A class of non-holomorphic modular forms I

<p>This paper studies examples of real analytic functions on the upper half plane satisfying a modular transformation property of the form</p> <p>(0.1) f(az + b/cz + d)= (cz + d)^r (cz + d)^s f(z)</p> <p>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:</p> <p>(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.</p> <p>(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.</p>