arXiv subjects

D. Logachev

Publications and source records attributed to D. Logachev.

8 recordsLinked to original sources

arXiv2025

Analytic ranks of twists of Carlitz modules -- a survey of results

We give in this paper a survey of results obtained in our earlier papers, and state explicitly some problems of further research, for example: are the analytic ranks bounded, or not? Twists of Carlitz modules are parametrized by polynomials over finite fields $\Bbb F_q$. The analytic rank of a twist is the order of zero of its L-function at a point. The set of polynomials of degree $\le m$ such that the analytic ranks of the corresponding twists are $\ge i$ is $X(m,i)(\Bbb F_q)$ where $X(m,i)$ is an affine variety defined over $\Bbb F_p$ (we do not know what is its dimension). We consider also a related invariant of a twist, namely, the behaviour of its L-function at infinity (the rank at infinity). We know much more on varieties corresponding to twists of a fixed rank at infinity and on their lifts from $\Bbb F_p$ to $\Bbb Z$ . For example, for $q=2$ the irreducible components of these varieties are described in terms of finite rooted weighted binary trees. A similar description for $q>2$ is not found yet.

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arXiv2024

Non-injectivity of the lattice map for non-mixed Anderson t-motives, and a result towards its surjectivity

Let $M$ be an uniformizable Anderson t-motive and $L(M)$ its lattice. First, we prove by an explicit construction that for the non-mixed $M$ the lattice map $M\mapsto L(M)$ is not injective. Second, we show that some lattices which do not belong to the set $L(M)$ of pure $M$, are lattices of non-pure $M$. This is a result towards surjectivity of the lattice map. The t-motives used in the proofs are non-pure t-motives of dimension 2, rank 3. Finally, we start calculations in order to answer a question whether all these t-motives are uniformizable, or not.

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arXiv2023

$h^1$, $h_1$ of Anderson t-motives, systems of affine equations and non-commutative determinants

The authors defined in "$h^1\ne h_1$ for Anderson t-motives" the notion of an affine equation associated to a t-motive $M$. Here we define two systems of affine equations associated to a t-motive $M$, used for calculation of $H^1(M)$ and $H_1(M)$. We describe the process of elimination of unknowns in these systems. This is an analog of the corresponding theory of systems of linear differential equations. It gives us a notion of a non-commutative determinant $det_{i,c}(M)$ which belongs to the Anderson ring $\Bbb C_\infty[T,\tau]$ of non-commutative polynomials. Finally, we calculate $det_{i,c}(M)$ for $M=$ a Drinfeld module or its 1-dual. Also, some explicit calculations are made for Anderson t-motives of dimension $n$, rank $2n$. Some problems of future research are formulated.

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arXiv2020

Introduction to Anderson t-motives: a survey

This is a survey on Anderson t-motives -- high-dimensional generalizations of Drinfeld modules. They are the functional field analogs of abelian varieties with multiplication by an imaginary quadratic field. We describe their lattices, their groups $H^1$ and $H_1$, their tensor products, the duality functor and the duality theorem, endomorphisms of Drinfeld modules in finite characteristic, and their L-functions of a certain type. Further on, we introduce the notion of affine equations, $T$-divisible $\Bbb F_q[[T]]$-modules, holonomic sequences in the functional field case, analogs of Siegel matrices as elements of flag varieties, and some other notions (to be continued). Many examples of explicit calculations are given, some elementary research problems are stated. Some results (Sections 16; 19) are apparently new.

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arXiv2018

Lattice of the dual of an Anderson $t$-motive in terms of a map to a flag variety

Let $M$ be an uniformizable Anderson $t$-motive of rank $r$, $L$ its lattice and $l_*:=\{l_1,\dots, l_r\}$ its basis. We define a map $\delta$ from the set of these bases to a flag variety (the present text gives the definition of $\delta$ only for elements of the maximal Schubert cell, and for few other cases). If $l_*$ belongs to the maximal Schubert cell then $\delta(l_*)$ is described as a set of matrices parametrized by integer points of a tetrahedron; they are called the Siegel element of $l_*$. We give explicit formulas for a Siegel element for $M'$ -- the dual of $M$. As a by-product, we get another proof of the theorem that the lattice of $M'$ is the dual of the lattice of $M$, independent of the proof obtained by U. Hartl and A.-K. Juschka. Generalizations of this result to non-maximal Schubert cells, other tensor operations (tensor product and Hom) and t-motives having non-trivial endomorphism rings, are subjects of further research.

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arXiv2009

Anderson T-motives and abelian varieties with MIQF: results coming from an analogy

Analogy between Anderson T-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: 1. A description of abelian varieties with MIQF of dimension $r$ and signature $(n, r-n)$ in terms of "lattices" of dimension $r$ in $\Bbb C^n$; 2. A construction of exterior powers of abelian varieties with MIQF having $n=1$.

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arXiv2007

Kolyvagin's trace relations for Siegel sixfolds

In his earlier paper the author offered a program of generalization of Kolyvagin's result of finiteness of SH to the case of some motives which are quotients of cohomology motives of Shimura varieties. The present paper is devoted to the first step of this program -- finding of an analog of Kolyvagin's trace relations for Siegel sixfolds and for the Hecke correspondence related to the matrix diag $(p,1,1,p,p^2,p^2)$. This is the first non-trivial case where we can expect that this program can be realised. Some results for other types of Siegel varieties and Hecke correspondences are obtained. Ideas and methods of the present paper open a large new area of research: results given here constitute a tiny part of what can be done.

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arXiv2007

Duality of Anderson T-motives

Let $M$ be a T-motive. We introduce the notion of duality for $M$. Main results of the paper (we consider uniformizable $M$ over $F_q[T]$ of rank $r$, dimension $n$, whose nilpotent operator $N$ is 0): 1. Algebraic duality implies analytic duality (Theorem 5). Explicitly, this means that the lattice of the dual of $M$ is the dual of the lattice of $M$, i.e. the transposed of a Siegel matrix of $M$ is a Siegel matrix of the dual of $M$. 2. Let $n=r-1$. There is a 1 -- 1 correspondence between pure T-motives (all they are uniformizable), and lattices of rank $r$ in $C^n$ having dual (Corollary 8.4).

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