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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:

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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?

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    $\begingroup$ 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
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    $\begingroup$ 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
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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.)

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

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