5 research outputs found
Zagier's weight mock modular form
Mock modular forms have their origins in Ramanujan's pioneering work on mock
theta functions. In a 1975 paper, Zagier proved certain transformation
properties of the generating function of the Hurwitz class numbers for
the discriminant . In the modern framework, these results show that the
generating function of is a mock modular form of weight 3/2 with the
theta function being the shadow. In this expository paper, we provide a
detailed proof of Zagier's result.Comment: 24 page
Mock Modularity In CHL Models
Dabholkar, Murthy and Zagier (DMZ) proved that there is a canonical
decomposition of a meromorphic Jacobi form of integral index for
with poles on torsion points
into polar and finite parts, and showed that
the finite part is a mock Jacobi form. In this paper we generalize the results
of DMZ to meromorphic Jacobi forms of rational index for congruence subgroups
of . As an application, we establish that a large
class of single-centered black hole degeneracies in CHL models are given by the
Fourier coefficients of mock Jacobi forms. In this process we refine the result
of DMZ regarding the set of charges for which the single-centered black hole
degeneracies are given by a mock modular form. In particular, in the case
studied by DMZ, we present examples of charges for which the single-centered
degeneracies are not captured by the mock modular form of the expected index.Comment: 64 Page
Class Numbers of Quadratic Fields
We present a survey of some recent results regarding the class numbers of quadratic field