# The $\heartsuit$ operator on $\mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})$

In Goldfeld's text Automorphic forms and L-functions for GL(n,R), for a fixed prime $$p$$ the operator $$\heartsuit \colon \mathcal{L}^2(SL_2(\mathbb{Z})\backslash \mathbb{H})\to\mathcal{L}^2_{cusp}(SL_2(\mathbb{Z})\backslash \mathbb{H})$$ is introduced to prove the existense of inifinitely many even cusp forms and consequently even maass forms.

Cant we just just take the collection of functions given by $$h(y)\cos (2\pi jx)$$ where $$h(y)$$ is real valued smooth function supported in $$[1,2]$$ and extend it to functions on $$SL_2(\mathbb{Z})\backslash \mathbb{H}$$. As $$j$$ varies through positive integers this already gives infinitely many independent even cuspidal functions. Why introduce $$\heartsuit$$ ?

• For one thing, your example... although it would indeed persuade many people... has the subtle flaw that it does not make the constant term identically 0, but only 0 in the usual fundamental domain. The fully automorphic version of this function, that is, made $SL_2(\mathbb Z)$-periodic by averaging/winding-up, has rather messy constant term at heights below $y=1$. That is, "cuspform" does not mean just "vanishes in the cusp", but something a bit more complicated. – paul garrett Oct 16 '18 at 1:58
• ah yes ! Had only the translates in my mind. Thanks. – pks Oct 16 '18 at 2:35