I came onto mathoverflow for a reference on an unrelated topic, but since I noticed this question I thought I would chime in: the image of the J homomorphism in the stable homotopy groups of spheres, by the work of Adams, is dictated by the denominators of Bernoulli numbers $-B_n/n$ (i.e. $\zeta(1-2n)$) and it is these orders that are what are fundamentally related to the vector fields on spheres problem.

The image of $J$ is the $n = 1$ instance of some general $v_n$-periodic families in the stable stems called Greek letter elements (See Miller-Ravenel-Wilson's Annals paper "Periodic phemonena in the Adams-Novikov spectral sequence"). The n=2 analog is called the divided beta family.

Adams's identification of the Image of J with denomenators of Bernoulli numbers was through his "e"-invariant, which took values in Q/Z equal to the images of the Bernoulli numbers.
Gerd Laures, in his paper "The topological q-expansion paper" introduced a higher form of the "e" invariant called the "f"-invariant, which takes values in the quotient of the Katz's ring of divided congruences of modular forms by the sum of the integral q-expansions, and the classical modular forms over Q. (A kind of higher analog of Q/Z) He showed that his f invariant gave an injection from the divided beta family to this quotient, but did not characterize the image.

In my paper "congruences of modular forms and the divided beta family in homotopy theory" I showed that the divided beta family was actually in bijective correspondence with a set of congurence conditions between modular forms. Gerd and I later showed in another paper that these congruences I wrote down were precisely the ones coming from his f invariant.

But what I do not know, and would be curious if any user had an idea concerning this, is if there is a natural family of modular forms, kind of like a higher form of Eisenstein series, which would realize these beta elements in its image in the quotient of the ring of divided congruences.

The phenomenon I describe is not unique to n = 2, abstractly there is some kind of "congruence condition" amongst holomophic automorphic forms on U(1,n-1) that describes the nth Greek letter family in the stable homotopy groups of spheres - I just don't know how to express this congruence condition in clasical terms.