Abstract. Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's theory, we prove a structure theorem for automorphic forms of all orders. This allows us to define an analogue of a mixed Hodge structure on a space of higher order automorphic forms.

We show that the essential dimension of a non-trivial Abelian variety over a number field is infinite. To cite this article: P. Brosnan, R. Sreekantan, C. R. Acad. Sci. Paris, Ser. I 346 (2008). RésuméDimension essentielle d'une variété abélienne sur un corps de nombres. On montre que la dimension essentielle d'une variété abélienne non-triviale définie sur un corps de nombres est infinie. Pour citer cet article : P. Brosnan, R. Sreekantan, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

In this paper we show that the mapis surjective, where E 1 and E 2 are two non-isogenous semistable elliptic curves over a local field, CH 2 (E 1 × E 2 , 1) is one of Bloch's higher Chow groups and P CH 1 (X v ) is a certain subquotient of a Chow group of the special fibre X v of a semi-stable model X of E 1 × E 2 . On one hand, this can be viewed as a non-Archimedean analogue of the Hodge-Dconjecture of Beilinson -which is known to be true in this case by the work of Chen and Lewis (J. the case when the elliptic curves have split multiplicative reduction.

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.