Beschreibung:
<p>We use the theory of Jacobi forms to study the number of elements in a maximal order of a definite quaternion algebra over the field of rational numbers whose characteristic polynomial equals a given polynomial. A certain weighted average of such numbers equals (up to some trivial factors) the Hurwitz class number <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis 4 n minus r squared right-parenthesis">
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<mml:mi>H</mml:mi>
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<mml:mn>4</mml:mn>
<mml:mi>n</mml:mi>
<mml:mo>−<!-- − --></mml:mo>
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<mml:mi>r</mml:mi>
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<mml:annotation encoding="application/x-tex">H(4n-r^2)</mml:annotation>
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</inline-formula>. As a consequence we obtain new proofs for Eichler’s trace formula and for formulas for the class and type number of definite quaternion algebras. As a secondary result we derive explicit formulas for Jacobi Eisenstein series of weight <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2">
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<mml:mn>2</mml:mn>
<mml:annotation encoding="application/x-tex">2</mml:annotation>
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</inline-formula> on <inline-formula content-type="math/mathml">
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<mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi>
<mml:mn>0</mml:mn>
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<mml:annotation encoding="application/x-tex">\Gamma _0(N)</mml:annotation>
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</inline-formula> and for the action of Hecke operators on Jacobi theta series associated to maximal orders of definite quaternion algebras.</p>