Automorphic functions and Bass--Milnor--Serre's homomorphism,~I

N. V. Proskurin

Zapiski Nauchnykh Seminarov POMI, Automorphic functions and number theory. Part I, Tome 129 (1983), pp. 85-126

Résumé

Let $\mathcal O$ be the ring of intergers in $\mathbb Q(\sqrt{-3})$ and let $SL_m(\mathcal O, q)$ be the congruence subgroup $\mod q$ in $SL_m(\mathcal O)$; $q=(3)$ is the ideal of $\mathcal O$. In [6] for solution of the congruence subgroup problem Bass, Milnor and Serre have constracted the homomorphish $\chi\colon SL_m(\mathcal O, q)\to\mathbb C^*$. For this aim the cubic residue sumbol is used. We consider $\chi$ as multiplier system. The object of our investigation is the Bisenstein series on $X\cong SL_3(\mathbb C)/SU(3)$ which is automorphic with respect to the $SL_3(\mathcal O, q)$ with $\chi$ as the multiplier system. We have calculated some coefficients of the expansion in the sense of [2], [3] for this Eisenstein series.