From the Eichler-Shimura relation, we have a formula for $T_p$ when we reduce $\textrm{End}(\textrm{Jac}(X))$ mod $p$. Explicity, $T_p=\textrm{Frob}_p+p\textrm{Frob}_p^{-1}$. Is there a way to define the Hecke operator as a lift of this operator satisfying certain other properties? Is there a definition of $T_p$ which does not rely on a moduli space interpretation or double coset operators, but "inherently" from the Jacobian? Excuse the vague formulation of this question; I am just learning about this stuff.
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$\begingroup$ Hi, Philip! You might be interested in some of the answers I got when I asked about the Eichler-Shimura relation: mathoverflow.net/questions/19390/… $\endgroup$– Qiaochu YuanCommented Jul 29, 2010 at 4:51
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$\begingroup$ ... or in the answer to this question: mathoverflow.net/questions/26871/… $\endgroup$– algoriCommented Jul 29, 2010 at 5:16
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1$\begingroup$ You don't say exactly what $X$ is, but from the context it must be something like $X_0(N)$ or $X_1(N)$ -- i.e., a moduli space of elliptic curves. So it would seem to be hard to get away from the moduli interpretation (but it is possible: there are other ways to think about these curves). How are you thinking about $X_0(N)$ if not as a moduli space or as $\Gamma_0(N) \backslash \overline{\mathcal{H}}$? $\endgroup$– Pete L. ClarkCommented Jul 29, 2010 at 6:28
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$\begingroup$ Optimally, $X$ would be any Riemann surface such that the endomorphism ring of its Jacobian is defined over $\mathbb{Q}$. In this case, the definition of $T_p$ couldn't rely an interpretation as a moduli space or quotient of the upper half-plane. This definition would coincide with the one we know for modular curves. $\endgroup$– Philip EngelCommented Jul 29, 2010 at 15:22
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3$\begingroup$ No. The Hecke operator is useful because relates to mod. forms (and one point of E-S relation is to relate Hecke e-values to Frob e-values on Tate module). The connection between Fourier coeffs of eigenforms and Frobenius e-values is not a matter of clever def'ns, so ad hoc def'n as a lift of endomorphism of the reduction would surely be useless (& baffling how to make such a lift aside from via moduli-theoretic def'n & using that to compute its effect on the special fiber). Method in Stein's answer surely will hit a brick wall; hard work can't be defined away. Moduli method is ubiquitous. $\endgroup$– BCnrdCommented Aug 19, 2010 at 2:18
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Philip remarks that he wants to define the Hecke operator as an endomorphism of the Jacobian of "any Riemann surface $X$ such that the endomorphism ring is defined over $\mathbf{Q}$", and at the same time he wants the Hecke operator to reduce to $\newcommand{\Frob}{\operatorname{Frob}} \newcommand{\Ver}{\operatorname{Ver}} \Frob_p + p \Frob_p^{-1}$. I think that for a generic Riemann surface $X$, the endomorphism ring is $\mathbf{Z}$. However, $\Frob_p + \Ver_p$ is very unlikely to be an integer, so for a general $X$ it is highly unlikely that there exists an element of ${\rm End}(X)$ that lifts $\Frob_p + \Ver_p$. (I'm using $\Ver_p$ to denote the dual of Frobenius.)
Here is another closely related point. When $A$ is an abelian variety with good reduction at a prime $p$, there is a natural map ${\rm End}(A) \to {\rm End}(A_{{\mathbf F}_p})$. (See my remark here). I think this map is injective (consider the induced map on Tate modules at some good prime $\ell$). Thus you could define the Hecke operator $T_p$ to be the unique (if it exists!) lift of $\Frob_p + \Ver_p$. That's intrinsic and makes no reference to any moduli space.
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$\begingroup$ WS: Yes, the reduction map always induces an injection on the endomorphism ring. This can be seen, for instance, by thinking about the Neron models: if the map were not injective, there would be a nonzero endomorphism which agrees on the generic fiber with the zero endomorphism, contradicting density of the generic fiber. $\endgroup$ Commented Aug 19, 2010 at 3:05
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1$\begingroup$ Pete, your suggested argument sounds a bit puzzling, insofar as you're not mentioning the most crucial part, which is the role of finite etale torsion levels which are fiberwise "schematically dense" and provide a bridge between the two fibers. That is, in your 3rd line it seems like you should have said "special fiber", not "generic fiber", and then you need to haul out the finite etale torsion levels to promote that to vanishing on the generic fiber, etc. (The example of $[p]$ on an elliptic curve with additive reduction shows the necessity of using finite etale torsion.) $\endgroup$– BCnrdCommented Aug 19, 2010 at 16:41
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$\begingroup$ Brian -- thanks for clarifying Pete's argument, which is definitely backwards. $\endgroup$ Commented Nov 20, 2010 at 18:17
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$\begingroup$ Additional remark: I just carefully reread Section 1 of Serre-Tate (Good Reduction of Abelian Varieties), and that makes everything crystal clear. $\endgroup$ Commented Nov 20, 2010 at 19:10