A class of non-holomorphic modular forms I

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.Comment: Based on a talk given at Zagier's 65th birthday conference `modular forms are everywhere'. What was formerly the appendix has now turned into arXiv:1710.0791

A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals

We introduce a new family of real analytic modular forms on the upper half plane. They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q, \overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first non-trivial functions in this class are real analytic Eisenstein series.Comment: Introduction rewritten in version 2, and other minor edit

A multi-variable version of the completed Riemann zeta function and other LL-functions

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure on its residues. The special case of two variables can be written as a partial Mellin transform of a real analytic Eisenstein series, which enables us to relate its values at pairs of positive even points to periods of (simple extensions of symmetric powers of the cohomology of) the CM elliptic curve corresponding to the Gaussian integers. In general, the totally even values of these functions are related to new quantities which we call multiple quadratic sums. More generally, we cautiously define multiple-variable versions of motivic $L$-functions and ask whether there is a relation between their special values and periods of general mixed motives. We show that all periods of mixed Tate motives over the integers, and all periods of motivic fundamental groups (or relative completions) of modular groups, are indeed special values of the multiple motivic $L$-values defined here.Comment: This is the second half of a talk given in honour of Ihara's 80th birthday, and will appear in the proceedings thereo

Depth-graded motivic multiple zeta values

We study the depth filtration on multiple zeta values, the motivic Galois group of mixed Tate motives over $\mathbb{Z}$ and the Grothendieck-Teichm\"uller group, and its relation to modular forms. Using period polynomials for cusp forms for $\mathrm{SL}_2(\mathbb{Z})$, we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo $\zeta(2)$ and modulo lower depth. We formulate a single conjecture about the homology of this Lie algebra which implies conjectures due to Broadhurst-Kreimer, Racinet, Zagier and Drinfeld on the structure of multiple zeta values and on the Grothendieck-Teichm\"uller Lie algebra.Comment: Rewritten introduction, added brief section explaining the depth-spectral sequence, and made a few proofs more user-friendly by adding some more detail

Mixed Tate motives over Z\Z

We prove that the category of mixed Tate motives over $\Z$ is spanned by the motivic fundamental group of \Pro^1 minus three points. We prove a conjecture by M. Hoffman which states that every multiple zeta value is a \Q-linear combination of $\zeta(n_1,..., n_r)$ where $n_i\in \{2,3\}$