Die Nullstellen der Thetareihen zu positiven, binär-quadratischen Formen. (Zeros of theta series for positive binary quadratic forms)
type: Item
instance of: scholarly article
title: Die Nullstellen der Thetareihen zu positiven, binär-quadratischen Formen. (Zeros of theta series for positive binary quadratic forms)
zbMATH Open document ID: 0757.11013
DOI: 10.1007/BF02567661
author: Carl Friedrich Hermann
published in: Manuscripta Mathematica
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/155686
review text: Let \(Q\) be a primitive, positive definite, binary quadratic form with discriminant \(4N\), \(N\equiv 3\pmod 4\). For \(N\) small or \(Q\) diagonalizable the zeros of the associated \(\vartheta\)-function with characteristic \((\nu,\mu)\) on the upper half-plane \(\mathbb{H}\) is well-known; moreover for odd characteristics the \(\vartheta\)-function vanishes identically on \(\mathbb{H}\). In embedding \(\mathbb{H}\) in the Siegel-space \(\mathbb{H}_ 2\) and applying results on theta-constants of degree 2 the author shows for the product \(\Theta\) of the 10 \(\vartheta\)'s with even characteristic \(\Theta(Q,z):=\prod \vartheta(Q,(\nu,\mu))\) the following: (1) \(\Theta\) vanishes identically on \(\mathbb{H}\) if and only if \(Q\) is diagonalizable, (2) If \(Q\) is not diagonalizable \(\Theta\) has only simple zeros, which are characterized by a quadratic equation with integer coefficients. As an application new relations on class numbers for imaginary-quadratic number fields are found which in special cases trace back to the work of Eichler.
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11E45
zbMATH DE Number: 60090
zbMATH Keywords: theta series
zbMATH Keywords: binary quadratic form
zbMATH Keywords: class numbers
zbMATH Keywords: imaginary-quadratic number fields
MaRDI profile type: Publication
cites work: Q3229464
cites work: Q3863951
cites work: Q3963086
cites work: On the Graded Ring of Siegel Modular Forms of Genus Two
cites work: Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus
cites work: Q5664012
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description: scientific article; zbMATH DE number 60090