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Lynne H. Walling - One of the best experts on this subject based on the ideXlab platform.
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Hecke operators on Hilbert-Siegel Theta Series
arXiv: Number Theory, 2019Co-Authors: Dan Fretwell, Lynne H. WallingAbstract:We consider the action of Hecke-type operators on Hilbert-Siegel Theta Series attached to lattices of even rank. We show that average Hilbert-Siegel Theta Series are eigenforms for these operators, and we explicitly compute the eigenvalues.
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ACTION OF HECKE OPERATORS ON SIEGEL Theta Series, II
International Journal of Number Theory, 2008Co-Authors: Lynne H. WallingAbstract:We apply the Hecke operators T(p)2 and (1 ≤ j ≤ n ≤ 2k) to a degree n Theta Series attached to a rank 2k ℤ-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the Theta Series under these Hecke operators as a sum of Theta Series attached to certain sublattices of , thereby generalizing the Eichler Commutation Relation. We then show that the average Theta Series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average Theta Series, extending previous work where we had the restrictions that χ(p) = 1 and n ≤ k. We also show that for j > k when χ(p) = 1, and for j ≥ k when χ(p) = -1, and that θ(gen L) is an eigenform for T(p)2.
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Action of Hecke operators on Siegel Theta Series II
arXiv: Number Theory, 2007Co-Authors: Lynne H. WallingAbstract:Given a Siegel Theta Series and a prime p not dividing the level of the Theta Series, we apply to the Theta Series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average Theta Series is an eigenform and we compute the eigenvalues.
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ACTION OF HECKE OPERATORS ON SIEGEL Theta Series I
International Journal of Number Theory, 2006Co-Authors: Lynne H. WallingAbstract:We apply the Hecke operators T(p) and [Formula: see text] to a degree n Theta Series attached to a rank 2k ℤ-lattice L, n ≤ k, equipped with a positive definite quadratic form in the case that L/pL is hyperbolic. We show that the image of the Theta Series under these Hecke operators can be realized as a sum of Theta Series attached to certain closely related lattices, thereby generalizing the Eichler Commutation Relation (similar to some work of Freitag and of Yoshida). We then show that the average Theta Series (averaging over isometry classes in a given genus) is an eigenform for these operators. We show the eigenvalue for T(p) is ∊(k - n, n), and the eigenvalue for T′j(p2) (a specific linear combination of T0(p2),…,Tj(p2)) is pj(k-n)+j(j-1)/2β(n,j)∊(k-j,j) where β(*,*), ∊(*,*) are elementary functions (defined below).
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The Eichler Commutation Relation for Theta Series with spherical harmonics
Acta Arithmetica, 1993Co-Authors: Lynne H. WallingAbstract:It is well known that classical Theta Series which are attached to positive definite rational quadratic forms yield elliptic modular forms, and linear combinations of Theta Series attached to lattices in a fixed genus can yield both cusp forms and Eisenstein Series whose weight is one-half the rank of the quadratic form. In contrast, generalized Theta Series—those augmented with a spherical harmonic polynomial—will always yield cusp forms whose weight is increased by the degree of the spherical harmonic. A recent demonstration of the far-reaching importance of generalized Theta Series is Hijikata, Pizer and Shemanske’s solution to Eichler’s Basis Problem [4] (cf. [2]) in which character twists of such Theta Series are used to provide a basis for the space of newforms. In this paper we consider Theta Series with spherical harmonics over a totally real number field. We show that such Theta Series are Hilbert modular cusp forms whose weight is integral or half-integral, depending on the rank of the associated lattice. We explicitly describe the action of the Hecke operators on these Theta Series in terms of other Theta Series, yielding a generalization of the well-known Eichler Commutation Relation. Finally, we use these Theta Series to construct Hilbert modular forms which are invariant under a subalgebra of the Hecke algebra. We are able to show that if the quadratic form has rankm and the spherical harmonic has degree l, then the Theta Series attached to the genus of a lattice is identically zero whenever l is small relative to m; in particular, the associated collection of Theta Series are linearly dependent.
Andrew Waldron - One of the best experts on this subject based on the ideXlab platform.
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Minimal Representations, Spherical Vectors¶and Exceptional Theta Series
Communications in Mathematical Physics, 2002Co-Authors: David Kazhdan, Boris Pioline, Andrew WaldronAbstract:Theta Series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrodinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic Theta Series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the Theta Series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.
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minimal representations spherical vectors and exceptional Theta Series
Communications in Mathematical Physics, 2002Co-Authors: David Kazhdan, Boris Pioline, Andrew WaldronAbstract:Theta Series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrodinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic Theta Series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the Theta Series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.
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Minimal representations, spherical vectors, and exceptional Theta Series I
Communications in Mathematical Physics, 2002Co-Authors: David Kazhdan, B. Pioline, Andrew WaldronAbstract:Theta Series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group $G$ is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of $G$, generalizing the Schr\\ödinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic Theta Series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the $p$-adic number fields provides the summation measure in the Theta Series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born-Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.
Boris Pioline - One of the best experts on this subject based on the ideXlab platform.
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Indefinite Theta Series and generalized error functions
Selecta Mathematica, 2018Co-Authors: Sergei Alexandrov, Jan Manschot, Sibasish Banerjee, Boris PiolineAbstract:Theta Series for lattices with indefinite signature $$(n_+,n_-)$$(n+,n-) arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($$n_+=1$$n+=1), but have remained obscure when $$n_+\ge 2$$n+≥2. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of ‘conformal’ holomorphic Theta Series ($$n_+=2$$n+=2). As an application, we determine the modular properties of a generalized Appell–Lerch sum attached to the lattice $${{\text {A}}}_2$$A2, which arose in the study of rank 3 vector bundles on $$\mathbb {P}^2$$P2. The extension of our method to $$n_+>2$$n+>2 is outlined.
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Indefinite Theta Series and generalized error functions
Selecta Mathematica (New Series), 2018Co-Authors: Sergey Alexandrov, Jan Manschot, Sibasish Banerjee, Boris PiolineAbstract:Theta Series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic Theta Series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice ${\operatorname A}_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.
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Indefinite Theta Series and generalized error functions
Selecta Mathematica, 2018Co-Authors: Sergei Alexandrov, Jan Manschot, Sibasish Banerjee, Boris PiolineAbstract:Theta Series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic Theta Series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice $A_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.
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Theta Series, wall-crossing and quantum dilogarithm identities
Letters in Mathematical Physics, 2016Co-Authors: Sergey Alexandrov, Boris PiolineAbstract:Motivated by mathematical structures which arise in string vacua and gauge theories with N=2 supersymmetry, we study the properties of certain generalized Theta Series which appear as Fourier coefficients of functions on a twisted torus. In Calabi-Yau string vacua, such Theta Series encode instanton corrections from $k$ Neveu-Schwarz five-branes. The Theta Series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich-Soibelman symplectomorphisms. This effectively provides a quantum version of these transformations, where the quantization parameter is inversely proportional to the five-brane charge $k$. Consistency with wall-crossing implies a new five-term relation for Faddeev's quantum dilogarithm $\Phi_b$ at $b=1$, which we prove. By allowing the torus to be non-commutative, we obtain a more general five-term relation valid for arbitrary $b$ and $k$, which may be relevant for the physics of five-branes at finite chemical potential for angular momentum.
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D3-instantons, Mock Theta Series and Twistors
Journal of High Energy Physics, 2013Co-Authors: Sergey Alexandrov, Jan Manschot, Boris PiolineAbstract:The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2,Z). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2,Z) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain Theta Series, which can be canceled by a contact transformation generated by a holomorphic mock Theta Series.
David Kazhdan - One of the best experts on this subject based on the ideXlab platform.
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Minimal Representations, Spherical Vectors¶and Exceptional Theta Series
Communications in Mathematical Physics, 2002Co-Authors: David Kazhdan, Boris Pioline, Andrew WaldronAbstract:Theta Series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrodinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic Theta Series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the Theta Series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.
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minimal representations spherical vectors and exceptional Theta Series
Communications in Mathematical Physics, 2002Co-Authors: David Kazhdan, Boris Pioline, Andrew WaldronAbstract:Theta Series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of G, generalizing the Schrodinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic Theta Series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the p-adic number fields provides the summation measure in the Theta Series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born–Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.
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Minimal representations, spherical vectors, and exceptional Theta Series I
Communications in Mathematical Physics, 2002Co-Authors: David Kazhdan, B. Pioline, Andrew WaldronAbstract:Theta Series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantum supermembrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group $G$ is split (or complex) and simply laced. Specifically, we review and construct explicitly the minimal representation of $G$, generalizing the Schr\\ödinger representation of symplectic groups. We compute the spherical vector in this representation, i.e. the wave function invariant under the maximal compact subgroup, which plays the role of the summand in the automorphic Theta Series. We also determine the spherical vector over the complex field. We outline how the spherical vector over the $p$-adic number fields provides the summation measure in the Theta Series, postponing its determination to a sequel of this work. The simplicity of our result is suggestive of a new Born-Infeld-like description of the membrane where U-duality is realized non-linearly. Our results may also be used in constructing quantum mechanical systems with spectrum generating symmetries.
Riccardo Salvati Manni - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Theta Series and the Kodaira dimension of $\mathcal{A}_6$
arXiv: Algebraic Geometry, 2019Co-Authors: Moritz Dittmann, Riccardo Salvati Manni, Nils R. ScheithauerAbstract:We construct a basis of the space ${\text S}_{14}({\text{Sp}}_{12}({\mathbb Z}))$ of Siegel cusp forms of degree $6$ and weight $14$ consisting of harmonic Theta Series. One of these functions has vanishing order $2$ at the boundary which implies that the Kodaira dimension of $\mathcal{A}_6$ is non-negative.
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Octavic Theta Series
arXiv: Algebraic Geometry, 2014Co-Authors: Eberhard Freitag, Riccardo Salvati ManniAbstract:Let L be the even unimodular lattice of signature (2,10), In the paper [FS] we considered the subgroup O(L)^+ of index two in the orthogonal group. It acts biholomorphically on a ten dimensional tube domain H_{10}. We found a 715 dimensional space of modular forms with respect to the principal congruence subgroup of level two O^+(L)[2]. It defines an everywhere regular birational embedding of the related modular variety into the 714 dimensional projective space. In this paper, we prove that this space of orthogonal modular forms is related to a space of Theta Series. The main tool is a modular embedding of H_{10} into the Siegel half space of degree 16. As a consequence the modular forms in the 715 dimensional space can be obtained as restrictions of the simplest among all Theta Series.
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ON THE Theta Series ATTACHED TO $D_{m}^{+}$–LATTICES
International Journal of Number Theory, 2006Co-Authors: Winfried Kohnen, Riccardo Salvati ManniAbstract:We show that the Theta Series attached to the -lattice for any positive integer divisible by 8 can be explicitly expressed as a finite rational linear combination of products of two Eisenstein Series.
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On Theta Series vanishing at ∞ and related lattices
Acta Arithmetica, 2006Co-Authors: Christine Bachoc, Riccardo Salvati ManniAbstract:We consider Theta Series with the highest possible order of vanishing at infinity when the level is a power of 2 or 3 and the lattices associated to these Theta Series. We prove that these lattices are constructed from binary, respectively ternary codes.
