Parabolic Points and Zeta–functions of Modular Curves

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“…But any class {α, β} is a sum of distinguished classes and therefore the homology is generated by the distinguished classes. This is proved in [6] in the rational case. The proof over K is very similar (see [5]).…”

“…It is described for the rational case in [6]. Grunewald, Mennicke and others extended the method to calculate the homology of the space…”

“…Let (n, p) denote one of the pairs (4, 3), (5,5) or (6,9). Consider the embedding of A n in PSL 2 (F p ), Denote by a n the element in H 2 (A n ) corresponding to the central extension of A n → PSL 2 (F p ) contained in SL 2 (F p ).…”

Multiple Gamma Functions and Multiple $q$-Gamma Functions

“…As we can see from these studies, the multiple gamma functions are fundamental for the analytic number theory: See also [16], [17]. However we do not think that the theory of the multiple gamma functions has been fully explored.…”

A combinatorial interpretation of Serre's conjecture on modular Galois representations

“…ad -bc = n, d -1 -c--0 mod N). For every n > 1 we define Hecke [9] and [17) proved (see [7]) in the more general setting of any subgroup r C SL2 (Z) of finite index (instead of rl (N)) and the setting of correspondences defined as double cosets (instead of the concrete Hecke or diamond operators).…”