arXiv subjects

Masha Vlasenko

Publications and source records attributed to Masha Vlasenko.

At least 19 recordsLinked to original sources

arXiv2024

Cohomology and congruences

These are notes of my lecture courses given in the summer of 2024 in the School on Number Theory and Physics at ICTP in Trieste and in the 27th Brazilian Algebra Meeting at IME-USP in S\~ao Paulo. We give an elementary account of $p$-adic methods in de Rham cohomology of algebraic hypersurfaces with explicit examples and applications in number theory and combinatorics. These lectures are based on the series of our joint papers with Frits Beukers entitled \emph{Dwork crystals} (\cite{DCI,DCII,DCIII}). These methods also have applications in mathematical physics and arithmetic geometry (\cite{IN,Cartier0}), which we overview here towards the end. I am grateful to the organisers of both schools and to the participants of my courses whose questions stimulated writing these notes.

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arXiv2023

Frobenius structure and $p$-adic zeta values

For differential operators of Calabi-Yau type, Candelas, de la Ossa and van Straten conjecture the appearance of $p$-adic zeta values in the matrix entries of their $p$-adic Frobenius structure expressed in the standard basis of solutions near a MUM-point. We prove that this phenomenon holds for simplicial and hyperoctahedral families of Calabi-Yau hypersurfaces in $n$ dimensions, in which case the Frobenius matrix entries are rational linear combinations of products of $\zeta_p(k)$ with $1 < k < n$.

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arXiv2022

Frobenius constants for families of elliptic curves

The paper deals with a class of periods, Frobenius constants, which describe monodromy of Frobenius solutions of differential equations arising in algebraic geometry. We represent Frobenius constants related to families of elliptic curves as iterated integrals of modular forms. Using the theory of periods of modular forms, we then witness some of these constants in terms of zeta values.

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arXiv2021

On $p$-integrality of instanton numbers

We show integrality of instanton numbers in several key examples of mirror symmetry. Our methods are essentially elementary, they are based on our previous work in the series of papers called Dwork crystals I, II and III.

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arXiv2021

Dwork crystals III: from excellent Frobenius lifts towards supercongruences

This paper is a continuation of our Dwork crystals series. Here we exploit the Cartier operation to prove supercongruences for expansion coefficients of rational functions. In the process it appears that excellent Frobenius lifts are a driving force behind supercongruences. Originally introduced by Dwork, these excellent lifts have occurred rather infrequently in the literature, and only in the context of families of elliptic curves and abelian varieties. In the final sections of this paper we present a list of examples that occur in the case of families of Calabi-Yau varieties.

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arXiv2019

Hodge structures and differential operators

This text can be considered as a non-technical and arithmetically motivated introduction to the definition of the limiting mixed Hodge structure. We state several assertions in terms natural to the classical theory of ordinary differential operators and prove them using elementary arguments.

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arXiv2019

Gamma functions, monodromy and Frobenius constants

In their paper on the gamma conjecture in mirror symmetry, Golyshev and Zagier introduce what we refer to as Frobenius constants associated to an ordinary linear differential operator L with a reflection type singularity. These numbers describe the variation around the reflection point of Frobenius solutions to L defined near other singular points. Golyshev and Zagier show that in certain geometric cases Frobenius constants are periods, and they raise the question quite generally how to describe these numbers motivically. In this paper we give a relation between Frobenius constants and Taylor coefficients of generalized gamma functions, from which it follows that Frobenius constants of Picard--Fuchs differential operators are periods. We also study the relation between these constants and periods of limiting Hodge structures. This is a major revision of the previous version of the manuscript. The notion of Frobenius constants and our main result are extended to the general case of regular singularities with any sets of local exponents. In addition, the generating function of Frobenius constants is given explicitly for all hypergeometric connections.

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arXiv2019

Dwork crystals II

We give a generalization of $p$-adic congruences for truncated period functions, that were originally discovered for a class of hypergeometric functions by Bernard Dwork.

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arXiv2019

Dwork crystals I

We present an elementary elaboration of Dwork's idea of explicit $p$-adic limit formulas for zeta functions of toric hypersurfaces.

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arXiv2018

Formal groups and congruences

We give a criterion of integrality of an one-dimensional formal group law in terms of congruences satisfied by the coefficients of the canonical invariant differential. For an integral formal group law a p-adic analytic formula for the local characteristic polynomial at p is given. We demonstrate applications of our results to formal group laws attached to L-functions, Artin--Mazur formal groups of algebraic varieties and hypergeometric formal group laws. (This is the final version submitted to a journal; new parts are added in the section on hypergeometric formal group laws.)

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arXiv2018

Higher Hasse--Witt matrices

We prove a number of p-adic congruences for the coefficients of powers of a multivariate polynomial f(x) with coefficients in a ring R of characteristic zero. If the Hasse--Witt operation is invertible, our congruences yield p-adic limit formulas which conjecturally describe the Gauss--Manin connection and the Frobenius operator on the unit-root crystal attached to f(x). As a second application, we associate with f(x) formal group laws over R. Under certain assumptions these formal group laws are coordinalizations of the Artin--Mazur functors. (This is a final version which we send for a publication.)

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arXiv2015

Explicit p-adic unit-root formulas for hypersurfaces

We prove a statement on p-adic continuity of matrices of coefficients of the logarithm of the Artin-Mazur formal group law associated to the middle cohomology of a hypersurface. As Jan Stienstra discovered in 1986, the entries of these matrices are coefficients of powers of the equation of the hypersurface, and in certain cases they satisfy congruences of Atkin and Swinnerton-Dyer type. These congruences imply that eigenvalues of our limiting matrices are eigenvalues of Frobenius of zero p-adic valuation on the middle crystalline cohomology of the fibre at p.

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arXiv2014

Linear Mahler Measures and Double L-values of Modular Forms

We consider the Mahler measure of the polynomial 1+x_1+x_2+x_3+x_4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the L-function of a modular modular form of weight 3. We prove that this Mahler measure is equal to a linear combination of double L-values of certain meromorphic modular forms of weight 4.

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arXiv2013

Dwork's congruences for the constant terms of powers of a Laurent polynomial

We prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a p-adic variable satisfies a non-trivial analytic continuation property, similar to what B. Dwork showed for a class of hypergeometric series.

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arXiv2012

Equations D3 and spectral elliptic curves

We study modular determinantal differential equations of orders 2 and 3. We show that the expansion of the analytic solution of a non-degenerate modular equation of type D3 over the rational numbers with respect to the natural parameter coincides, under certain assumptions, with the q-expansion of the newform of its spectral elliptic curve and therefore possesses a multiplicativity property. We compute the complete list of D3 equations with this multiplicativity property and relate it to Zagier's list of non-degenerate modular D2 equations.

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arXiv2012

k-Run Overpartitions and Mock Theta Functions

In this paper we introduce k-run overpartitions as natural analogs to partitions without k-sequences, which were first defined and studied by Holroyd, Liggett, and Romik. Following their work as well as that of Andrews, we prove a number of results for k-run overpartitions, beginning with a double summation q-hypergeometric series representation for the generating functions. In the special case of 1-run overpartitions we further relate the generating function to one of Ramanujan's mock theta functions. Finally, we describe the relationship between k-run overpartitions and certain sequences of random events, and use probabilistic estimates in order to determine the asymptotic growth behavior of the number of k-run overpartitions of size n.

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arXiv2011

Nahm's Conjecture: Asymptotic Computations and Counterexamples

We consider certain q-series depending on parameters (A,B,C), where A is a positive definite r times r matrix, B is an r-vector and C is a scalar, and ask when these q-series are modular forms. Werner Nahm conjectured a criterion for which A's can occur, in terms of torsion in the Bloch group. The conjecture was proved by Don Zagier and Michael Terhoeven for r=1. We develop their approach for r>1 and find several new examples of modular forms as well as some counterexamples to Nahm's conjecture.

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arXiv2011

Lines crossing a tetrahedron and the Bloch group

We consider a simple modification of the Chow group CH^2(Spec(k),3) using only linear subvarieties in affine spaces and show that it maps surjectively to the Bloch group B(k) for any infinite field k. We also describe the kernel of this map.

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