Is there a way to define Hecke operators "inherently" as certain endomorphisms of the Jacobian? 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.
 A: 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 $\{\rm Frob}_p + p{\rm Frob}_p^{-1}$.   I think that for a generic Riemann surface $X$, the endomorphism ring is $\mathbf{Z}$.  However, $\{\rm Frob}_p + {\rm Ver}_p$ 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 $\{\rm Frob}_p + {\rm Ver}_p$.   (I'm using ${\rm Ver}_p$ 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 ${\rm Frob}_p + {\rm Ver}_p$.  That's intrinsic and makes no reference to any moduli space.
