arXiv subjects

Aleksandr Grishkov

Publications and source records attributed to Aleksandr Grishkov.

6 recordsLinked to original sources

arXiv2025

Explicit statement of a conjecture on resultantal varieties

The paper [GLZ] "L-functions of Carlitz modules, resultantal varieties and rooted binary trees" is devoted to a description of some resultantal varieties related to L-functions of Carlitz modules. It contains a conjecture that some of these varieties coincide. This conjecture can be formulated in terms of polynomials, namely, in terms of a fact that an explicitly defined polynomial belongs to the radical of the ideal generated by some other polynomials. We give an explicit statement of this conjecture and a numerical result.

math.NT↗

arXiv2021

L-functions of Carlitz modules, resultantal varieties and rooted binary trees, II

We continue study of some algebraic varieties (called resultantal varieties) started in a paper of A. Grishkov, D. Logachev "Resultantal varieties related to zeroes of L-functions of Carlitz modules". These varieties are related with the Sylvester matrix for the resultant of two polynomials, from one side, and with the L-functions of twisted Carlitz modules, from another side. Surprisingly, these varieties are described in terms of finite weighted rooted binary trees. We give a (conjecturally) complete description of them, we find parametrizations of their irreducible components and their invariants: degrees, multiplicities, Jordan forms, Galois actions. Proof of the fact that this description is really complete is a subject of future research. Maybe a generalization of these results will give us a solution of the problem of boundedness of the analytic rank of twists of Carlitz modules.

math.AG↗

arXiv2021

Reductions of Hecke correspondences on Anderson modular objects

We formulate some properties of a conjectural object $X_{fun}(r,n)$ parametrizing Anderson t-motives of dimension $n$ and rank $r$. Namely, we give formulas for $\goth p$-Hecke correspondences of $X_{fun}(r,n)$ and its reductions at $\goth p$ (where $\goth p$ is a prime of $\Bbb F_q[θ]$). Also, we describe their geometric interpretation. These results are analogs of the corresponding results of reductions of Shimura varieties. Finally, we give conjectural formulas for Hodge numbers (over the fields generated by Hecke correspondences) of middle cohomology submotives of $X_{fun}(r,n)$.

math.NT↗

arXiv2021

$h^1\ne h_1$ for Anderson t-motives

Let $M$ be an Anderson t-motive of dimension $n$ and rank $r$. Associated are two $\Bbb F_q[T]$-modules $H^1(M)$, $H_1(M)$ of dimensions $h^1(M)$, $h_1(M)\le r$ - analogs of $H^1(A,\Bbb Z)$, $H_1(A,\Bbb Z)$ for an abelian variety $A$. There is a theorem (Anderson): $h^1(M)=r \iff h_1(M)=r$; in this case $M$ is called uniformizable. It is natural to expect that always $h^1(M)=h_1(M)$. Nevertheless, we explicitly construct a counterexample. Further, we answer a question of D.Goss: is it possible that two Anderson t-motives that differ only by a nilpotent operator $N$ are of different uniformizability type, i.e. one of them is uniformizable and other not? We give an explicit example that this is possible.

math.NT↗

arXiv2020

Calculation of $h^1$ of some Anderson t-motives

We consider Anderson t-motives $M$ of dimension 2 and rank 4 defined by some simple explicit equations parameterized by $2\times2$ matrices. We use methods of explicit calculation of $h^1(M)$ -- the dimension of their cohomology group $H^1(M)$ ( = the dimension of the lattice of their dual t-motive $M'$) developed in our earlier paper. We calculate $h^1(M)$ for $M$ defined by all matrices having 0 on the diagonal, and by some other matrices. These methods permit to make analogous calculations for most (probably all) t-motives. $h^1$ of all Anderson t-motives $M$ under consideration satisfy the inequality $h^1(M)\le4$, while in all known examples we have $h^1(M)=0,1,4$. Do exist $M$ of this type having $h^1=2,3$? We do not know, this is a subject of further research.

math.NT↗

arXiv2017

Lattice map for Anderson T-motives: first approach

There exists a lattice map from the set of pure uniformizable Anderson t-motives to the set of lattices. It is not known what is the image and the fibers of this map. We prove a local result that sheds the first light to this problem and suggests that maybe this map is close to 1 -- 1. Namely, let $M(0)$ be a t-motive of dimension $n$ and rank $r=2n$ \ --- \ the $n$-th power of the Carlitz module of rank 2, and let $M$ be a t-motive which is in some sense "close" to $M(0)$. We consider the lattice map $M \mapsto L(M)$, where $L(M)$ is a lattice in $C^n$. We show that the lattice map is an isomorphism in a "neighborhood" of $M(0)$. Namely, we compare the action of monodromy groups: (a) from the set of equations defining t-motives to the set of t-motives themselves, and (b) from the set of Siegel matrices to the set of lattices. The result of the present paper gives that the size of a neighborhood, where we have an isomorphism, depends on an element of the monodromy group. We do not know whether there exists a universal neighborhood. Method of the proof: explicit solution of an equation describing an isomorphism between two t-motives by a method of successive approximations using a version of the Hensel lemma.

math.AG↗