Download Arithmetic of Hyperbolic Three-Manifolds by Colin Maclachlan, Alan W. Reid PDF

By Colin Maclachlan, Alan W. Reid

Lately there was enormous curiosity in constructing suggestions in keeping with quantity thought to assault difficulties of 3-manifolds; includes many examples and plenty of difficulties; Brings jointly a lot of the present literature of Kleinian teams in a transparent and concise approach; at the present no such textual content exists

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Example text

If v is non-Archimedean, then v belongs to a place corresponding to a J>rime ideal P. The field kp is usually referred to as a P-adic field. The valuation ring of kp with respect to the extended valuation fJp is the ring of P-adic integers and is denoted by Rp . Recall that the valuation ring ll( vp) of k with respect to Vp is a discrete valuation ring whose unique r naximal ideal is generated by an element 1r E Rk. The same can be proved for the ring Rp. More precisely, the following holds: an The valuation ring Rp of the completion kp is a discrete valuation ring whose unique maximal ideal is generated by ip (1r ) .

I ) . 20 8. 0. 82 , . . , ,Bd } · Prove that N ( I ) 2 = discr{ /h , /h . . 8d } �k . 3. 14 . 9. Let l I k be an abelian extension. Show that there are infinitely many primes P with splitting pattern ([l : k] , 1) if and only if g is cyclic. 10. Show that in any finite extension l I k of number fields, there are infinitely many primes P of k which split completely in l. 1 1 . Deduce from Dirichlet 's density theorem that there are infinitely many rational primes in any arithmetic progression { an + d I ( a, d) = 1 } .

I f k is a number field, then from the preceding section we can embed k in l . l u · completions kv for each finite and infinite place of k. Thus if (V, B) regular quadratic space over k, then it gives rise to regular quadratic :;paees over the local fields, CC , JR, and It> for each finite place P. ltcgular quadratic spaces over CC are classified up to isometry by their di­ I f 'nsion and over lR by their dimension and signature, which is the number , f positive eigenvalues minus the number of negative eigenvalues.

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