scholarly journals WEIGHT CHANGING OPERATORS FOR AUTOMORPHIC FORMS ON GRASSMANNIANS AND DIFFERENTIAL PROPERTIES OF CERTAIN THETA LIFTS

2016 ◽  
Vol 228 ◽  
pp. 186-221 ◽  
Author(s):  
SHAUL ZEMEL
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Orthogonal Groups ◽  
Basic Properties ◽  
Differential Properties

We define weight changing operators for automorphic forms on Grassmannians, that is, on orthogonal groups, and investigate their basic properties. We then evaluate their action on theta kernels, and prove that theta lifts of modular forms, in which the theta kernel involves polynomials of a special type, have some interesting differential properties.

scholarly journals Some constructions of modular forms for the Weil representation of SL2(ℤ)

2015 ◽  
Vol 220 ◽  
pp. 1-43
Author(s):  
Nils R. Scheithauer
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Root Systems ◽  
Weil Representation ◽  
Orthogonal Groups ◽  
Singular Weight ◽  
Leech Lattice ◽  
Explicit Constructions

AbstractModular forms for the Weil representation of SL2(ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systemsand to the Leech lattice.

scholarly journals Some constructions of modular forms for the Weil representation of SL2(ℤ)

2015 ◽  
Vol 220 ◽  
pp. 1-43 ◽  
Author(s):  
Nils R. Scheithauer
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Root Systems ◽  
Weil Representation ◽  
Orthogonal Groups ◽  
Singular Weight ◽  
Leech Lattice ◽  
Explicit Constructions

AbstractModular forms for the Weil representation of SL2 (ℤ) play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac-Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems and to the Leech lattice.

scholarly journals On the Bloch–Kato conjecture for Hilbert modular forms

2021 ◽  
Author(s):  
Matteo Tamiozzo
Keyword(s):  
Modular Forms ◽  
Automorphic Forms ◽  
Euler System ◽  
Central Point ◽  
Shimura Curves ◽  
Hilbert Modular Forms ◽  
Reciprocity Laws ◽  
Hilbert Modular ◽  
Special Points

AbstractThe aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.

scholarly journals Higher pullbacks of modular forms on orthogonal groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Jacobi Form ◽  
Siegel Modular Forms ◽  
Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

scholarly journals Boundary behaviour of special cohomology classes arising from the Weil representation

2012 ◽  
Vol 12 (3) ◽  
pp. 571-634 ◽  
Author(s):  
Jens Funke ◽  
John Millson
Keyword(s):  
Modular Forms ◽  
Symmetric Spaces ◽  
Theta Functions ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Boundary Behaviour ◽  
Orthogonal Groups ◽  
Local Coefficients ◽  
Generating Series ◽  
Vector Valued

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

scholarly journals Integral canonical models for Spin Shimura varieties

2015 ◽  
Vol 152 (4) ◽  
pp. 769-824 ◽  
Author(s):  
Keerthi Madapusi Pera
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
K3 Surfaces ◽  
Shimura Varieties ◽  
Good Reduction ◽  
Intersection Numbers ◽  
Orthogonal Groups ◽  
Canonical Models ◽  
Tate Conjecture ◽  
Precise Sense

We construct regular integral canonical models for Shimura varieties attached to Spin and orthogonal groups at (possibly ramified) primes$p>2$where the level is not divisible by$p$. We exhibit these models as schemes of ‘relative PEL type’ over integral canonical models of larger Spin Shimura varieties with good reduction at$p$. Work of Vasiu–Zink then shows that the classical Kuga–Satake construction extends over the integral models and that the integral models we construct are canonical in a very precise sense. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla’s program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras

1991 ◽  
Vol 121 ◽  
pp. 35-96 ◽  
Author(s):  
Siegfried Böcherer ◽  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Orthogonal Group ◽  
Automorphic Forms ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Theta Correspondence ◽  
Quaternion Algebras ◽  
Metaplectic Group

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras II

1997 ◽  
Vol 147 ◽  
pp. 71-106 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze-Pillot
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Automorphic Forms ◽  
Multiplicative Group ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Quaternion Algebras

AbstractWe continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

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