We define theta functions attached to indefinite quadratic forms over real number fields and prove that these theta functions are Hilbert modular forms by regarding them as specializations of symplectic theta functions. The eighth root of unity which arises under modular transformations is determined explicitly.

We first generalize a classical iteration formula for one variable holomorphic mappings to a formula for higher dimensional holomorphic mappings. Then, as an application, we give a short and intuitive proof of a classical theorem, due to H. Poincaré, for the condition under which a singularity of a holomorphic vector field is an isochronous center. 1. An iteration formula Let f be a holomorphic...

We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-ReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the...

Let the field F contain all /?-power roots of unity for some prime p and suppose that the absolute Galois group G of F is a one-relator pro-p group. We use Merkurjev-Suslin's theorem on the power norm residue symbol to show that G is an extension of a Demushkin group by a free pro-/) group. Let F be a field, Fs its separable closure, and G = Ga\(Fs/F) its absolute Galois group. Let n be an inte...

Quantum toroidal algebras (or double affine quantum algebras) are defined from quantum affine Kac-Moody algebras by using the Drinfeld quantum affinization process. They are quantum groups analogs of elliptic Cherednik algebras (elliptic double affine Hecke algebras) to whom they are related via Schur-Weyl duality. In this review paper, we give a glimpse on some aspects of their very rich repre...

the idea that the distinction of sciences has to be based on topics is facing serious criticisms and objections, since both arguments of the followers of this approach are violable and the history of science is replete with many examples that do not fit into this tight framework. this issue prompted some scholars such as muḥaqqiq khurāsānī to move toward distinction based on objective. this app...