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.
Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
|
4
1
|
||||||||||||||||||||
|
|
3
|
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. |
||||||||||||
|
