# Is there an integral pairing between quaternionic Hecke algebras and cusp forms?

Let $F$ be a totally real number field with integers $\mathcal{O}_F$ and $B$ a quaternion algebra over $F$ split at exactly one infinity place.Fix $n\geq 1$ and like in the special case $F=\mathbb{Q}, B=M_2(\mathbb{Q}), \Gamma_1(p^n)$ one can associate to a finite place $\mathfrak{p}$ of $F$, where $B$ is split a geometric object:

$M_{n,H}(B)\:=B^\times\backslash (B\otimes_\mathbb{Q} A)^\times/ U_1(\mathfrak{p}^n)\times H$, where $A$ is the ring of rational adeles, $U_1(\mathfrak{p}^n)$ is the standard compact open congruence subgroup of $(B\otimes F_\mathfrak{p})^\times$ (depending on a choice of orders) and $H$ is 'sufficiently small'. In the $F\neq \mathbb{Q}$ case this is called a Shimura curve. Carayol has shown that this curve has good reduction at $\mathfrak{p}$: It admits a smooth proper model over the $\mathfrak{p}$-integers of $F$.

But: it has no cusps, therefore no q-expansion principle is available. And the points of this curve have no direct moduli interpretation, so there does not live a universal family of 'abelian varieties+level structure + X' on it. One can still, though, by the usual adelic machinery, define good automorphic forms on these curves. By the lack of cusps, I guess we can call them all "cusp forms".

In the "classical" (modular curve) case $Y_{\Gamma,\mathbb{Z}}$, there is another way to define modular forms: let $\tau\colon\mathcal{E}\to Y_{\Gamma,\mathbb{Z}}$ denote the universal family of elliptic curves over it and denote $\omega:=\tau_*\Omega_{\mathcal{E}/Y}^1$. This line bundle extends to the cusps and modular forms can be defined as sections of the higher tensor powers of this extended bundle.

Question 1: Is there something like this construction for Shimura curves? For example, from what I understand, Carayol proved the existence of good models for this curves also via a moduli interpretation (of unitary Shimura varieties).

Now for the title: If you have q-expansions at hand, you can prove that the classical (complex) pairing of cusp forms with the complex Hecke algebra, gives rise to an integral p-adic perfect pairing:

$T_{\Gamma,\mathbb{Z}_p}\times S_k(\Gamma,\mathbb{Z}_p(\zeta_N))\to \mathbb{Z}(\zeta_N)$.

For Shimura curves I only know of an adelic way to introduce the Hecke operators.

Question 2: Is there a similar pairing as above?

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I am very suspicious of your claim that $M_{n, H}(B)$ has a model with good reduction at $\mathfrak{p}$. This is certainly false for $F = \mathbf{Q}$ and $B = M_2(\mathbf{Q})$. –  David Loeffler May 22 '14 at 16:34
@David: Actually, that case is not treated in Carayol's paper and I meant the claim to include only the Shimura curve setting. –  Konrad May 22 '14 at 16:52
It's still wrong for $B$ a division algebra. Check your sources again. –  David Loeffler May 22 '14 at 19:51
Well, I guess there is a reason, why certain papers are named the way they are... Sorry and thanks for your scepticism, @David. I will edit the question to reflect this. Do you happen to know if there is work on the above problems, smooth model or not? I guess Buzzard's question here on coherent sheafes for Shimura varieties goes some way for the first one. –  Konrad May 23 '14 at 13:24