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Siegel modular forms of genus 2 with the simplest divisor
Proceedings of the London Mathematical Society. 2011. Vol. 102. No. 6. P. 1024-1052.
Gritsenko V., Cléry F.
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.
Gritsenko V., Poor C., Yuen D. S., Journal of Number Theory 2015 Vol. 148 P. 164-195
We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that constructs holomorphic Borcherds Products from certain Jacobi forms that are theta blocks without theta denominator. The proof uses generalized valuations from formal series to partially ordered abelian semigroups of closed convex sets. We present nine infinite families of paramodular Borcherds Products that are simultaneously Gritsenko ...
Added: February 26, 2015
Gritsenko V., Wang H., Sbornik Mathematics 2019 Vol. 210 No. 12
The problem on the construction of antisymmetric paramodular forms of canonical weight $3$ was open since 1996. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer surfaces associated to $(1,t)$-polarised abelian surfaces. In this paper, we construct the first infinite family of antisymmetric paramodular forms of weight $3$ as ...
Added: November 12, 2019
Gritsenko V., Ванг Х., Математический сборник 2019 Т. 210 № 12 С. 43-66
Задача о построении антисимметричных парамодульных форм канонического веса 3 была поставлена в 1996 году. Любая параболическая форма этого типа определяет каноническую диф- ференциальную форму на любой гладкой компактификации про- странство модулей куммеровых поверхностей, отвечающих (1,t)- поляризованным абелевым поверхностям. В этой статье мы строим первое бесконечное семейство антисимметричных парамодулярных форм веса 3 как автоморфные произведения Борчердса, чьи пер- вые коэффициенты Фурье–Якоби ...
Added: October 29, 2019
Gritsenko V., Cléry F., Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 2013 Vol. 83 No. 2 P. 187-217
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 ...
Added: February 26, 2015
Poor C., Gritsenko V., Yuen D., / Cornell University. Series arXiv "math". 2016.
We define an algebraic set in 23 dimensional projective space whose Q-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree two paramodular groups. Weight 3 examples provide ...
Added: September 21, 2016
Gritsenko V., Ванг Х., Успехи математических наук 2017 Т. 72 № 5 С. 191-192
In this paper we prove the conjecture above in the last case of known theta-blocks of weight 2. This gives a new intereting series of Borcherds products of weight 2. ...
Added: October 11, 2017
Valery Gritsenko, Wang H., Russian Mathematical Surveys 2017 Vol. 72 No. 5 P. 968-970
In this paper we prove the indicated conjecture in the last case of known infinite series of theta-blocks of weight 2. ...
Added: January 29, 2018
Gritsenko V., Poor C., Yuen D. S., International Mathematics Research Notices 2020 Vol. 2020 No. 20 P. 6926-6946
We define an algebraic set in 23-dimensional projective space whose ℚ-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight 3examples provide antisymmetric canonical differential forms on ...
Added: October 29, 2019
Adler D., Функциональный анализ и его приложения 2020 Т. 54 № 3 С. 8-25
We prove the polynomiality of the bigraded ring $J_{*,*}^{w, W}(F_4)$ of weak Jacobi forms for the root system $F_4$ which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms related to the root ...
Added: November 6, 2020
Gritsenko V., Wang H., European Journal of Mathematics 2018 Vol. 4 No. 2 P. 561-584
We show that the eighth power of the Jacobi triple product is a Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function
$\Delta(\tau)=\eta^{24}(\tau)$, $\eta^{12}(\tau)$ and $\eta^{8}(\tau)$ in terms ...
Added: October 11, 2017
Sakharova N., Arnold Mathematical Journal 2018 Vol. 4 No. 3-4 P. 301-313
In this paper we construct the modular Cauchy kernel $\Xi_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=\gamma z_1, ~\gamma \in \Gamma_0(N) \right\}.$$
The function $\Xi_N(z_1, z_2)$ is used in two cases and for ...
Added: November 23, 2018
Adler D., Gritsenko V., / Cornell University. Series math "arxiv.org". 2019.
We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower corresponds to the tower of strongly reflective modular forms on the orthogonal groups of signature (2,n) ...
Added: November 5, 2019
Adler D., Gritsenko V., Journal of Geometry and Physics 2020
We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower corresponds to the tower of strongly reflective modular forms on the orthogonal groups of signature (2,n) ...
Added: October 30, 2019
Gritsenko V., / Cornell University. Series math "arxiv.org". 2012. No. 6503.
The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra ...
Added: March 3, 2015
Gritsenko V., Nikulin V. V., Transactions of the Moscow Mathematical Society 2017 Vol. 78 P. 75-83
Using our results about Lorentzian Kac--Moody algebras and arithmetic mirror symmetry, we give six series of examples of lattice-polarized K3 surfaces with automorphic discriminant. ...
Added: January 29, 2018
Adler D., / Cornell University. Series math "arxiv.org". 2020.
We prove the polynomiality of the bigraded ring J_{*,*}^{w, O}(F_4) of weak Jacobi forms for the root system F_4 which are invariant with respect to the corresponding Weyl group. This work is a continuation of the joint article with V.A. Gritsenko, where the structure of algebras of the weak Jacobi forms related to the root systems ...
Added: September 24, 2020
Gritsenko V., Wang H., Journal of Number Theory 2020 Vol. 214 P. 382-398
We determine the structure of the bigraded ring of weak Jacobi forms with integral Fourier coefficients. This ring is the target ring of a map generalising the Witten and elliptic genera and a partition function of (0, 2)-model in string theory. We also determine the structure of the graded ring of all weakly holomorphic Jacobi ...
Added: October 26, 2020
Sakharova N., / Cornell University. Series math "arxiv.org". 2018. No. 1802.03299.
In this paper we construct the modular Cauchy kernel $\Xi_N(z_1, z_2)$, i.e. the modular invariant function of two variables, $(z_1, z_2) \in \mathbb{H} \times \mathbb{H}$, with the first order pole on the curve $$D_N=\left\{(z_1, z_2) \in \mathbb{H} \times \mathbb{H}|~ z_2=\gamma z_1, ~\gamma \in \Gamma_0(N) \right\}.$$
The function $\Xi_N(z_1, z_2)$ is used in two cases and for ...
Added: February 23, 2018
Springer Publishing Company, 2020
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) ...
Added: September 9, 2020
Gritsenko V., Wang H., Proceedings of the American Mathematical Society 2020 Vol. 148 P. 1863-1878
In this paper we construct an infinite family of paramodular forms of weight 2 which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta-block conjecture of Gritsenko--Poor--Yuen (2013) related to the only known infinite series of theta-blocks of weight 2 and q-order 1. We also consider some applications of this result. ...
Added: October 29, 2019
Adler D., Gritsenko V., Journal of Geometry and Physics 2020 Vol. 150 P. 103616
We construct a tower of arithmetic generators of the bigraded polynomial ring J_{*,*}^{w, O}(D_n) of weak Jacobi modular forms invariant with respect to the full orthogonal group O(D_n) of the root lattice D_n for 2\le n\le 8. This tower corresponds to the tower of strongly reflective modular forms on the orthogonal groups of signature (2,n) ...
Added: November 1, 2019
191574970, Functional Analysis and Its Applications 2006 Vol. 40 No. 2 P. 81-90
It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...
Added: September 23, 2016
Losev A. S., Slizovskiy S., JETP Letters 2010 Vol. 91 P. 620-624
Added: February 27, 2013
Ilyashenko Y., Яковенко С. Ю., М. : МЦНМО, 2013
Предлагаемая книга—первый том двухтомной монографии, посвящённой аналитической теории дифференциальных уравнений.
В первой части этого тома излагается формальная и аналитическая теория нормальных форм и теорема о разрешении особенностей для векторных полей на плоскости.
Вторая часть посвящена алгебраически разрешимым локальным задачам теории аналитических дифференциальных уравнений , квадратичным векторным полям и проблеме локальной классификации ростков векторных полей в комплексной области ...
Added: February 5, 2014