13
$\begingroup$

Given a (quasi-) modular form $f(\tau)$ for some congruence subgroup (say) $\Gamma(k)$, we know that $f(N\tau)$ is a (quasi-) modular form for $\Gamma(N k)$. Is there anything known about when we can do a partial reverse, that is, when we can take linear combinations of (quasi-) modular forms for some higher level subgroup to obtain one of stricktly lower level?

An example is the following:

Let $E(q) = \sum_{k=0}^\infty \sigma_1(2k+1)q^{2k+1}$ and $A(q) = \sum_{k=1}^\infty \sigma_1(k)q^k$. Then $E(q)$ is modular with respect to a non-trivial character, and both $A(q^2)$ and $A(q^4)$ are quasi-modular of level 2 and 4, respectively (though not of pure weight).

However: it turns out that $$ E(q) + 3A(q^2) - 2A(q^4) = A(q) $$ which shows that a linear combination of higher level terms (and one which is modular with respect to a non-trivial character) yields one of lower level.

Is this simply random chance? Are there known relations of this type?

$\endgroup$

1 Answer 1

2
$\begingroup$

There are (at least) two answers: there is a Galois theory of modular functions (as in G. Shimura's 1971 book, or in Lang's book on elliptic functions), and a theory of newforms" (Atkin-Lehner, and also Casselman in a representation-theoretic context).

Thinking in terms of general Galois theory, it ought not be so surprising that various sums of non-invariant things become invariant, I suppose, but the particulars are often non-trivial.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.