Abstract
Almost 40 years ago, H. Cohen formulated a conjecture about the modularity of a certain infinite family of functions involving the generating function of the Hurwitz class numbers of binary quadratic forms. We use techniques from the theory of modular, mock modular, and Jacobi forms. In this paper, we prove a slight improvement of Cohen’s original conjecture. From our main result, we derive so far unknown recurrence relations for Hurwitz class numbers. 11E41; 11F37; 11F30
Highlights
It has been an important problem in number theory to determine the class numbers of binary quadratic forms
The main idea of the proof of Theorem 1.2 is to relate both summands in the coefficient of the above power series to objects which in accordance to the nomenclature in [9] should be called quasi mixed mock modular forms, complete them, such that they transform like modular forms and show that the completion terms cancel each other out
As a mock modular form, the function H is rather peculiar since it is basically the only example of such an object which is holomorphic at the cusps of Γ0(4)
Summary
3 2 mock modular form, i.e. the holomorphic part of a harmonic weak Maaß form (see Section 2 for a definition) Using this theory, some quite unexpected connections to combinatorics occur, as for example in [2], where class numbers were related to ranks of so-called overpartitions. For l > 0 the coefficient of Xl in (1.4) is a cusp form This obviously implies new relations for Hurwitz class numbers which to the author’s knowledge have not been proven so far. The main idea of the proof of Theorem 1.2 is to relate both summands in the coefficient of the above power series to objects which in accordance to the nomenclature in [9] should be called quasi mixed mock modular forms, complete them, such that they transform like modular forms and show that the completion terms cancel each other out. More detailed proofs will be available in the author’s PhD thesis [18]
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