8. The p-adic family of Eisenstein series
We close Part I with a brief detour illustrating a second flavour of p-adic
variation in number theory: the p-adic variation of modular forms. Almost all
the work has already been done. In building the Kubota–Leopoldt p-adic
L-function we constructed a pseudo-measure \zeta_p on \Zpx interpolating
the values \zeta(1-k); we will now see that these values are exactly the
constant terms of a family of Eisenstein series, and that the remaining Fourier
coefficients of that family are far easier to interpolate. Assembling the two
gives a single power series — a \Lam-adic modular form — whose specialisations
are honest modular forms.
Throughout, p is a fixed prime, \uhp = \set{z \in \C : \mathrm{Im}(z) > 0}
is the upper half-plane and q = e^{2\pi i z}.