On the Fourier–Jacobi expansion of the unitary Kudla lift

This chapter first studies the automorphic forms that are defined as global sections of certain invertible sheaves on the toroidal compactifications. The local structures of toroidal compactifications lead naturally to the theory of Fourier–Jacobi expansions and the Fourier–Jacobi expansion principle. The chapter also obtains the algebraic construction of arithmetic minimal compactifications (of the coarse moduli associated with moduli problems), which are projective normal schemes defined over the same integral bases as the moduli problems are. As a by-product of codimension counting, we obtain Koecher's principle for arithmetic automorphic forms (of naive parallel weights). Furthermore, this chapter shows the projectivity of a large class of arithmetic toroidal compactifications by realizing them as normalizations of blowups of the corresponding minimal compactifications.

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A Fourier-Neumann series and its application to the reduction of triple cosine series

Glasgow Mathematical Journal ◽

10.1017/s0017089500001968 ◽

1973 ◽

Vol 14 (2) ◽

pp. 198-201 ◽

Cited By ~ 1

Author(s):

C. J. Tranter ◽

J. C. Cooke

Keyword(s):

Generating Function ◽

Neumann Series ◽

Definite Integral ◽

Cosine Series ◽

Jacobi Expansion ◽

Image Position ◽

The Right

The Jacobi expansionis well known and easily obtained from the generating function of the Besselcoefficients. The sum of the series on the right of equation (1) when sin (n+½)x is replaced by cos (n+½)x cannot be found in this way but it can be expressed in terms of a definite integral as shown below. The result so obtained is useful in reducing certain triple cosine series to dual series and so simplifying the solution given by one of us for such series in an earlier paper [1].

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On the Coble quartic and Fourier–Jacobi expansion of theta relations

International Journal of Mathematics ◽

10.1142/s0129167x15500196 ◽

2015 ◽

Vol 26 (02) ◽

pp. 1550019

Author(s):

Francesco Dalla Piazza ◽

Riccardo Salvati Manni

Keyword(s):

Algebraic Geometry ◽

American Mathematical Society ◽

Theta Functions ◽

Mathematical Society ◽

Jacobi Expansion ◽

The Ideal

In [Q. Ren, S. Sam, G. Schrader and B. Sturmfels, The universal Kummer threefold, Experiment Math.22(3) (2013) 327–362], the authors conjectured equations for the universal Kummer variety in genus 3 case. Although, most of these equations are obtained from the Fourier–Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic, cf. [A. Coble, Algebraic Geometry and Theta Functions, American Mathematical Society Colloquium Publications, Vol. 10 (American Mathematical Society, 1929)] was obtained differently, cf. [S. Grushevsky and R. Salvati Manni, On Coble's quartic, preprint (2012), arXiv:1212.1895] too. The aim of this paper is to show that Coble's quartic can be obtained as Fourier–Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in [Experiment Math.22(3) (2013) 327–362].

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