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The paper contributes to building algebraic foundations of self-organized criticality answering a previously unsolved question about the limiting structure of the extended sandpile group as well as relating it to another limit at the level…

We study how the field of definition of a rational function changes under iteration. We provide a complete classification of polynomials with the property that the field of definition of one of their iterates drops in degree (over a given…

In this paper we provide an algorithm to classify groups of points on abelian threefolds over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $\mathbb{F}_q$-isogeny class. This work…

We use the theory of trianguline $(\varphi,\Gamma)$-modules over pseudorigid spaces to prove a modularity lifting theorem for certain Galois representations which are trianguline at $p$, including those with characteristic $p$ coefficients.…

In this note, we propose the modular height of an abelian variety defined over a field of finite type over Q. Moreover, we prove its finiteness property.

In this paper, we investigate Weil polynomials and their relationship with isogeny classes of abelian varieties over finite fields. We give a necessary condition for a degree 12 polynomial with integer coefficients to be a Weil polynomial.…

We present new criteria that obstruct an isogeny class of abelian varieties over a finite field with a given Weil polynomial from containing a Jacobian of a genus-3 hyperelliptic curve. Based on our analysis of the Weil polynomials of…

In our previous paper we describe the Galois module structures of $p$th-power class groups $K^\times/{K^{\times p}}$, where $K/F$ is a cyclic extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. Our…

We develop Kummer theory for algebraic function fields in finitely many transcendental variables. We consider any finitely generated Kummer extension (possibly, over a cyclotomic extension) of an algebraic function field, and describe the…

By a theorem of Chevalley the image of a morphism of varieties is a constructible set. The algebraic version of this fact is usually stated as a result on "extension of specializations" or "lifting of prime ideals". We present a difference…

Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic polynomial f of the Frobenius endomorphism…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some…

The computation of the characters of supercuspidal representations of a p-adic group involves some 4th roots of unity whose values are defined in terms of orbits of the Galois group of a p-field on a root system. The part of the definition…

Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n-1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomology group. In…

We study homomorphisms of multiplicative groups of fields preserving algebraic dependence and show that such homomorphisms give rise to valuations.

We provide a simple method of constructing isogeny classes of abelian varieties over certain fields $k$ such that no variety in the isogeny class has a principal polarization. In particular, given a field $k$, a Galois extension $\ell$ of…

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur…

In this work, we classify the group gradings on finite-dimensional incidence algebras over a field, where the field has characteristic zero, or the characteristic is greater than the dimension of the algebra, or the grading group is…

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…