Hecke operators acting as correspondences?

Question

This question is inspired by Relation between Hecke Operator and Hecke Algebra

I remember having heard of yet another way of looking at Hecke operators acting on the spaces of modular forms for classical congruence subgroups of $SL_2(\mathbf{Z})$, and I would like to ask if anyone knows a reference for that. Here are some details and a related question.

There is a universal elliptic curve $U$ over $\mathbf{H}$, the upper half-plane. It is an analytic manifold obtained by taking the quotient of $\mathbf{C}\times\mathbf{H}$ by the action of $\mathbf{Z}^2$ given by $(n,m)\cdot (z,\tau)=(z+n+m\tau,\tau)$ where $n,m\in\mathbf{Z},z\in \mathbf{C}$ and $\tau\in\mathbf{H}$.

A torsion free finite index subgroup $\Gamma$ of $SL_2(\mathbf{Z})$ (congruence or not) acts on $U$. Here is the first question: for which $\Gamma$ is the quotient $U/\Gamma$ algebraic?

In any case $U/\Gamma$ is algebraic if $\Gamma=\Gamma(N)=\ker (SL_2(\mathbf{Z})\to SL_2(\mathbf{Z}/N))$ and $N\geq 3$. Denote the corresponding quotient $U/\Gamma(N)$ by $U(N)$. This is the universal elliptic curve with a level $N$ structure. (The notation $U(N)$ may be non-standard, in which case please let me know.)

There is a natural map $U(N)\to Y(N)=\mathbf{H}/\Gamma(N)$. Now take the $n$-th fibered cartesian power $U^n(N)$ of $U(N)$ over $Y(N)$ and let $p:U^n(N)\to Y(N)$ be the projection.

All derived direct images of the constant sheaf under $p$ decompose as direct sums of the Hodge local systems, which correspond to the symmetric powers of the standard representation of $\Gamma(N)$. Using the Eichler-Shimura isomorphism one can construct classes in $H^1(Y(N),V_k)$ from modular forms of weight $k+2$ for $\Gamma(N)$ where $V_k$ is the local system on $Y(N)$ that comes the $k$-th symmetric power of the standard representation. (In fact these classes can be interpreted in terms of the Hodge theory, see Eichler-Shimura isomorphism and mixed Hodge theory) So modular forms give elements in the $E_2$ sheet of the Leray spectral sequence for $p$.

I would like to ask: can one interpret the Hecke operators acting on modular forms as correspondences acting on $U^n(N)$ over $Y(N)$ (and hence, on the Leray spectral sequence for $p$)?

Answer 1

The answer to your last question is yes, modulo my own misunderstandings. In Scholl's "Motives for modular forms", §4 (DigiZeitschreiften), the Hecke operators are defined in this way, and I think the equivalence of these definitions is implied by the diagram (3.16) and proposition 3.18 of Deligne's "Formes modulaires et representations l-adiques" (NUMDAM).

Let p be a prime not dividing N, and let Y(N,p) be the moduli scheme parametrizing elliptic curves with full level N structure and a choice of a cyclic subgroup C of order p. There are two natural "forgetful" maps q₁ and q₂ from Y(N,p) to Y(N) -- the former forgets the cyclic subgroup, the latter takes the induced level N structure on the quotient by C.

From these two maps, one gets two different families of elliptic curves over Y(N,p). Explicitly, the pullback of Uⁿ(N) along q₁ is isomorphic to the n:th fibered power of the universal elliptic curve over Y(N,p); let us denote it Uⁿ(N,p). Let Q(N,p) be the quotient of U(N,p) by the cyclic subgroup C, and take its n:th fibered power as well. Then similarly Qⁿ(N,p) is the pullback of Uⁿ(N) along q₂. Finally, the quotient map gives us $\phi : U^n(N,p) \to Q^n(N,p)$.

But this data gives us the right correspondence on Uⁿ(N) over Y(N), namely, one takes the composite $q_{1\ast} \phi^\ast q_2^\ast$ (where I use q₁ and q₂ also for the induced maps on fibered powers of universal curves).