8.4. Lambda-adic modular forms and weight space🔗

The series \mathbf{E} is the prototypical example of a \Lam-adic modular form. Informally it says: Eisenstein series vary p-adically continuously in the weight — if k and k' are close p-adically then the q-expansions of E_k and E_{k'} are close p-adically. This point of view originates with Serre, who used p-adic families of Eisenstein series to give a new construction of the p-adic zeta function of a totally real field: if one can interpolate all the non-constant coefficients — which, as we have just seen, is easy — then one automatically interpolates the constant term, namely the p-adic zeta function, which is far harder to interpolate directly.

These results are often phrased over the weight space \cW, the rigid analytic space whose \Cp-points parametrise continuous characters of \Zpx. The integers embed via \kappa_k : x \mapsto x^k, and k \equiv k' \pmod{p-1} iff \kappa_k and \kappa_{k'} lie in the same connected unit ball. Writing \cO^+(\cW) for the bounded rigid analytic functions on \cW (which correspond to measures on \Zpx) and Q(\cW) for the rigid meromorphic functions with at worst a simple pole at the trivial character (corresponding to pseudo-measures), one rewrites \mathbf{E} = \sum_{n \geq 0} B_n q^n \in Q(\cW)[\![q]\!] with B_n \in \cO^+(\cW) for n > 0, satisfying \mathbf{E}(\kappa_k) = E_k^{(p)} for all even k \geq 4. Thus \mathbf{E} is literally a p-adic interpolation of the Eisenstein series across weight space.

Pioneering work of Hida pushed this much further, producing analogous Hida families for far more general modular forms; these were in turn generalised to Coleman families and eigenvarieties, parametrising the p-adic variation of modular and automorphic forms over weight space. Such families are central to the modern construction and study of p-adic L-functions, foreshadowing the \mathrm{GL}(2) picture taken up in Part II.