (2002), 91--110; math.AG/0103203) the author introduced a certain explicit construction of superelliptic jacobians, whose endomorphism ring is the ring of integers in the $p$th cyclotomic…

Let $\mathbb{F}_q[T]$ be the polynomial ring over a finite field $\mathbb{F}_q$. We study the endomorphism rings of Drinfeld $\mathbb{F}_q[T]$-modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings…

In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\widehat\varphi}={\phi}_{1}^{e_1}\times \dots \times…

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This…

In this paper, we initiate the study of endomorphisms and modular theory of the graph C*-algebras $\O_{\theta}$of a 2-graph $\Fth$ on a single vertex. We prove that there is a semigroup isomorphism between unital endomorphisms of…

It is well known that there is a one-to-one correspondence between supersingular $j$-invariants up to the action of $\text{Gal}(\mathbb{F}_{p^2}/\mathbb{F}_p)$ and type classes of maximal orders in $B_{p,\infty}$ by Deuring's theorem.…

We characterize the distribution of 2-Selmer ranks of quadratic twists of elliptic curves over $\mathbb{Q}$ with full rational 2-torsion. We propose a new type of random alternating matrix model $M_{*,\mathbf…

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in even residual characteristic obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a…

We study a new parametric family of distributions on the ordered simplex $\nabla^{d-1} = \{y \in \mathbb{R}^d: y_1 \geq \dots \geq y_d \geq 0, \sum_{k=1}^d y_k = 1\}$, which we call Generalized Rank Dirichlet (GRD) distributions. Their…

We study the isogenies of certain abelian varieties over finite fields with non-commutative endomorphism algebras with a view to potential use in isogeny-based cryptography. In particular, we show that any two such abelian varieties with…