Exact formulas for traces of singular moduli of higher level modular functions

In this course we introduce the main notions relative to the classical theory of modular forms. A complete treatise in a similar style can be found in the author's book joint with F. Str{\"o}mberg [1].

In this paper, we study traces of Hecke operators on Drinfeld modular forms of level 1 in the case $A = \mathbb{F}_q[T]$. We deduce closed-form expressions for traces of Hecke operators corresponding to primes of degree 1 and provide algorithms for primes of higher degree. We improve the Ramanujan bound and deduce the decomposition of cusp forms of level $\Gamma_0(T)$ into oldforms and newforms, as conjectured by Bandini-Valentino, under the hypothesis that each Hecke eigenva...

The present notes contain the material of the lectures given by the author at the summer school on ``Modular Forms and their Applications'' at the Sophus Lie Conference Center in the summer of 2004.

We compute the divisor of the modular equation on the modular curve $\Gamma_0(N) \backslash \mathbb H^*$ and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup $\Gamma_0(N)$ of genus zero. We also introduce the notions and properties of $\Gamma$-equivalence and $\Gamma$-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for $N = 2, 3, 4, 5$.

This doctoral thesis studies the overlap between two well-known collections of results in number theory: the theory of periods and period polynomials of modular forms as developed by Eichler, Shimura and Manin and its extensions by K\"ohnen and Zagier, and the theory of 'multiple zeta values' (MZV's) as initied by Euler and studied by many authors in the last two decades. These two theories had been linked by Manin, who introduced 'multiple L-values' (MLV's). We propose to st...

Recently, Bruinier and Ono found an algebraic formula for the partition function in terms of traces of singular moduli of a certain non-holomorphic modular function. In this paper we prove that the rational polynomial having these singuar moduli as zeros is (essentially) irreducible, settling a question of Bruinier and Ono. The proof uses careful analytic estimates together with some related work of Dewar and Murty, as well as extensive numerical calculations of Sutherland.

The elliptic modular j function enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. In this article, we are concerned with analogues of Kaneko's result for the McKay-Thompson series of square-free level.

This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Cl\'ery and Carel Faber.

In this paper, for a square-free integer l>1, a even positive integer k and a positive integer N, we give a trace formula of the Hecke operator T(l) on the space S_k^0(N) of all newforms of weight k and level \Gamma_0(N). Moreover, we give a decomposition S_k^0(N)=\oplus_{i|N}S_k^0(N;i) and also give a trace formula of T(l) on S_k^0(N;i).

We derive finite rational formulas for the traces of cycle integrals of certain meromorphic modular forms. Moreover, we prove the modularity of a completion of the generating function of such traces. The theoretical framework for these results is an extension of the Shintani theta lift to meromorphic modular forms of positive even weight.