12
$\begingroup$

Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective homomorphism $$\phi\colon \text{SL}(n,O_K) \longrightarrow (\prod_{i=1}^r \text{SL}(n,\mathbb{R})) \times (\prod_{i=1}^s \text{SL}(n,\mathbb{C}))$$ arising from these different embeddings. The image of $\phi$ is a lattice; it is easy to see that it is discrete (that's kind of the whole point of dealing with all of the embeddings at once), but it is nontrivial to see that it has finite covolume.

Question: Who was the first person to prove that the image of $\phi$ is a lattice? It certainly follows from work of Borel-Harish Chandra since the image of $\phi$ is an arithmetic subgroup, but I'm sure this special case was known long before their work. Also, who was the first person to consider a map like $\phi$?

$\endgroup$
1
  • 6
    $\begingroup$ Hilbert's student Blumenthal treated examples of this for $SL_2$ very early in the 20th century, to study Hilbert-Blumenthal modular forms. Siegel's arguments of the 1930s and 1940s for computing volumes of the quotients attached to $SL_n(\mathbb Z)$ and $Sp_n(\mathbb Z)$ immediately apply to number fields, imitating some aspects of Hecke's 1910s and 1920s treatments of $L$-functions attached to number fields. $\endgroup$ Sep 28, 2014 at 20:22

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.