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Intuition behind the Eichler-Shimura relation?


L-functions and higher-dimensional Eichler-Shimura relation

Answering the first question above, Matt Emerton gives a sketch of a proof of the Eichler-Shimura congruence relation which tells that the Hecke correspondence mod p is a sum of the graph of the Frobeniusm and it's transpose.

I am wondering how the statement and the proof can be generalized to moduli of higher dimensional abelian varieties with level structure (and maybe some more structure, like PEL).

It seems like the reason for having only two isogenies $E \to E'$ in Matt's answer is that p-isogenies correspond to subgroups of order p in E[p] as a scheme, and E[p] (if we assume E to be ordinary) in char p is a product of $Z/p$ and the dual group, $\mu_p$, and apparently these two groups: $Z/p$ and $\mu_p$ are the only nontrivial subgroups in E[p], so taking quotients we come up with the Frobenius and Verschiebung (dual isogeny).

Now let's say A is an abelian variety of dimension g in char p, which has maximal p-rank, i.e. $A[p] = Z/p^g \times \mu_{p^g}$. What are the subgroups of order $p^g$ of such a group? I am not familiar with how local groups behave, but I can assume there will be g+1 isogenies $A \to A'$, each one having the Kernel of the kind $H_1 \times H_2$, where $H_1$ and $H_2$ are subgroups in $Z/p^g$ and in $\mu_{p^g}$ respectively.

Also one probably needs a statement that abelian varieties with maximal p-rank are Zarisky open and dense in the moduli space, which is true in dimension 1.

Then the reduction mod p of the Hecke correspondence $T_p$, appropriately defined will be a sum of these g+1 cycles? Is that making any sense?



Dear Evgeny,

The paper Congruence relations on some Shimura varieties of Wedhorn (and some follow-up papers) proves what is (I think) the state of the art on Eichler--Shimura relations on higher dimensional Shimura varieties. Namely, he establishes the congruence relation for a wide class of PEL Shimura varieties. Just as you surmise, the density of the ordinary locus plays an important role in his argument.



  • $\begingroup$ Matt, thanks for the reference! The sum on page 25 in that paper looks more or less like I was thinking: in the case of Siegel modular varieties, T_p mod p is a sum of g+1 cycles (with multiplicities), each one with the prescribed type of isogeny's kernel. $\endgroup$ – Evgeny Shinder Jan 12 '11 at 3:32

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