Related papers: On Neron class groups of abelian varieties

Let $D$ be a totally definite quaternion algebra over a totally real number field $F$, and $\mathcal{O}$ be an $O_F$-order (of full rank) in $D$. The type number $t(\mathcal{O})$ is an important arithmetic invariant of $\mathcal{O}$ that…

We study the class of finite groups $G$ satisfying $\Phi (G/N)= \Phi(G)N/N$ for all normal subgroups $N$ of $G$. As a consequence of our main results we extend and amplify a theorem of Doerk concerning this class from the soluble universe…

The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian…

A finite group $G$ is called a Schur group if every Schur ring over $G$ is schurian, i.e. associated in a natural way with a subgroup of the symmetric group $Sym(G)$ that contains all right translations of $G$. The list of all possible…

Let $G$ be a finite group and $S$ a subset of $G$. Then $S$ is product-free if $S \cap SS = \emptyset$, and complete if $G^{\ast} \subseteq S \cup SS$. A product-free set is locally maximal if it is not contained in a strictly larger…

Let $G$ be a countable cancellative amenable semigroup and let $(F_n)$ be a (left) F{\o}lner sequence in $G$. We introduce the notion of an $(F_n)$-normal element of $\{0,1\}^G$. When $G$ = $(\mathbb N,+)$ and $F_n = \{1,2,...,n\}$, the…

Let $\ell\geq 5$ be prime. Let $\mathcal{F}_\ell$ be the collection of (isomorphism classes of) pure number fields $\mathbb{Q}(\sqrt[\ell]{a})$ of degree $\ell$, ordered by the absolute value of their discriminant. In 2018, Benli proved a…

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example: Theorem: For $n\ge 2$, there is a…

We characterize the maximum $r$-local index of a Schur algebra over an abelian number field $K$ in terms of global information determined by the field $K$, for $r$ an arbitrary rational prime. This completes and unifies previous results of…

This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to…

Given an elliptic curve $E$ over a perfect defectless henselian valued field $(F,\mathrm{val})$ with perfect residue field $\textbf{k}_F$ and valuation ring $\mathcal{O}_F$, there exists an integral separated smooth group scheme…

We prove an analogue of the Tate isogeny conjecture and the semi-simplicity conjecture for overconvergent crystalline Dieudonn\'e modules of abelian varieties defined over global function fields of characteristic $p$. As a corollary we…

In representation theory of finite groups, there is a well-known and important conjecture due to M. Brou\'e. He conjectures that, for any prime $p$, if a $p$-block $A$ of a finite group $G$ has an abelian defect group $P$, then $A$ and its…

We classify, up to equivalence, all finite-dimensional simple graded division algebras over the field of real numbers. The grading group is any finite abelian group.

Let $\chi(x)\in \mathbb{Z}[x]$ be a monic polynomial whose roots are distinct integers. We study the ideal class monoid and the ideal class group of the ring $\mathbb{Z}[x]/(\chi(x))$. We obtain formulas for the orders of these objects, and…

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any…

We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\Aut(G)$.…

Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let l be a prime number different from p. We develop a theory of l-modular types for the group…

Let~$A$ be a quotient of $J_0(N)$ associated to a newform $f$ such that the special $L$-value of $A$ (at $s=1$) is non-zero. We give a formula for the ratio of the special $L$-value to the real period of $A$ that expresses this ratio as a…

A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We…