This question is inspired by [Relation between Hecke Operator and Hecke Algebra](https://mathoverflow.net/questions/19684/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](https://mathoverflow.net/questions/9599/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 $p$)?



Hecke operators acting as correspondences?
The answer to your last question is yes, modulo my own misunderstandings. In Scholl's "Motives for modular forms", §4 ([DigiZeitschreiften](http://www.digizeitschriften.de/dms/img/?PID=GDZPPN00210749X)), 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](http://www.numdam.org/item?id=SB_1968-1969__11__139_0)).


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). 


