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## Modular forms of orthogonal type and Jacobi theta-series

In this paper we consider Jacobi forms of half-integral index for any positive definite lattice L (classical Jacobi forms from the book of Eichler and Zagier correspond to the lattice A_1=<2>). We give a lot of examples of Jacobi forms of singular and critical weights for root systems using Jacobi theta-series. We describe the Jacobi lifting for Jacobi forms of half-integral indices. In some case it gives additive lifting construction of new reflective modular forms.

In this paper we consider moduli spaces of polarized and numerically polarized Enriques surfaces. The moduli spaces of numerically polarized Enriques surfaces can be described as open subsets of orthogonal modular varieties of dimension 10. One of the consequences of our description is that there are only finitely many isomorphism classes of moduli spaces of polarized and numerically polarized Enriques surfaces. We use modular forms to prove for a number of small degrees that the Kodaira dimension of the moduli space of numerically polarized Enriques surfaces is negative. Finally we prove that there are infinitely many polarizatons for which the moduli space of numerically polarized Enriques surfaces is birational to the moduli space of unpolarized Enriques surfaces with a level 2 structure.

Abstract This mini course is an additional part to my semester course on the theory of Jacobi modular forms given at the mathematical department of NRU HSE in Moscow (see Gritsenko Jacobi modular forms: 30 ans après; COURSERA (12 lectures and seminars), 2017–2019). This additional part contains some applications of Jacobi modular forms to the theory of elliptic genera and Witten genus. The subject of this course is related to my old talk given in Japan (see Gritsenko (Proc Symp “Automorphic forms and L-functions” 1103:71–85, 1999)).

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference "Black holes, Black Rings and Modular Forms" (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.

This book offers an introduction to the research in several recently discovered and actively developing mathematical and mathematical physics areas. It focuses on: 1) Feynman integrals and modular functions, 2) hyperbolic and Lorentzian Kac-Moody algebras, related automorphic forms and applications to quantum gravity, 3) superconformal indices and elliptic hypergeometric integrals, related instanton partition functions, 4) moonshine, its arithmetic aspects, Jacobi forms, elliptic genus, and string theory, and 5) theory and applications of the elliptic Painleve equation, and aspects of Painleve equations in quantum field theories. All the topics covered are related to various partition functions emerging in different supersymmetric and ordinary quantum field theories in curved space-times of different (d=2,3,…,6) dimensions. Presenting multidisciplinary methods (localization, Borcherds products, theory of special functions, Cremona maps, etc) for treating a range of partition functions, the book is intended for graduate students and young postdocs interested in the interaction between quantum field theory and mathematics related to automorphic forms, representation theory, number theory and geometry, and mirror symmetry.

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebras for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G2, and also establish the relations of our extremal degrees to minimal q-degrees of the extremal terms of the Kostant q-partition function; they coincide with the latter only for some root systems. © 2015 Elsevier Inc.

We show how to analyse the cusp forms of small weights for the moduli spaces of polarised K3 surfaces with these degrees 2d.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.