  The Fourier coefficients of the Siegel-Eisenstein series are p-adically
continued for all primes p, as meromorphic functions, using the reciprocal of a
product of L-functions. A construction of p-adic meromorphic families of such
series is given in relation to the geometry of homogeneous spaces. Applications
are given to p-adic L-functions, to Siegel's Mass Formula, to p-adic analytic
families of automorphic representations. Based on author's talk for the
Conference Automorphic Forms and Related Geometry, Assessing the Legacy of I.I.
Piatetski-Shapiro
