I am reading the paper Geometric construction of cohomology for arithmetic groups I by Millson and Raghunathan, and unfortunately I know almost no number theory, so I am confused by the following proposition:

enter image description here

Specifically, what does it mean when they say "Let $S=\{p_1, \ldots, p_m\}$ be a finite set of primes of $E$ and let $G(O_S)$ be the group of $S$ integral matrices over $E$ that preserve $Q$."

  1. What is $O_S$ exactly? I found the definition of $O_R$ for $R$ a ring to be be the ring of roots of monic polynomials with integer coefficients. But $S$ is obviously not a ring. Is the intent her to look at the subring generated by $S$ and find the roots within this subring?

  2. What do they mean be a "finite set of primes of $E$"? When I look up the definition of a prime element of a ring it says that an element $p$ of a ring $R$ is prime "if it is nonzero, has no multiplicative inverse and satisfies the following requirement: whenever $p$ divides the product $xy$ of two elements of $R$, it also divides at least one of $x$ or $y$." But this doesn't make sense in this context since every non-zero element of a field is invertible. Do you think that the authors mean that these are irreducible elements instead? Or is there some other meaning of prime in this context?

  • 1
    $\begingroup$ planetmath.org/RingOfSintegers $\endgroup$ – Qiaochu Yuan Nov 22 '18 at 3:03
  • $\begingroup$ @QiaochuYuan Thanks! I've added a second question regarding terminology. Could you help me with this as well? $\endgroup$ – ಠ_ಠ Nov 22 '18 at 7:28
  • 1
    $\begingroup$ I think "prime of $E$" means a prime element of the ring of integers of $E$. $\endgroup$ – Qiaochu Yuan Nov 22 '18 at 20:21

Here $O_S$ will be the ring of integers of $E$ with the inverses of the $p_i$ adjoined. That is, $$O_S=O_E[p_1^{-1},\ldots,p_m^{-1}]$$ where $O_E$ is the ring of algebraic integers in $E$. (This is standard jargon in algebraic number theory.)

  • $\begingroup$ Thanks! Do you happen to know of a good reference for this sort of stuff? $\endgroup$ – ಠ_ಠ Nov 22 '18 at 3:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.