For any non-square 1 < D ≡ 0,1 (mod 4), Zagier defined $F_k(D;x):={\sum }_{\begin{array}{c}a,b,c\in \mathbb{Z},a<0\\ b^2-4ac=D\end{array}}max(0,{(a{x}^2+bx+c)}^{k-1})$. Here we use the theory of periods to give identities and congruences which relate various values of $F_k(D;x)$.

We investigate the vertical version of the Sato-Tate conjecture for some GL₂ automorphic representations over totally real fields with specified local components at a finite set of finite places.

Let $\Lambda$ be a subgroup of an arithmetic lattice in $\mathrm{SO}(n+1,1)$. The quotient ${ℍ}^{n+1}/\Lambda$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $PS{L}_2\left(\mathbb{Z}\right)\setminus PS{L}_2\left(ℝ\right)$. This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $PS{L}_2\left(\mathbb{Z}\right)\setminus ℍ$. The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of $S{L}_2\left(ℝ\right)$. In the proof the key estimates come from applying...

Quasimodular forms were the heroes of a Summer school held June 20 to 26, 2010 at Besse et Saint-Anastaise, France. We give a short introduction to quasimodular forms. More details on this topics may be found in [1].

This paper is based on lectures delivered at the Workshop on quasimodular forms held in June, 2010 in Besse, France, and it provides a survey of some recent work on quasimodular forms.

In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the...

Fix $K$ a $p$-adic field and denote by $G_K$ its absolute Galois group. Let $K_{\infty }$ be the extension of $K$ obtained by adding $p^n$-th roots of a fixed uniformizer, and $G_{\infty }\subset G_K$ its absolute Galois group. In this article, we define a class of $p$-adic torsion representations of $G_{\infty }$, calledquasi-semi-stable. We prove that these representations are “explicitly” described by a certain category of linear algebraic objects. The results of this note should be considered as a first step in the understanding of the structure of quotient...

We introduce and study the Rademacher-Carlitz polynomial $R(u,v,s,t,a,b):={\sum }_{k=\lceil s\rceil }^{\lceil s\rceil +b-1}{u}^{\lfloor (ka+t)\rfloor }/b_{v^k}$ where $a,b\in {\mathbb{Z}}_{>0}$, s,t ∈ ℝ, and u and v are variables. These polynomials generalize and unify various Dedekind-like sums and polynomials; most naturally, one may view R(u,v,s,t,a,b) as a polynomial analogue (in the sense of Carlitz) of the Dedekind-Rademacher sum $r_t(a,b):={\sum }_{k=0}^{b-1}\left(((ka+t)/b)\right)\left((k/b)\right)$, which appears in various number-theoretic, combinatorial, geometric, and computational contexts. Our results come in three flavors: we prove a reciprocity theorem for Rademacher-Carlitz...