It is an important philosophy in the Langlands program that (roughly) the zeta function of a Shimura variety can be written as a product of L-functions of automorphic representations. The first evidence, also the most simplest example I think, is the classical result of Eichler and Shimura that (roughly) the zeta function of a modular curve is a product of L-functions of certain cusp forms. In their proof a key ingredient is the Eichler-Shimura relation relating the Hecke operator and the Frobenius map through the reduction mod $p$ at good primes. Later on Langlands in the Antwerp proceedings did similar things by comparing the Lefschetz and Selberg trace formulas to count points on the reduction mod $p$, which is referred as the Langlands-Kottwitz method later, which has many related works (generalizations, etc).

Now my problem is, I am not majoring in arithmetic stuff, yet I am very interested in this topic, I just wondered is there any reference demonstrating the simplest examples by using modern language? I think I can get a flavor of this subject by learning the most simplest examples such as modular curves, but I found the language in the original paper (Langland’s Antwerp 1972) is somehow too old. So is there any new reference reformulating them with some more advanced language, for example in terms of schemes, etale cohomology, shimura varieties, etc? I am avoiding learning the general theory of shimura varieties directly because it is much too heavy, actually I would like to use modular curve case as an inspiring example for learning the general shimura varieties. So is that possible to do that? By the way, in the series of Peter Scholze’s first three papers, the first paper named “The Langlands-Kottwitz approach for modular curves”, of course this paper is too difficult for me, so is it possible for me to know what he did, what is the difference between his work and the classical results on modular curves mentioned above, and what advanced knowledge do I need to finally understand his work?

Thanks for help!

  • $\begingroup$ The Hecke algebra $\mathbb{T}$ is an algebra of endomorphisms of the Jacobian $J(X_0(N))$ and those are defined over $\mathbb{Q}$, so they commute why the action of $\sigma \in Gal(\overline{\mathbb{Q}}/K)$ on $J(X_0(N))$, so the kernel of a representation $\mathbb{T} \to \mathbb{C}$ defines a subvariety $V$ of $Jac(X_0(N))$ that is sent to itself by the $\sigma$, and hence we have a Galois representation $Gal(\overline{\mathbb{Q}}/K)\to End(V)$ ? Finally any newform $\in S_2(\Gamma_0(N))$ is a representation of the Hecke algebra. $\endgroup$ – reuns Nov 23 '18 at 2:32

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