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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

$$S(K):=V(K)/W(K).$$

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

$$f\big\vert_{\alpha_p}(g):=f(g\alpha_p^{-1}).$$

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? –  plusepsilon.de Jan 20 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 at 17:51
    
Also they only claim that the kernel can be computed via strong approximation, not that the kernel is zero. –  plusepsilon.de Jan 21 at 10:35
    
So are there cases when this map is not injective? –  user45758 Jan 21 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 at 16:39

1 Answer 1

"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" (http://www.math.dartmouth.edu/~jvoight/articles/hmf-crm-bcn-080813.pdf), 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 ("...is 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 (http://homepages.warwick.ac.uk/staff/J.E.Cremona/courses/TCC_MF0910/lecture14.pdf), 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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