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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 (which we will take to be odd, but the case 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 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 and . But now note that , a subring of which can be equipped with the structure of a Banach algebra via the ‘sup norm’ (the distance between and is the largest value of ).

It thus makes sense to consider the *closure* of with respect to this topology, and it is this which Serre defines to be the algebra of -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 . 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 then their weights are also congruent modulo a large power of and also modulo . From this one can easily attach a weight in the group to a p-adic modular form (and since we are working at level 1, all weights arising will be *even* (so the subgroup 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 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 , except we only know that for all the coefficients converge (but know nothing about the *constant* coefficients). Suppose further we also know the weights converge to some *nonzero* .

Well then the weight zero modular form and 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 , where is some constant depending on how far is from zero. Applying this to our sequence we learn in particular that the $a^i_0$ lie in a closed bounded subset of , so there is a convergent subsequence, and passing to the corresponding subsequence of modular forms we deduce that 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 Eisenstein series, whose q-expansions are given in terms of the Bernoulli numbers and the “sum of th powers of all divisors of ” function :

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If we take a p-adically convergent sequence of weights that also tends to infinity in the archimedean metric, then one sees explicitly that p-adically:

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Therefore the sequence 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 , which is well-defined because depends only on . In particular, we deduce the existence of a well-defined continuous p-adic function such that for all . 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.