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The Class Number Formula

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The Class Number Formula The Class Number Formula Version 1.0—22nd November, 2013 klokken 11:26. A fundamental tool in the study of an algebraic number field is the Dedekind Zeta function. It is a generating function encoding a lot of the arithmetic properties of the field. We fix a number field K whose ring of integers is denoted by A, and we let d = [K : Q]. Then the Dedekind Zeta function of K is X 1 ζ (s) = (c) K N (a)s a⊆ A where a runs through all nontrivial, proper ideals of A. If we specialize K to be the field Q of rational numbers, all ideals are principal with a unique positive generator and so they are is in a one-one correspondence with the natural numbers, and we get back the Riemann Zeta function. We shall in this section establish the basic facts about the Dedekind Zeta function, and the first question that arises is: For which s is it defined? It turns out that the series in (c) converges for Re > 1, but one may show that the function can be analytically continued to the whole complex plain, except at s = 1 where the zeta function has a simple pole. For our modest needs it will be sufficient to know that the series converges for all real s > 1. Just like the Riemann Zeta function, the Dedekind Zeta function enjoys the property of having an Euler product. In the Riemann case the Euler product is a consequence of the fundamental theorem of arithmetic, stating that integers have a unique factorization in primes. Unique factorization is true for ideals in Dedkind rings, and therefore the Dedekind Zeta functions also has an Euler product: Y 1 ζ (s) = ; K 1 − N (p)−s p where p runs through all prime ideals in A. The proof follows the same lines as for the Riemann Zeta function, and we shall later on give it. There are several results and many conjectures about the values of the Dedekind Zeta function at the integers, the classic result being a formula for the residue of ζk(s) at s = 1 which involves the class number of K and several other basic invariants of K. It reads 2r+tπtRh lim (s − 1)ζK (s) = p (e) s!1+ µ j∆j Where r and t are the number of real and complex conjugate pairs of embeddings of K, R is the regulator of K and µ the number of roots of unity contained in K, and of course we have the old acquaintances, the class number h and the discriminant ∆. The Class Number Formula MAT4250 — Høst 2013 This formula is a main tool in computing the class number for many fields, the other invariants occurring in the formula being more accessible, and therefore several call it the class number formula. The first instance of the class number formula was found by Dirichlet already in 1837. He proved a version for quadratic fields formulated in terms of quadratic forms. The general formula as presented here was proven by Dedekind. Let σ1; : : : ; σr+t be embeddings of K, chosen like we did in section xxx. That is the first r are the real embeddings, in some order, and remaining t are chose one from each pair of complex conjugate embeddings. Dirichlet’s unit theorem gives us a set of fundamental units η1; : : : ; ηr+t−1. Then the regulator is the absolute value any of the minors (they are all equal up to sign since the row-sums of the matrix vanish, see lemma3 on page 11 below) of the (r + t) × (r + t − 1)-matrix R• = (j log jσj(ηi)j) (P) The regulator plays a more conceptual role as the volume of the fundamental parallel- lotopes of the logarithmic unit lattice in the trace-zero hyperplane H in Rr+t−1. We closely follow the presentation in chapter 5 of Borevich and Shafarevisch [BorShaf]. Introducing the class group The sum (c) above defining the Dedekind Zeta function may be split into a sum P ζK (s) = c ζc(s) of functions ζc(s), each corresponding to a class c of the class group CK . Indeed, one may take X 1 ζ (s) = c N (a)s a2c where the ideals a over which the sum is takes, are integral and, as indicated, confined to the class c. The functions ζc(s) all have the same residue at s = 1. This explains that the class number h appears as a factor in the class group formula, and the proof of the class group formula reduces to a computation of the common residue at s = 1 of the functions ζc(s), the result being the formula 2r+tπtR lim ζc(s) = p : s!1+ µ j∆j Now, let us fix an ideal a0 2 c. Any other element a of the class c is of the form (f)a0 for some f 2 K∗. For a to be an integral ideal a necessary and sufficient condition is −1 −1 that f belongs to a0 (this is nothing else but the definition of a0 ). Introducing this in the sum defining ζc(s), we find X 1 X 1 1 X 1 ζ (s) = = = : c N (a)s N ((f)a )s N (a )s N ((f))s a2c −1 0 0 −1 (f)⊆ a0 (f)⊆ a0 −1 The two last sums are taken over all integral and principal ideals (f) contained in a0 . Now, two principal ideals (f) and (f 0) are equal if and only if the elements f and f 0 are — 2 — The Class Number Formula MAT4250 — Høst 2013 associate, that is, f = uf 0 for a unit u. So, letting S be a set of representatives of the −1 associates, i.e., S contains exactly one element from each class of associates in a0 —or phrased in a slightly different manner, S is a fundamental domain for the action of the ∗ −1 unit group U = A on a0 —one has X 1 ζc(s) = s ; NK= (f) f2S Q where we as well has replaced the counting norm N ((f)) by the the norm NK=Q(f) (they are equal!). The proof of this formula has two main ingredients. First we establish the formula lims!1+ ζc(s) = γ=Γ where Γ is the volume of the Minkowski type lattice L −1 in a0 r t −1 K ⊗Q R = R ⊕ C associated to a0 , and γ is—with a friendly interpretation— the volume of the part of the quotient a0=U where the norm is at most one in absolute value (the absolute value of the norm is a well defined function on the quotient since N(u) = ±1 for units u). This is a special case of more general statement about counting lattice elements belonging to subsets of a certain type. We know the volume of the lattice L −1 . It was computed in xxx: a0 −1 −tp Γ = N (a0) 2 j∆K j The second ingredient of the proof is the computation of γ, and we shall find: γ = 2rπtR/µ. All together, this gives: −1 X lim (s − 1)ζK (s) =N (a0) lim (s − 1)ζc(s) = s!1+ s!1+ c2CK r t r+t t −1 γ −1 2 π R/µ 2 π RH =N (a0) h = N (a0) h = : −1 −tp p Γ N (a0) 2 j∆K j µ j∆K j The Minkowski setting We recall the setting from the section on Minkowski’s geometry of numbers. Let r t V = K ⊗Q R ' R ⊕ C ; the isomorphism being an isomorphism of algebras with multiplication in Rr ⊕ Ct being defined componentwise. There is the embedding Σ of K into V which is given by Σ(α) = (σj(α)) where the σi-s are r + t chosen embeddings, chosen according to the usual rule that the r first be the real embeddings and among the t next there should be one from each pair of complex conjugate embeddings. — 3 — The Class Number Formula MAT4250 — Høst 2013 r+t The logarithmic map. Recall the logarithmic map l : V0 ! R given by l(x) = (j log jxjj) where x = (xj) and the j are the weights corresponding to the embeddings σj; with j = 1 when σj is real, and j = 2 when σj is complex. The set V0 where the logarithm is defined, consists of the elements in V = Rr ⊕Ct all of whose coordinates are non-zero. Q i Norm and trace. The norm map N(x) on V is just the product xi of the weighted coordinates. It is multiplicative, and if ρx is the endomorphism of V given by multi- r+t plication by x, then det ρx = N(x).The trace map tr(y) on R is just the sum of the coordinates of y. One has log jN(x)j = tr(l(x)). The group of units. The group U of units is completely described by Dirichlet’s r+t−1 ∗ unit theorem. It decomposes in a product U ' µK × Z where µK ⊆ K is the group of roots of unity in K. We choose a basis η1; : : : ; ηr+t−1 for the free part, or as ones says, the units η1; : : : ; ηr+t−1 form a fundamental set of units. We have shown (and that was the hard part of proof of Dirichlet’s unit theorem) that the vectors r+t e1 = l(η1); : : : ; er+t−1 = l(ηr+t−1) are linearly independent in R ; indeed, we showed they form a basis for the trace-zero hyperplane H.
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