Is there a way to define Hecke operators "inherently" as certain endomorphisms of the Jacobian? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:21:58Z http://mathoverflow.net/feeds/question/33754 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33754/is-there-a-way-to-define-hecke-operators-inherently-as-certain-endomorphisms-of Is there a way to define Hecke operators "inherently" as certain endomorphisms of the Jacobian? Philip Engel 2010-07-29T04:48:25Z 2010-08-19T01:00:18Z <p>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.</p> http://mathoverflow.net/questions/33754/is-there-a-way-to-define-hecke-operators-inherently-as-certain-endomorphisms-of/36043#36043 Answer by William Stein for Is there a way to define Hecke operators "inherently" as certain endomorphisms of the Jacobian? William Stein 2010-08-19T01:00:18Z 2010-08-19T01:00:18Z <p>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.)</p> <p>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 <a href="http://mathoverflow.net/questions/8887/legitimacy-of-reducing-mod-p-a-complex-multiplication-action-of-an-elliptic-curve" rel="nofollow">here</a>). I <em>think</em> 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.</p>