Speciality:
01.01.06 (Mathematical logic, algebra, and number theory)
Birth date:
18.05.1966
E-mail: Keywords: modular forms,
group representations,
algebraic number theory,
algebraic groups.
Subject:
I have studied one special class of modular forms which we can define by the conditions: elements of this class are cusp forms of integer weight with characters, they are eigenforms of all Hecke operators and have no zeros outside of the cusps. The full list of these functions was obtained. They are products of Dedekind $/eta$-functions. We shall call them multiplicative $/eta$-functions. The arithmetic interpretation for the Fourier coefficients of some of these forms by Hurwitz quartenions and Cayley algebra was found. The expression of Ramanujan characters by Weil characters for some of these cusp forms was obtained. One can associate a product of $/eta$-functions with an element of finite order in a group by a linear representation. The problem of finding all finite groups such that all modular forms associated with elements of these groups by means of some faithful representations are multiplicative $/eta$-products was considered. All groups of order 24, finite subgroups in SL(5,C), metacyclic, in particular dihedral, groups were investigated. It was proved that there is no such solvable group that one can assign with all its elements by an exact representation all multiplicative $/eta$-products and only them. The coefficients of these functions were studied as central functions on a group. Also elliptic curves over finite fields were studied : graphs of 2 isogenies were constructed; the formula connecting the number of elliptic curves with the fixed group of $F_q$ rational points and the number of classes of equivalence of positive definite quadratic forms of two variables was found.
Main publications:
Voskresenskaya G. V. One special class of modular forms and group representations // Journal de Thoer.des Nombres Bordeaux, 1999, 11, 247–262.
