On differential modular forms and some analytic relations between Eisenstein series

Abstract

In the present article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight 2, 4 and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein series and obtain them in a natural way as coefficients of a family of elliptic curves. The fact that a complex manifold over the moduli of polarized Hodge structures in the case h 10 = h 01 = 1 has an algebraic structure with an action of an algebraic group plays a basic role in all of the proofs.

Metric in the moduli of SU(2) monopoles from spectral curves and Gauss-Manin connection in disguise

“…In this case r 2 = π 2 /4, as we show below (15), and there is no free parameter. This is the case 13,14 where a 2-monopole is simply a superposition of two 1-monopoles both centered at the origin of R 3 and it agrees with the fact that the dimension of the reduced moduli M 0 1 is zero.…”

Quasi-modular forms attached to elliptic curves: Hecke operators

“…In this section we recall some definitions and theorems in [6,7]. The reader is also referred to [9] for a complete account of quasi-modular forms in a geometric context.…”

“…For the passage from the third to fourth equality we have used the functional equation of f (and hence the SL(2, Z)-invariant function F ) with respect to the actions in (6) and (7), see [7] Proposition 6. One can take the representatives…”

“…One can describe quasi-modular forms in the framework of the algebraic geometry of elliptic curves, and in particular, the Ramanujan differential equation between Eisenstein series can be derived from the Gauss-Manin connection of families of elliptic curves, see for instance [7] and [9]. We call this the Gauss-Manin connection in disguise.…”

Quasimodular forms and Poincaré series

“…Quasimodular forms were introduced by Kaneko and Zagier in [5] and have been studied actively since then in connection with various topics in number theory (see e.g. [1], [7], [9], [11]). They are also linked to some problems in applied mathematics (cf.…”