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In this post I want to advertise Serre’s lovely note, following the opening section which gives a simple definition of p-adic modular forms (via q-expansions) and uses this and some elementary congruences to construct the classical p-adic L-functions. Serre’s writing is excellent so I give only a brief account and let the interested reader consult the master.

The definition goes as follows. Fix a level (in fact, let’s just take level 1), and a prime p (which we will take to be odd, but the case p=2 appears no harder). By taking q-expansions, one can consider the algebra of classical modular forms of level 1 (all weights and indeed mixed weights) as a subalgebra \mathcal{M} \subseteq \mathbb{C}[[q]]. In fact, flat base change and the construction of models for modular curves implies one can actually do this on the integral and rational levels in such a way as loses no information: one is naturally led to study \mathcal{M}_\mathbb{Z} \subseteq \mathbb{Z}[[q]] and \mathcal{M}_\mathbb{Q} = \mathcal{M}_\mathbb{Z} \otimes \mathbb{Q}. But now note that \mathbb{Z}[[q]]\otimes \mathbb{Q} \subset \mathbb{Z}_p[[q]][1/p], a subring of \mathbb{Q}_p[[q]] which can be equipped with the structure of a Banach algebra via the ‘sup norm’ (the distance between \sum a_n q^n and \sum b_n q^n is the largest value of |a_n-b_n|).

It thus makes sense to consider the closure of \mathcal{M}_\mathbb{Q} with respect to this topology, and it is this which Serre defines to be the algebra \mathcal{M}_p of p-adic modular forms. In other words, a p-adic modular form is just a power series with p-adic coefficients which can be approximated uniformly coefficient-wise by the q-expansions of classical modular forms.

Pure classical modular forms have a weight k. What about p-adic modular forms? One crucial computation in Serre’s paper that makes the theory work says that if two modular forms are congruent modulo a large power of p then their weights are also congruent modulo a large power of p and also modulo p-1. From this one can easily attach a weight in the group X = \mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_p to a p-adic modular form (and since we are working at level 1, all weights arising will be even (so the subgroup 2\mathbb{Z}/(p-1)\mathbb{Z} on the first factor is all that is hit). We should remark that this computation has as an input the classical theorem of Clausen-von Stadt on the denominators of Bernoulli numbers.

Another interesting consequence of this which Serre notes is the following. Suppose we have a power series f= a_0+a_1q+a_2q^2+... and we would like to show it’s a p-adic modular form. Maybe we can write it as a putative limit of some modular forms f^i=a^i_0+a^i_1q+a^i_2q^2+..., except we only know that for all n \geq 1 the coefficients a_n^i \rightarrow a_n converge (but know nothing about the constant coefficients). Suppose further we also know the weights k^i converge to some nonzero k \in X.

Well then the weight zero modular form a_0 and f cannot be too close p-adically, where the length scale implicit in the phrase ‘too close’ is given with reference to one of the modular forms in question, so one gets a bound of the form sup_{n \geq 1} |a_n| \geq C |a_0|, where C is some constant depending on how far k is from zero. Applying this to our sequence f^i we learn in particular that the $a^i_0$ lie in a closed bounded subset of \mathbb{Q}_p, so there is a convergent subsequence, and passing to the corresponding subsequence of modular forms we deduce that f is a p-adic modular form after all.

One could complain that this is a silly theorem: when will we have that all but the constant coefficient is known to converge? Recall the classical weight k Eisenstein series, whose q-expansions are given in terms of the Bernoulli numbers B_k = -k \zeta(1-k) and the “sum of (k-1)th powers of all divisors of n” function \sigma_{k-1}(n):

G_k = -\frac{B_k}{2k} + \sum_{n\geq 1} \sigma_{k-1}(n)q^n.

If we take a p-adically convergent sequence k_1,k_2,... \rightarrow k \in X of weights that also tends to infinity in the archimedean metric, then one sees explicitly that p-adically:

\sigma_{k_i-1}(n) = \sum_{d|n} d^{k_i-1} \rightarrow \sum_{d|n, p\not|d} d^{k-1} =: \sigma^*_{k-1}(n).

Therefore the sequence G_{k_i} of Eisenstein series fit exactly in the situation of the above theorem, and we deduce that there is a p-adic Eisenstein series of any weight k \in X, which is well-defined because \sigma^*_{k-1}(n) depends only on k. In particular, we deduce the existence of a well-defined continuous p-adic function \zeta_p:X-\{1\} \rightarrow \mathbb{Q}_p such that \zeta_p(k) = \frac{1}{2} \zeta(1-k) for all k \in \mathbb{Z}_{\geq 2}. One can check that this is exactly the p-adic zeta function constructed classically by Kubota-Leopoldt and featuring on the analytic side of the main conjecture of Iwasawa theory.