The Eisenstein family for the modular group along closed geodescis.
The Equations Defining Abelian Varieties and Modular Functions.
The Equations for Modular Function Fields of Principal congruence subgroups of prime level.
The equidistribution of Fourier coefficients of half integral weight modular forms on the plane
Let be a nonzero cuspidal Hecke eigenform of weight and the trivial nebentypus , where the Fourier coefficients are real. Bruinier and Kohnen conjectured that the signs of are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies , where is a squarefree integer such that . Let and be natural numbers such that . In this work, we show that is equidistributed over any arithmetic progression .
The evaluation of two-dimensional lattice sums via Ramanujan's theta functions
We analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many explicit examples are given.
The Farey graph.
The finite subgroups of maximal arithmetic kleinian groups
Given a maximal arithmetic Kleinian group , we compute its finite subgroups in terms of the arithmetic data associated to by Borel. This has applications to the study of arithmetic hyperbolic 3-manifolds.
The first negative Hecke eigenvalue of a Siegel cusp form of genus two
The Fixed Points of the Symmetrie Hilbert Modular Group of a Real Quadratic Field with Arbitrary Discriminant.
The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight, II
The Fourier coefficients of modular forms and Niebur modular integrals having small positive weight, I
The Fourier expansion of Epstein's zeta function over an algebraic number field and its consequences for algebraic number theory
The Fourier transforms of weighted orbital integrals on simisimple groups of real rank one.
The Fourth Moment of Ramanujan ..-Function.
The fundamental lemma of a relative trace formula for
The geometry and spectrum of the one holed torus.
The geometry of non-unit Pisot substitutions
It is known that with a non-unit Pisot substitution one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization...
The Group of Automorphisms of the Modular Function Field.
The Hilbert Modular Group for the Field ...(...13).
The Hilbert modular group, resolution of the singularities at the cusps and related problems