Higher order modular forms and mixed Hodge theory

Abstract

In this paper we introduce a certain space of higher order modular forms of weight 0 and show that it has a Hodge structure coming from the geometry of the fundamental group of a modular curve. This generalises the usual structure on classical weight 2 forms coming from the cohomology of the modular curve. Further we construct some higher order Poincaré series to get higher order higher weight forms and using them we define a space of higher weight, higher order forms which has a mixed Hodge structure as well.

Iterated Integrals and Higher Order Invariants

“…In Section 3 we first show that homotopy invariant iterated integrals always give higher order invariants. For the case of surfaces, this was proven by Sreekantan in [18]. We then use the result from Section 3 to deduce the main result, saying that higher order invariants can always be expressed by iterated integrals.…”

“…Modular forms of higher order have been studied extensively in recent years [4,5,[8][9][10][11][12][13][14]18]. To construct them, one often uses iterated integrals.…”

“…From dimension formulae [13] it is clear that in the case of holomorphic forms, iterated integrals do not give all higher order forms. But there are strong indicators [11,12,14,18] that in the case of smooth functions, all higher order forms are indeed obtainable via iterated integrals. This has been an implicit open question for a while that is answered affirmatively in this paper.…”

Higher-order Maass forms

“…If it were, the value of the derivative at 1 would be expressed as the value of the actual L-function of second-order Γ 0 (N) at 1. That could be advantageous for the study of L ′ f (1) in terms of the outstanding conjectures, especially since there is now evidence that a motivic structure underlies higher order forms (see [10] and [22]).…”