A class of non-holomorphic modular forms I

Journal article

This paper studies examples of real analytic functions on the upper half plane satisfying a modular transformation property of the form

(0.1) f(az + b/cz + d)= (cz + d)^r (cz + d)^s f(z)

for integers r, s. They do not satisfy a simple condition involving the Laplacian. The raison d’être for this class of functions is two-fold:

(1) Holomorphic modular forms f with rational Fourier coefficients correspond to certain pure motives Mf over Q. Using iterated integrals, we can construct non-holomorphic modular forms which are associated to iterated extensions of the pure motives Mf . Their coefficients are periods.

(2) In genus one closed string perturbation theory, one assigns a lattice sum to a graph [14], which defines a real-analytic function on the upper half plane invariant under SL2(Z). It is an open problem to give a complete description of this class of functions and prove their conjectured properties.

Authors: Brown, F

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Research in the Mathematical Sciences
• Publication date: 2018-01-01
• Acceptance date: 2017-12-18
• EISSN: 2197-9847