arXiv subjects

Vasily Golyshev

Publications and source records attributed to Vasily Golyshev.

10 recordsLinked to original sources

arXiv2026

The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves

For certain integers $u$, we investigate the 2-part of the Bloch-Kato conjecture for $L(E_u,2)$, where $E_u: y^2=x(x+1)(x+u^2)$ is part of a (twisted) Legendre family that is 2-isogenous to a family studied by Boyd. For this, we first work out the corresponding 2-parts of the Tamagawa factors and Galois invariants. Then we give an explicit description of the 2-torsion in the Selmer group $H_f^1(\mathbb{Q},E_u[2^\infty](-1))$. We construct a specific element in the kernel of the tame symbol for $K_2$ on an integral model of $E_u$, with non-vanishing real and 2-adic regulators. Using techniques involving the norm residue isomorphism of Merkur'ev-Suslin, we prove indivisibility of this element by 2 in that kernel, even modulo torsion, even though it is explicitly divisible by 2 in the kernel of the tame symbol for $K_2$ on $E_u$. We also bound the 2-divisibility of the images of these elements under the 2-adic regulator map. Finally, in many cases we investigate numerically the validity of the 2-part of the Bloch-Kato conjecture.

math.NT↗

arXiv2025

Fibering out Calabi-Yau motives

We prove the modularity of mixed periods associated with singular fibers of specific families of Calabi-Yau threefolds. This is done by "fibering out", i.e. by expressing these periods as integrals of periods of families of K3 surfaces and by using modularity properties of the latter. Besides classical periods of holomorphic modular forms and meromorphic modular forms with vanishing residues, the computations lead to new interesting periods associated with meromorphic modular forms with non-vanishing residues as well as contours between CM points.

math.AG↗

arXiv2024

The arithmetic of Calabi-Yau motives and mobile higher regulators

We construct elements in the motivic cohomology of certain rank 4 weight 3 Calabi--Yau motives, and write down explicit expressions for the regulators of these elements in the context of conjectures on $L$-values such as those of Beilinson or Bloch-Kato. We apply a combination of three ideas: (i) that a motive can be made to vary in a family in such a way that a desired motivic cohomology class is realized by relative cohomology; (ii)~that there are ways to construct higher-rank (such as $2\times 2$) regulators from a single family; and (iii) that one can arrange elements in $H^4_{\text{Mot}}(X,\mathbb{Z}(p))$ with different $p$'s by choosing hypergeometric families with different local exponents. Following background material on Hodge theory, algebraic cycles, differential equations, and hypergeometric variations, we work out two cases in detail where $p=3,4$. Regarding our Calabi-Yau motives $X_t$ as fibers in a suitable total space $\mathcal{X}_U\to U\subset \mathbb{P}^1$, each Hodge class in $\mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^4(\mathcal{X}_U,\mathbb{Q}(p)))$ produces a family of extension classes in $\mathrm{Ext}^1_{\mathrm{MHS}}(\mathbb{Q}(0),H^3(X_t,\mathbb{Q}(p)))$ called a normal function. Our main results for these cases, which are essentially independent, are the explicit computation of the normal functions, and the construction of motivic cohomology cycles realizing the Hodge classes, thereby proving Beilinson's Hodge-type conjecture and providing the first numerical checks of Beilinson's conjectures on special values of $L$-functions for such motives.

math.AG↗

arXiv2023

Apéry extensions

The Apéry numbers of Fano varieties are asymptotic invariants of their quantum differential equations. In this paper, we initiate a program to exhibit these invariants as (mirror to) limiting extension classes of higher cycles on the associated Landau-Ginzburg models -- and thus, in particular, as periods. We also construct an ``Apéry motive'', whose mixed Hodge structure is shown, as an application of the decomposition theorem, to contain the limiting extension classes in question. Using a new technical result on the inhomogeneous Picard-Fuchs equations satisfied by higher normal functions, we illustrate this proposal with detailed calculations for LG-models mirror to several Fano threefolds. By describing the ``elementary'' Apéry numbers in terms of regulators of higher cycles (i.e., algebraic $K$-theory/motivic cohomology classes), we obtain a satisfying explanation of their arithmetic properties. Indeed, in each case, the LG-models are modular families of $K3$ surfaces, and the distinction between multiples of $ζ(2)$ and $ζ(3)$ (or $(2π\mathbf{i})^3$) translates ultimately into one between algebraic $K_1$ and $K_3$ of the family.

math.AG↗

arXiv2015

Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class G_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by G_F Ch(E_i) for an exceptional collection {E_i} in the derived category of coherent sheaves D^b_coh(F). Gamma Conjecture II refines part (3) of Dubrovin's conjecture. We prove Gamma Conjectures for projective spaces and Grassmannians.

math.AG↗

arXiv2012

Mirror Symmetry and Fano Manifolds

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas.

math.AG↗

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.

math.NT↗

arXiv2009

Mirror symmetry for abelian varieties

We work out the notion of mirror symmetry for abelian varieties and study its properties. Our construction are based on the correspondence between two $Q$--algebraic groups. One is the Hodge (or special Mumford--Tate) group. The second group $\bar{Spin(A)}$ is defined as follows: the group of autoequivalences of the bounded derived category of coherent sheaves acts on the total cohomology $H(A,Q)$ of an abelian variety $A$ via algebraic correspondences. The group $\bar{Spin(A)}$ is now the Zariski closure of its image in $GL(H(A,Q))$. Our constructions are compatible with the picture of mirror symmetry sketched by Kontsevich, Morrison, and others.

math.AG↗

arXiv2007

Fuchsian equations of type DN

We prove that a generic differential operator of type DN is irreducible, regular, (anti)self-adjoint, and has quasiunipotent local monodromies. We prove that the defining matrix of a DN operator can be recovered from the expression of the operator as a polynomial in t and d/dt.

math.AG↗

arXiv2006

Hypergeometric Equations and Weighted Projective Spaces

We compute the Hodge numbers of the variation of Hodge structure of the middle cohomology (with compact support) of the Landau-Ginzburg model dual to a weighted projective space. We state a conjectural formula for the Hodge numbers of general hypergeometric variations. We show that a general Landau-Ginzburg fibre is birational to a Calabi-Yau variety if and only if a general anticanonical section of the weighted projective space is Calabi-Yau. We analyse the 104 weighted 3-spaces with canonical singularities, and show that a general anticanonical section is not a K3 surface exactly in those 9 cases where a generic Landau-Ginzburg fibre is an elliptic surface of Kodaira dimension 1.

math.AG↗