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The Eichler Commutation Relation for theta series with spherical harmonics

Lynne H. Walling (1993)

Acta Arithmetica

It is well known that classical theta series which are attached to positive definite rational quadratic forms yield elliptic modular forms, and linear combinations of theta series attached to lattices in a fixed genus can yield both cusp forms and Eisenstein series whose weight is one-half the rank of the quadratic form. In contrast, generalized theta series - those augmented with a spherical harmonic polynomial - will always yield cusp forms whose weight is increased by the degree of the...

The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Soufiane Mezroui (2020)

Czechoslovak Mathematical Journal

Let f = ∑ n = 1 ∞ a ( n ) q n ∈ S k + 1 / 2 ( N , χ 0 ) be a nonzero cuspidal Hecke eigenform of weight k + 1 2 and the trivial nebentypus χ 0 , where the Fourier coefficients a ( n ) are real. Bruinier and Kohnen conjectured that the signs of a ( n ) are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies { a ( t n 2 ) } n , where t is a squarefree integer such that a ( t ) ≠ 0 . Let q and d be natural numbers such that ( d , q ) = 1 . In this work, we show that { a ( t n 2 ) } n is equidistributed over any arithmetic progression n ≡ d mod q .

The evaluation of two-dimensional lattice sums via Ramanujan's theta functions

Ping Xu (2014)

Acta Arithmetica

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.

Jones, Gareth A. (1987)

Séminaire Lotharingien de Combinatoire [electronic only]

The finite subgroups of maximal arithmetic kleinian groups

Ted Chinburg, Eduardo Friedman (2000)

Annales de l'institut Fourier

Given a maximal arithmetic Kleinian group Γ ⊂ PGL ( 2 , ℂ ) , we compute its finite subgroups in terms of the arithmetic data associated to Γ by Borel. This has applications to the study of arithmetic hyperbolic 3-manifolds.

The geometry of non-unit Pisot substitutions

Milton Minervino, Jörg Thuswaldner (2014)

Annales de l’institut Fourier

It is known that with a non-unit Pisot substitution σ 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...

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