On the generalized Bernoulli numbers that belong to unequal characters

Abstract

The study of class number invariants of absolute abelian fields, the investigation of congruences for special values of $L$-functions, Fourier coefficients of half-integral weight modular forms Rubin's congruences involving the special values of $L$-functions of elliptic curves with complex multiplication, and many other problems require congruence properties of the generalized Bernoulli numbers (see [16-18, 12, 29, 3], etc.) The rst steps in this direction can be found in the papers of H W Leopoldt (see [15]) and L Carlitz (see [5]). For further studies see [22, 24, 29]. This paper presents some new examples extending both old author's results and recent investigations of H Lang (see [14]), A Balog, H Darmon, K Ono (see [3]), etc. On the whole the proved results are consequence of congruences connecting the generalized Bernoulli numbers that belong to unequal characters.       This paper presents some new examples ex tending both old authors results and recent investigations of H Lang see   A Balog  H Darmon K Ono see  etc On the whole the proved results are consequence of congruences connecting the generalized Bernoulli numbers that belong to unequal characters        etc The rst steps in this direction can be found in the papers of H W Leopoldt see   and L Carlitz see  For further studies see       This paper presents some new examples ex tending both old authors results and recent investigations of H Lang see   A Balog  H Darmon K Ono see  etc On the whole the proved results are consequence of congruences connecting the generalized Bernoulli numbers that belong to unequal characters