Anti-cyclotomic Katz $p$-adic $L$-functions and congruence modules

— The purpose of this paper is to prove the divisibility of the characteristic power series of the congruence module of a Hida /?-adic family of theta series coming from a CM-field (with fixed CM-type) by the anti-cyclotomic specialisation of the Katz ^-adic L-function with auxiliary conductor. This requires to construct first this^-adic L-function since in the original paper by Katz the auxiliary conductor was trivial. The divisibility proven here is one of two steps towards one of the two divisibilities predicted by the (anti-cyclotomic) Iwasawa main conjecture for CM-fields. The second step has been carried out by the authors and will be published elsewhere.