On the universal power series for Jacobi sums and the Vandiver conjecture (Q1262341)

This paper studies Ihara's series \(F_{\rho}(u,v)\), a unit in \({\mathbb{Z}}_{\ell}[[u,v]]\) related to an element \(\rho\) of the Galois group of \({\mathbb{Q}}\). It relates the image of \(\rho \to F_{\rho}\) to the Vandiver conjecture at \(\ell\). This image is contained in a multiplicative subgroup \({\mathcal F}\) defined by five conditions. The surjectivity of Gal(\({\bar {\mathbb{Q}}}/{\mathbb{Q}})\to {\mathcal F}\) is equivalent to it (theorem 1). The relation of the cokernel to a classical Iwasawa module is given in theorem 2. In conclusion, theorem 3 studies \(h_{\rho}(u)\), the first term in the v-expansion of \(F_{\rho}(u,v)\) à la Coleman.