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I am trying to read a paper in which the authors claim that a certain map between vector spaces is injective, and this follows from the strong approximation theorem. I do not understand how the injectivity follows from strong approximation and so could someone help me. In what follows, I shall describe the map.

Let $F$ be a totally real number field of even degree. Let $\mathscr{O}_F$ denote its ring of integers. Let $P$ denote the set of places of $F$ and let $P_f$ denote the set of finite places and $P_\infty$ the set of infinite places. Let $D$ be the unique quaternion algebra over $F$ which is non-split exactly at all the infinite places of $F$. Let $O$ be a maximal order in $D$. For each place $\nu$ of $F$, define $F_\nu$ to be completion of $F$ at $\nu$ and similarly $\mathscr{O}_\nu$ to be the completion of $\mathscr{O}_F$ at $\nu$. Also define

$$D_\nu:=D\otimes_FF_\nu\qquad\qquad O_\nu:=O\otimes_{\mathscr{O}_F}\mathscr{O}_\nu. $$

If $\nu$ is a finite place, then we fix an isomorphism $\phi_\nu:D_\nu\xrightarrow{\sim}M_2(F_\nu)$ such that $\phi_\nu(O_\nu)=M_2(\mathscr{O}_\nu)$.

For any $F$ algebra $R$, define $B(R):=(D\otimes_FR)^{*}$. Let $\mathbb{A}$ be the adeles of $F$. Consider the subgroup of $B(\mathbb{A})$ given by $$U_0(\mathscr{O}_F)=\Big(\prod_{\nu\in P_f}O_\nu\times \prod_{\nu\in P_\infty} D_\nu^*\Big)\cong \Big(\prod_{\nu\in P_f}GL_2(\mathscr{O}_\nu)\times \prod_{\nu\in P_\infty} D_\nu^*\Big).$$ For an ideal $\mathscr{N}$ in $\mathscr{O}_F$, there is a natural group homomorphism

$$U_0(\mathscr{O}_F)\to GL_2(\mathscr{O}_F/\mathscr{N})=\prod_{\nu<\infty}GL_2(\mathscr{O}_\nu/\mathscr{N}).$$

Define $U_1(\mathscr{N})$ as the inverse image of the unipotent subgroup of $GL_2(\mathscr{O}_F/\mathscr{N})$ and $U_0(\mathscr{N})$ to be the inverse image of the Borel subgroup of $GL_2(\mathscr{O}_F/\mathscr{N})$.

Let $p$ be a prime of $\mathscr{O}_F$ which is coprime to $\mathscr{N}$. For $K$ an open compact subgroup of $B(\mathbb{A})$ consider the space of maps $B(\mathbb{A})\to \mathbb{F}_p$ which satisfy

$\bullet f(hg)=f(g)$ for $h\in B(F)$

$\bullet f(gu)=f(g)$ for $u\in K$.

Denote the space of such maps by $V(K)$. We have the reduced norm map $N:B(\mathbb{A})\to \mathbb{A}^{\times}$ (the ideles). Denote by $W(K)$ the subspace of $V(K)$ containing maps which factor through the reduced norm. Define


If we take $K=U_1(\mathscr{N})$ then the Jacquet Langlands correspondence identifies the space of Hilbert modular forms of parallel weight 2 for the congruence subgroup $\Gamma_1(\mathscr{N})$ with the above space $S(K)$.

Clearly, if $K'\subset K$, then there is a natural restriction map $$S(K)\to S(K').$$

Let $\alpha_p\in B(\mathbb{A})$ denote the element which is 1 at places $\nu\neq p$ and at $p$ is given by the diagonal matrix $\rm{diag}(p,1)$. Define


This defines a map from $S(K)\to S(\alpha_p^{-1}K\alpha_p)$. Composing with the restriction map, we get a map $S(K)\to S(\alpha_p^{-1}K\alpha_p\cap K)$, which we continue to denote $f\vert_{\alpha_p}$.

Letting $K=U_1(\mathscr{N})$ in the above, we get that

$$\alpha_p^{-1}K\alpha_p\cap K=U_1(\mathscr{N})\cap U_0(p)$$ and so we get a map \begin{align*} S(U_1(\mathscr{N}))^{\oplus 2}&\to S(U_1(\mathscr{N})\cap U_0(p))\\ (f_1,f_2)&\mapsto f_1+f_2\big\vert_{\alpha_p}. \end{align*}

The authors say that the the above map is injective and this follows from strong approximation, however, it is not clear to me how this follows. I would be grateful if someone could explain to me how.

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Could you please give a reference to the paper? – Marc Palm Jan 20 '14 at 13:16
It is a paper by Edixhoven and Khare titled "Hasse Invariant and Group Cohomology". The statement I am referring to is in the second para of section 4 of their paper. – user45758 Jan 20 '14 at 17:51
Also they only claim that the kernel can be computed via strong approximation, not that the kernel is zero. – Marc Palm Jan 21 '14 at 10:35
So are there cases when this map is not injective? – user45758 Jan 21 '14 at 16:24
They do say that they want an analog of lemmas 1 and 3, and both these lemmas are injectivity statements. – user45758 Jan 21 '14 at 16:39

"Strong approximation" can be used in a number of ways!

In Lemma 3, they use strong approximation to know that the reduction map $\mathrm{SL}_2(\mathcal{O}_K) \to \mathrm{SL}_2(\mathcal{O}_K/n\mathcal{O}_K)$ is surjective for all $n \geq 1$. They then use this and Shapiro's lemma to reduce the group cohomology from $\Gamma_1(\mathcal{N}) \cap \Gamma_0(\wp)$ to $\Gamma_1(\mathcal{N})$, but with more complicated coefficients.

A similar thing works for definite quaternion algebras: by allowing the codomain to be more complicated, one can consider maps that are invariant under $\Gamma_1(\mathcal{N})$ with image in (a submodule of) $\mathbb{F}_p[\mathbb{P}^1(\mathcal{O}_F/p\mathcal{O}_F)]$. Dembele and I explain this in our paper "Explicit methods for Hilbert modular forms" (, page 16, in the global language, and then again on page 45 in the adelic language.

Edixhoven-Khare seem to claim that now the kernel should be visible, and I think if you follow the strategy from the other lemmas, along with their next sentence (" irreducible as a consequence of..."), you should be able to get what you need.

If you want something more explicit, it's a matter of tracing through what the degeneracy maps look like. Cremona and Dembele worked this out carefully in some lecture notes (, slide 17. The first degeneracy map is just the one that takes functions with values in $\mathbb{F}_p$ and considers them as $\mathbb{F}_p$-vector valued; the other has a similarly concrete description.

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