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 $p$)?


1 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).

  • $\begingroup$ I added links to free (legitimate) copies of the cited papers. $\endgroup$
    – David Roberts
    Jun 13, 2016 at 13:22

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