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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.