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Vector-valued Hirzebruch–Zagier series and class number sums

Vector-valued Hirzebruch–Zagier series and class number sums For any number $$m \equiv 0,1 \, (4)$$ m ≡ 0 , 1 ( 4 ) , we correct the generating function of Hurwitz class number sums $$\sum _r H(4n - mr^2)$$ ∑ r H ( 4 n - m r 2 ) to a modular form (or quasimodular form if m is a square) of weight two for the Weil representation attached to a binary quadratic form of discriminant m and determine its behavior in the Petersson scalar product. This modular form arises through holomorphic projection of the zero-value of a nonholomorphic Jacobi Eisenstein series of index 1 / m. When m is prime, we recover the classical Hirzebruch–Zagier series whose coefficients are intersection numbers of curves on a Hilbert modular surface. Finally, we calculate certain sums over class numbers by comparing coefficients with an Eisenstein series. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

Vector-valued Hirzebruch–Zagier series and class number sums

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References (18)

Publisher
Springer Journals
Copyright
Copyright © 2018 by SpringerNature
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
DOI
10.1007/s40687-018-0142-4
Publisher site
See Article on Publisher Site

Abstract

For any number $$m \equiv 0,1 \, (4)$$ m ≡ 0 , 1 ( 4 ) , we correct the generating function of Hurwitz class number sums $$\sum _r H(4n - mr^2)$$ ∑ r H ( 4 n - m r 2 ) to a modular form (or quasimodular form if m is a square) of weight two for the Weil representation attached to a binary quadratic form of discriminant m and determine its behavior in the Petersson scalar product. This modular form arises through holomorphic projection of the zero-value of a nonholomorphic Jacobi Eisenstein series of index 1 / m. When m is prime, we recover the classical Hirzebruch–Zagier series whose coefficients are intersection numbers of curves on a Hilbert modular surface. Finally, we calculate certain sums over class numbers by comparing coefficients with an Eisenstein series.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: May 7, 2018

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