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Dmitry Logachev

Publications and source records attributed to Dmitry Logachev.

12 recordsLinked to original sources
arXiv2022

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.

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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 $\Bbb F_q[\theta]$). 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 $X_{fun}(r,n)$.

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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.

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arXiv2018

$h^1\ne h_1$ 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.

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arXiv2016

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.

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arXiv2013

The set of jumping conics of a locally free sheaf of dimension 2 on $P^2$

We consider a locally free sheaf $F$ of dimension 2 on $P^2$. A conic $q$ on $P^2$ is called a jumping conic if the restriction of $F$ to $q$ is not the generic one. We prove that the set of jumping conics is the maximal determinantal variety of a skew form. Some properties of this skew form are found.

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arXiv2012

Resultantal varieties related to zeroes of L-functions of Carlitz modules

We show that there exists a connection between two types of objects: some kind of resultantal varieties over C, from one side, and varieties of twists of the tensor powers of the Carlitz module such that the order of 0 of its L-functions at infinity is a constant, from another side. Obtained results are only a starting point of a general theory. We can expect that it will be possible to prove that the order of 0 of these L-functions at 1 (i.e. the analytic rank of a twist) is not bounded --- this is the function field case analog of the famous conjecture on non-boundedness of rank of twists of an elliptic curve over Q. The paper contains a calculation of a non-trivial polynomial determinant.

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arXiv2011

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.

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arXiv2005

Reduction of a problem of finiteness of Tate-Shafarevich group to a result of Zagier type

Kolyvagin proved that the Tate-Shafarevich group of an elliptic curve over Q of analytic rank 0 or 1 is finite, and that its algebraic rank is equal to its analytic rank. A program of generalisation of this result to the case of some motives which are quotients of cohomology motives of Shimura varieties is offered. We prove some steps of this program, mainly for quotients of $H^7$ of Siegel sixfolds. For example, we "almost" find analogs of Kolyvagin's trace and reduction relations. Some steps of the present paper are new contribution, because they have no analogs in the case of elliptic curves. There are still a number of large gaps in the program. The most important of these gaps is a high-dimensional analog of a result of Zagier about ratios of Heegner points corresponding to different imaginary quadratic fields on a fixed elliptic curve. The author suggests to the readers to continue these investigations.

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arXiv2004

Fano threefolds of genus 6

This paper was written in 1982. Ideas and methods of "Clemens C.H., Griffiths Ph. The intermediate Jacobian of a cubic threefold" are applied to a Fano threefold X of genus 6 -- intersection of Grassmann sixfold with two hyperplanes and a quadric. We prove: 1. The Fano surface F(X) of X is smooth and irreducible. Hodge numbers and some other invariants of F(X) are calculated. 2. Tangent bundle theorem for X, and its geometric interpretation. It is shown that F(X) defines X uniquely. 3. The Abel - Jacobi map from the Albanese of F(X) to the middle Jacobian of X is an isogeny.

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arXiv2004

Relations between conjectural eigenvalues of Hecke operators on submotives of Siegel varieties

There exist conjectural formulas on relations between $L$-functions of submotives of Shimura varieties and automorphic representations of the corresponding reductive groups, due to Langlands -- Arthur. In the present paper these formulas are used in order to get explicit relations between eigenvalues of $p$-Hecke operators (generators of the $p$-Hecke algebra of $X$) on cohomology spaces of some of these submotives, for the case $X$ is a Siegel variety. Hence, this result is conjectural as well: methods related to counting points on reductions of $X$ using the Selberg trace formula are not used. It turns out that the above relations are linear, their coefficients are polynomials in $p$ which satisfy a simple recurrence formula. The same result can be easily obtained for any Shimura variety. This result is an intermediate step for a generalization of the Kolyvagin's theorem of finiteness of Tate -- Shafarevich group of elliptic curves of analytic rank 0, 1 over $Q$, to the case of submotives of other Shimura varieties, particularly of Siegel varieties of genus 3.

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