Do $p$-adic topological modular forms exist?

Question

Are there $p$-adic topological modular forms? What is the analogue of finite slope and overconvergent?

Answer 1

There are very natural analogues of $p$-adic modular forms in the study of topological modular forms ($tmf$)--they are $K(1)$-local topological modular forms.

One can obtain the classical ring of p-adic modular forms by first inverting the Eisenstein series $E_{p-1}$ and then $p$-adically completing (at least for large primes; at 2 and 3 we have to be a little more careful). The results are modular forms that are defined on ordinary elliptic curves over p-complete rings with no supersingular contributions; i.e. on a moduli of elliptic curves whose formal group has exact height $1$.

The keyword for the analogue in topology is called $K(1)$-localization, and it is carried out in roughly the same way--we would like to narrow in on the portion that carries exact height $1$, by inverting a class (topologists call this "$v_1$") and $p$-adically completing. The technical details are a little trickier because it applies to a wider class of objects than $tmf$, and every prime behaves like the prime 2 or 3 does in the classical case. The problem can be solved in a similar way: for every $k$, some power $v_1^{p^d}$ of $v_1$ exists mod $p^k$, and so it still makes sense to invert $v_1$ mod $p^k$. We can then assemble these into an inverse system and take the limit.

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As a sub-comment: the weight of a $p$-adic modular form is no longer necessarily an integer. On the topological side, weight corresponds to grading: the coefficient group $\pi_n tmf$ is a set of homotopy classes of maps $S^n \to tmf$. Generally, there are not spheres with non-integer dimensions, but in the $K(1)$-local category there are: the full set of gradings is called the $K(1)$-local Picard group. At odd primes it is $Hom(\Bbb Z_p^\times, \Bbb Z_p^\times) \cong \Bbb Z_p \times \Bbb Z/(p-1)$, and so we can define groups $\pi_n tmf_{K(1)}$, a group of weight-$n$ topological modular forms where $n$ is now drawn from this larger group that expresses something like $p$-adic interpolation. At $p=2$ there is an additional "exotic" part of the grading with no algebraic analogue.

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I don't have a good answer to the second half of your question. As far as I know, the analytic techniques you would need to give a definition of overconvergence are something that don't yet exist on the topological side. I think that this analytic point of view leads to really interesting questions, but while I've thought some about trying to answer them I don't have anything of substance I can offer right now.