Abstract. We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for Γ1(N ): an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

The existence of the limiting pair correlation for angles between reciprocal geodesics on the modular surface is established. An explicit formula is provided, which captures geometric information about the length of reciprocal geodesics, as well as arithmetic information about the associated reciprocal classes of binary quadratic forms. One striking feature is the absence of a gap beyond zero in the limiting distribution, contrasting with the analog Euclidean situation.

In [1], the authors prove a conjecture of Calegari and Stein regarding mod p congruences between cusp forms of weight four for Γ0(p) and the derivatives of cusp forms of weight two for the same congruence subgroup. In this paper, we investigate whether or not the result remains valid for cusp forms of level Np.

For any normal commutative Hopf subalgebra K = k G of a semisimple Hopf algebra we describe the ring inside kG obtained by the restriction of Hmodules. If G = Z p this ring determines a fusion ring and we give a complete description for it. The case G = Z p n and some other applications are presented.

We study analytic and arithmetic properties of the elliptic gamma functionin the regime p = q, in particular, its connection with the elliptic dilogarithm and a formula of S. Bloch. We further extend the results to more general products by linking them to non-holomorphic Eisenstein series and, via some formulae of D. Zagier, to elliptic polylogarithms. (see, for example, [3, Section 2]), where we definex = e 2π iẑ andq = e 2π iτ . Less is known about modular properties of the related product θ 1 (z; τ ) := ∞ m=0(1 − x −1 q m+1 ) m+1(1 − xq m ) m , 123