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1.1. Outline. In Section 2 we start by recalling the definition and basic properties of Milnor– Witt K-theory, before we move on to computing Milnor–Witt K-groups of the rationals as well as some local fields. We finish this preliminary section by defining valuations in the setting of Milnor–Witt K-theory, lifting the classical valuations on a number field. MW In Section 3 we put a topology on the first Milnor–Witt K-group K1 (Fv) of a completion MW × of the number field F , in such a way that K1 (Fv) becomes a covering of Fv . This is used in Section 4 where we define id`eles and id`ele class groups in the setting of Milnor–Witt K-theory. 0 We show that the associated volume zero id`ele class group CF is again a compact topological 0 group extending the classical compact group CF . See Proposition 4.8 for more details. MW MW In Section 5 we shift focus from K1 to K2 . We definee Hilbert symbols on Milnor–Witt K-groups and show a Moore reciprocity sequence in this setting; see Proposition 5.10. We then MW move on to the study of K2 of rings of integers in Section 6. Finally, in Section 7 we show an analog of Hasse’s norm theorem similar to the generalizations of Bak–Rehmann [BR84] and Østvær [Øst03].

1.2. Conventions and notation. Throughout we let F denote a number field of signature (r , r ), and we let Pl denote the set of places of F . For any v Pl , we let F denote the 1 2 F ∈ F v .completion of F at the place v, and we let iv : F ֒ Fv denote the embedding of F into Fv By Ostrowski’s theorem, Pl decomposes as a disjoint→ union Pl = Pl Pl of the finite and F F 0 ∪ ∞ infinite places of F , respectively. The set Pl∞ of infinite places of F decomposes further into the r c nc sets Pl∞ and Pl∞ of real and complex infinite places, respectively. Finally, let PlF denote the set of noncomplex places of F . M In to streamline the notation with the literature, we will often use the notations Kn (F ) and K (F ) interchangeably whenever n 0, 1, 2 . n ∈{ } 1.3. Acknowledgments. I thank Ambrus P´al for interesting discussions and for encouraging me to write this text. Furthermore, I am very grateful to Jean Fasel for particularly helpful comments on a draft, and to Kevin Hutchinson for his interest and for useful comments and remarks. This work was supported by the RCN Frontier Research Group Project no. 250399.

2. Preliminaries 2.1. Milnor–Witt K-theory. We start out by providing some generalities on Milnor–Witt K- groups, mostly following Morel’s book [Mor12].

MW Definition 2.1 (Hopkins–Morel). The Milnor–Witt K-theory K∗ (F ) of F is the graded asso- ciative Z-algebra with one generator [a] of degree +1 for each unit a F ×, and one generator η of degree 1, subject to the following relations: ∈ − (1) [a][1 a]=0 for any a F × 1. (2) [ab] =− [a] + [b]+ η[a][b].∈ \ (3) η[a] = [a]η. (4) (2 + η[ 1])η = 0. − NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 3

MW MW MW We let Kn (F ) denote the n-th graded piece of K∗ (F ). The product [a1] [an] Kn (F ) ··· ∈ MW may also be denoted by [a1,...,an]; by [Mor12, Lemma 3.6 (1)], these symbols generate Kn (F ). MW M For any n 0 there is a surjective map p:Kn (F ) Kn (F ) from Milnor–Witt K-theory to Milnor K-theory≥ given by killing η. Its is In+1(F ),→ the (n+1)-th power of the fundamental ideal in the Witt ring of F . Moreover, the negative Milnor–Witt K-groups of F are all isomorphic MW to the Witt group of F , while K0 (F ) recovers the Grothendieck–Witt group GW(F ) of F MW [Mor12, Lemma 3.10]. In fact, by [Mor04b], the group Kn (F ) fits in a pullback square

MW p M Kn (F )Kn (F )

In(F ) In(F )/In+1(F ) in which the left hand vertical homomorphism is given by mapping [a1,...,an] to the Pfister form a ,...,a . hh 1 nii M 2.1.1. The residue map. In Milnor K-theory, there is a residue map, or tame symbol, ∂v :K∗ (F ) M → K∗−1(k(v)) defined for each finite place v of F . These homomorphisms assemble to a total residue map ∂ :KM(F ) KM (k(v)) ∗ → ∗−1 vM∈Pl0 given as ∂ = ∂ . Morel shows in [Mor12, Theorem 3.15] that the same is true for Milnor– v∈Pl0 v Witt K-theory. More precisely, for each uniformizer πv for v, there is a unique homomorphism πv L ∂v giving rise to a graded homomorphism ∂ = ∂πv :KMW(F ) KMW(k(v)) v ∗ → ∗−1 vM∈Pl0 vM∈Pl0 πv which commutes with η and satisfies ∂v ([πv,u1,...,un]) = [u1,..., un] whenever the ui’s are πv units modulo πv. In contrast to the case for Milnor K-theory, the maps ∂v depend on the choice of uniformizer πv; this stems from the relation [uπv] = [u]+[πv]+η[u, πv]. One can however define πv a twisted version of Milnor–Witt K-theory in order to make the maps ∂v canonical. Indeed, MW MW for any field k and any one-dimensional k- V , let K∗ (k, V ) := K∗ (k) Z[k×] × MW ⊗ Z[V 0], where Z[k ] acts by u u on K∗ (k) and by multiplication on Z[V 0]. Then the \ MW MW 7→m h im2 ∨ \ map ∂v :K∗ (F ) K∗−1 (k(v), ( v/ v) ) given by ∂v([πv,u1,...,un]) = [u1,..., un] πv is independent of the→ choice of uniformizer [Mor12, Remark 3.21]. ⊗

2.1.2. Transfers in Milnor–Witt K-theory. If L/F is an extension of number fields, recall that there exist norm maps, or transfer maps in Milnor K-theory [Kat78], τ :KM(L) KM(F ), L/F n → n × × which generalize the classical norm NL/F : L F . Similar maps exist also for Milnor–Witt K-theory, and are constructed in the same way→ as for the Milnor K-groups. We briefly recall this construction. Let α be a primitive element for the extension of number fields L/F , so that L = F (α). Let P F [t] be the minimal polynomial of α. The transfer map ∈ τ :KMW(L) KMW(F ) L/F n → n is defined by using the split exact sequence [Mor12, Theorem 3.24]

0 KMW(F ) KMW(F (t)) ∂ KMW(k(x)) 0 → n+1 → n+1 −→ n → x∈(A1 )(1) MF NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 4 as follows. Let s := s denote the section of ∂, and let y (A1 )(1) be the closed point x x ∈ F corresponding to P . Then we define τL/F as the composition L ∼ s −∂−1/t τ :KMW(L) = KMW(k(y)) y KMW(F (t)) ∞ KMW(F ), L/F n −→ n −→ n+1 −−−−−→ n −1/t where ∂∞ is the residue map corresponding to the on F (t) with uniformizer 1/t. − 2.2. Milnor–Witt K-theory of local fields. Proposition 2.2. Let v Pl be a place of F . We have ∈ F M Kn (Fv) ∼= µ(Fv) Av, v Pl0, n 2 MW ⊕ ∈ r ≥ Kn (Fv) ∼= Z Av, v Pl∞, n 1 M⊕ ∈ c ≥ K (Fv), v Pl , n 1, n ∈ ∞ ≥ where the Av’s are uniquely divisible groups. c Proof. For v Pl∞, the statement follows from the fact that C is quadratically closed [Mor12, Proposition 3.13].∈ If v Pl we have I3(F ) = 0 by [MH73, p. 81], and hence the exact sequence ∈ 0 v 0 In+1(F ) KMW(F ) KM(F ) 0 → v → n v → n v → MW M yields Kn (Fv) = Kn (Fv) for each n 2. ∼ MW ≥ It remains to show that Kn (R) = Z A, where A is a uniquely divisible group and n 1. MW ∼ ⊕ MW ≥ Let Λ∞ K∗ (R) be the graded subring of K∗ (R) generated by the elements [ 1,..., 1] MW ⊆ m − − ∈ Km (R) for m 0, and let Λ∞ denote the m-th graded piece of Λ∞. Moreover, let Λ0 := × ≥ MW η, [a]: a R>0 be the graded subring of K∗ (R) generated by η and the elements [a] with a ∈ m a positive real number. For any m Z we let Λ0 denote the m-th graded piece of Λ0. We aim MW R n n ∈ n Z n to show that Kn ( ) ∼= Λ0 Λ∞; that Λ∞ ∼= ; and that Λ0 is a uniquely divisible . ⊕ To this end, first notice that the relation [ 1,a] = [a,a] implies that − ([ 1]+ x)([ 1] + y) = [ 1, 1]+ x[ 1] + [ 1]y + xy Λ Λ − − − − − − ∈ ∞ ⊕ 0 for any x, y Λ . Now KMW(R) is generated by the symbols [a ,...,a ] for which a R× ∈ 0 n 1 n i ∈ [Mor12, Lemma 3.6 (1)]. If ai R>0 for all i, then the observation above along with the relation ∈ MW n n [ x] = [ 1] + [x]+ η[x, x] implies that any generator [ a1,..., an]ofKn (R) lies in Λ∞ Λ0 . We− note− that this is indeed a direct sum, because if m±[ 1,...,± 1] = m [a ,...,a ] ⊕ Λn − − j j 1j nj ∈ 0 for some m,mj Z and aij > 0, then we obtain the relation of Pfister forms ∈ P m 1,..., 1 = m a ,...,a In(R). hh− − ii j hh 1j nj ii ∈ j X Since the signature (with respect to the standard ordering on R) of the left hand side is m 2n MW n · n while the signature of the right hand side is 0, we must have m =0. Thus Kn (R) = Λ∞ Λ0 . n n n ∼ ⊕ It remains to identify Λ∞ and Λ0 . By [MH73, p. 81], the group I (R) is the free abelian group n n n n generated by 1,..., 1 = 2 1 . It follows that the canonical map Λ∞ I (R) = 2 1 Z hh− − ii h i M → ∼ h i is an . Letting A denote the uniquely divisible subgroup of Kn (R), we now have a short exact sequence 0 In+1(R) Λn Λn Z/2 A 0. (2) → → ∞ ⊕ 0 → ⊕ → n+1 n+1 n n The first map sends the generator 1,..., 1 of I (R) to η[ 1] = 2[ 1] Λ∞, and nhh− − ii − − −n ∈  hence it has image contained in Λ∞. Thus the short exact sequence (2) yields Λ0 ∼= A. NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 5

Remark 2.3. In the case n = 2 we can give the following explicit description of the MW appearing in Proposition 2.2. If v Pl0, then the maps K2 (Fv) = K2(Fv) µ(Fv) are given ∈ ∼ → r by the classical local Hilbert symbols ( , )v [Neu99, V §3]. On the other hand, if v Pl∞, the MW − − ∈MW homomorphism K2 (R) Z is defined as follows. The canonical homomorphism K2 (R) I2(R) carries a generator→ [a,b] to the Pfister form a,b . Recall that I2(R) is the free abelian→ hh ii r MW group on 1, 1 = 4 1 . Using this we define, for v Pl∞, the map K2 (Fv) Z as the compositehh− − ii h i ∈ → ∼ ∼ ∼ ∼ KMW(F )K= MW(R) I2(R)= 4 1 Z = 4Z= Z. (3) 2 v 2 h i σ φ Here σ is the signature homomorphism with respect to the standard ordering on R. We can think of the composite homomorphism (3) as a “Z-valued ” extending the classical Z/2-valued Hilbert symbol on R. We will return to this point of view in Section 5.1. 2.3. Milnor–Witt K-theory of the rationals. Proposition 2.4. For each n Z, the residue map ∂ induces an isomorphism ∈ KMW(Q) = Z KMW(F ). n ∼ ⊕ n−1 p Mp≥2 In particular, KMW(Q) = Z GW(F ); KMW(Q) = Z F×, 1 ∼ ⊕ p 2 ∼ ⊕ p Mp≥2 Mp≥2 while KMW(Q) = Z for n 3. n ∼ ≥ Proof. The statement is clear for n 0 by the description of the Witt-, and the Grothendieck– Witt group of Q [MH73, IV §2]. Hence≤ we may assume that n 1. MW ≥ Upon defining graded subrings Λ∞ and Λ0 of K∗ (Q) as in the proof of Proposition 2.2 we MW n n n MW find Kn (Q)=Λ∞ Λ0 . Here Λ∞ = Z is the free abelian group on [ 1,..., 1] Kn (Q), n ⊕ ∼ MW − − ∈ × while Λ0 is generated by η along with the symbols [a1,...,an] Kn (Q) for a1,...,an Q>0. n ∈ ∈ The group Λ∞ is related to powers of the fundamental ideal of Q as follows: If n = 1, then Λ1 = Z lies split in the image of the injection • ∞ ∼ I2(Q) = Z Z/2 KMW(Q) ∼ ⊕ → 1 p>2 M given by a,b η[a,b]. hh ii 7→ If n 2, then In+1(Q) = Z = Λn . • ≥ ∼ ∼ ∞ Thus have a projection map q : In+1(Q) Z for each n 1, which we will use to identify Λn. → ≥ 0 Since KM(Q) = Z/2 KM (F ) for n 1, it follows that we have a commutative diagram n ∼ ⊕ p≥2 n−1 p ≥ with exact rows L n M 0 ker q Λ0 Kn−1(Fp) 0 Mp≥2

0 In+1(Q)KMW(Q) Z/2 KM (F ) 0 n ⊕ n−1 p Mp≥2 q π

0 Z Z Z/2 0. n Z Here the map π is the projection onto Λ∞ ∼= . We have the following cases: NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 6

For n = 1 the upper short exact sequence reads • 0 Z/2 Λ1 Z 0. → → 0 → → p>2 M Mp≥2 Since this sequence splits, we conclude that

Λ1 = Z (Z Z/2) = GW(F ). 0 ∼ ⊕ ⊕ ∼ p p>2 M Mp≥2 If n 2, then the map q is an isomorphism and the claim follows. • ≥ This finishes the proof. 

2.4. Valuations. Let v be a finite place of F . Classically, the v-adic discrete valuation on the local field Fv is a map ord : F × Z. v v → M M By definition, ordv coincides with the residue map ∂v :K1 (Fv) K0 (k(v)) = Z on Milnor K-theory. On the other hand, if v is an infinite real place of F , then→ we have a valuation ord : F × Z/2 v v → × R× under which x Fv ∼= maps to 0 if x> 0, and 1 if x< 0. Finally, by definition, ordv :=0 whenever v is an∈ infinite complex place. Definition 2.5. Let v be a place of F . (1) If v Pl , we define ∈ 0 ord :KMW(F ) GW(k(v), (m /m2)∨) v 1 v → v v by ordv := ∂v, whereg∂v is the residue map on Milnor–Witt K-theory. (2) If v Plr , we define the map ∈ ∞ g ord :KMW(F ) = KMW(R) Z v 1 v ∼ 1 → by ordv([x]) := φ(σ( xg)), where φ and σ are defined similarly as in diagram (3) in Remark 2.3. hh ii (3) Forgv Plc , we set ord := 0. ∈ ∞ v MW In any case, we let K ( v) := ker(ordv). 1 O g 2.4.1. By definition, we obtain commutative diagrams g

MW gordv MW gordv m m2 ∨ K1 (R) Z K1 (Fv ) GW(k(v), ( v / v) )

p p rk

× ordv × ordv R Z/2, Fv Z for any v Plr and v Pl , respectively. Here rk denotes the rank homomorphism. ∈ ∞ ∈ 0 Lemma 2.6. Let v Pl be a place of F . ∈ F MW × If v is either an infinite place or a nondyadic finite place, then K1 ( v) = v (where, • × × r × × c O ∼ O by definition, v := R>0 for v Pl∞ and v := C for v Pl∞). If v is a dyadicO place, then there∈ is a shortO exact sequence ∈ • 0 I2(F ) KMW( ) × 0. → v → 1 Ov → Ov → NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 7

Proof. The statement is clear for the infinite places by Proposition 2.2. Let v Pl0 be a finite place of F , and consider the commutative diagram with exact rows ∈ ∂ 0 KMW( )KMW(F )Kv MW(k(v)) 0 1 Ov 1 v 0 p′ p p′′ ord 0 × F × v Z 0. Ov v 2 ′′ If v ∤ 2, then the induced map ker p ∼= I (Fv) ∼= Z/2 ker p ∼= I(k(v)) ∼= Z/2 is an isomorphism and so we conclude by the snake lemma. → ′′ MW If on the other hand v 2, then k(v) is quadratically closed and hence the map p :K0 (k(v)) Z is an isomorphism.| In this case the snake lemma applied to the above diagram yields→ ′ 2  ker p ∼= ker p ∼= I (Fv). MW 3. Topology on K1 (Fv) MW We now aim to put a topology on K1 (Fv), for each place v of F , in such a way that the homomorphism p:KMW(F ) F × becomes a covering map. 1 v → v 3.0.1. In general, suppose that we are given an abelian group G (written additively) along with a subgroup H of G which is a topological group. We can then extend the topology on H to a topology on G as follows. Choose a set theoretical section s of the quotient map π : G G/H, and let G/H have the discrete topology. Then s defines a partition → G = (s(x)+ H) x∈aG/H of G, and there is a natural topology on each coset (s(x)+ H) ∼= H coming from that on H. We can then declare a subset U G to be open if and only if U (s(x)+ H) is open for every x G/H. This turns G into a topological⊆ group. ∩ ∈ 3.0.2. Now let v be a place of F . Then we have a short exact sequence 0 F ×2 KMW(F ) I(F ) 0 → v → 1 v → v → ×2 MW in which the map Fv K1 (Fv) is given by a (2 + η[ 1])[a]. In the notations above, →MW ×2 7→ − we can then let G := K1 (Fv) and H := Fv . The quotient map π : G G/H is given as → ×2 [a] a . Using the set theoretical section s( x ) := [x] of π along with the fact that Fv is 7→ hh ii hh ii MW a topological group, we obtain by the above a topology on K1 (Fv). MW × Proposition 3.1. For any v PlF , the map p:K1 (Fv) Fv is a covering of topological groups. Furthermore, if v is a finite∈ place of F , then we have→ the following: (1) The map p is proper. MW MW (2) The space K1 (Fv ) is locally compact and totally disconnected, and K1 ( v) is com- MW O pact in K1 (Fv). nc Proof. If v is a complex place there is nothing to show, so we may assume that v PlF . We have a commutative diagram with exact rows ∈

×2 MW 0 Fv K1 (Fv) I(Fv) 0

= p

×2 × 2 0 Fv Fv I(Fv)/I (Fv) 0. That p is a covering map follows from this diagram along with the fact that the quotient topology × ×2 2 on Fv /Fv ∼= I(Fv)/I (Fv) is the discrete topology. NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 8

2 Now let v be a finite place of F . Then I (Fv) ∼= Z/2, so that we have an exact sequence p 0 Z/2 KMW(F ) F × 0. → → 1 v −→ v → Thus p is in this case a two sheeted covering map, hence proper. The claim (2) follows from ×  Lemma 2.6 along with the properties of the topology on Fv . 4. Ideles` MW By using the topology on K1 (Fv ) defined in the previous section, we can construct id`eles and id`ele class groups in the setting of Milnor–Witt K-theory: Definition 4.1. For any finite set S of places of F , put J (S) := KMW(F ) KMW( ). F 1 v × 1 Ov vY∈S vY6∈S e The Milnor–Witt id`ele group JF of F is defined as the direct limit

JF := lim JF (S), e −→S where S ranges over all finite subsets ofe PlF . e Proposition 4.2. There is a short exact sequence p 0 I2(F ) J J 0, → v → F −→ F → vM∈PlF where the map p is induced by the projection mapse from Milnor–Witt K-theory to Milnor K- theory.

Proof. For the definition of the homomorphism p, note that we have projection maps JF (S) J (S) for any finite set of places of F . Here J (S) := F × ×. The map p is then→ F F v∈S v × v6∈S Ov the induced morphism on the colimit, which we notice is surjective. e Q Q Let Pl2 := v PlF : v 2 denote the dyadic places of F . It follows from Lemma 2.6 that { ∈ | } 2 the kernel of the projection map JF (S) JF (S) is I (Fv). Passing to the colimit as → v∈S∪Pl2 2  S varies, we thus find that the kernel of the map JF JF is v∈Pl I (Fv). e L→ F MW L 4.0.1. Topology. Using the topology on the groupse K1 (Fv), we can topologize JF similarly as in the classical case: e Definition 4.3. Let S be a finite set of places of F . We define a topology on JF (S) by taking the topology generated by the sets e W := U KMW( ), S v × 1 Ov vY∈S vY6∈S MW where the Uv’s are open subsets of K1 (Fv) for each v S. This defines a topology on JF via the direct limit topology. ∈ e Lemma 4.4. The Milnor–Witt id`ele group JF is a locally compact topological group. Proof. This follows from the fact that J is the restricted product of the groups KMW(F ) with F e 1 v respect to the KMW( ), which are compact for almost all v by Proposition 3.1.  1 Ov MWe Definition 4.5. Define the map i:K1 (F ) JF by i([x]) := ([iv(x)])v∈PlF . We refer to the cokernel → e CF := cokere i e of i as the Milnor–Witt id`ele class group of F . e e e NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 9

MW Lemma 4.6. The map i is injective. Hence CF = JF /i(K1 (F )). 2 Proof. By Proposition 4.2, the kernel of the map JF JF is I (Fv). Hence there is a e e e e v∈PlF commutative diagram → e L 2 2 I (F ) I (Fv) vM∈PlF

i MW e 0 ker i K1 (F ) JF CF 0 p e e e × i 0 F JF CF 0. 2 2 By the Hasse–Minkowski theorem, the map I (F ) I (Fv) is injective. It follows from → v∈PlF this and a diagram chase that ker i = 0.  L 4.0.2. Id`eles of volume zero. Recall that we have a volume map vol: J R× defined as e F >0 vol(x) := x . Here x = (x ) J , and is the v-adic absolute→ value on F . v∈PlF v Fv v v∈PlF F Fv v 0 | | ∈ | · | × 0 We let JF denote the kernel of the volume map. By the classical product formula, F JF . Q 0 0 × ⊆ The fact that the resulting quotient group CF := JF /F is compact is an equivalent formulation of Dirichlet’s unit theorem and the finiteness of the ideal class group [Gra03, Proposition 4.2.7].

Definition 4.7. Let p: JF JF denote the projection map. We define a volume map vol: JF × → 0 MW →0 R>0 on JF by vol := vol p, and put JF := ker(vol). By definition we have i(K1 (F )) JF . 0 e ◦ f e⊆ Let CF be the quotient group e f 0e 0 fMW e e CF := JF /i(K1 (F )). e Proposition 4.8. There is a short exact sequence e e e 0 Z/2 C0 C0 0. → → F → F → Hence C0 is a compact topological group. F e 2 Proof. By definition, the kernel I (Fv) of the projection map p: JF JF is contained e v∈PlF 0 → in JF . Consider the commutative diagram with exact rows L e i MW e 0 0 e 0 K1 (F ) JF CF 0 p e e × i 0 0 0 F JF CF 0. 2 2 By the snake lemma it suffices to show that the cokernel of the map I (F ) I (Fv) is → v∈PlF Z/2. But this follows from the snake lemma applied to the commutative diagram with exact rows L 0 I3(F ) I2(F ) I2(F )/I3(F ) 0

=∼ ι

3 2 2 3 0 I (Fv) I (Fv) I (Fv)/I (Fv) 0, vM∈PlF vM∈PlF vM∈PlF using the fact that the left hand vertical map ι is an isomorphism by Lemma 4.9 below, and that the cokernel of the right hand vertical map is Z/2 by [Mil70, Lemma A.1].  NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 10

3 3 Lemma 4.9. For any number field F , the canonical map ι: I (F ) I (Fv) induced by → v∈PlF .the inclusions iv : F ֒ Fv is an isomorphism → L Proof. The map is injective by the Hasse–Minkowski theorem. We must show that it is surjective. c 3 If v Pl0 Pl∞, then I (Fv) = 0 and there is nothing to show. On the other hand, if v is an ∈ ∪ 3 3 infinite real place, then I (Fv) ∼= I (R) ∼= Z. By strong approximation [Neu99, p. 193] we can find an element a F which is negative in the i-th ordering on F and positive otherwise. Then 3 ∈ 3 1, 1,a I (F ) maps to the i-th unit vector in r I (R).  hh− − ii ∈ v∈Pl Remark 4.10. One can speculate on whether thereL is a variant of the abelianized ´etale funda- mental group which is the recipient of a reciprocity map defined on the Milnor–Witt id`ele class group C and which lifts the classical Artin reciprocity map ρ: C Gal(L/F )ab. F F →

e 5. A Moore reciprocity sequence for Milnor–Witt K-theory The classical result of Moore on uniqueness of reciprocity laws states that there is an exact sequence 0 WK (F ) K (F ) h µ(F ) π µ(F ) 0. → 2 → 2 −→ v −→ → v∈Plnc MF Here h denotes the global Hilbert symbol, and the group WK2(F ) is known as the wild kernel [Gra03, §7]. Moreover, the map π is defined as

mv /m π((ζv)v) := ζv , v∈Plnc YF MW where mv := #µ(Fv) and m := #µ(F ). In this section we will show that also K2 (F ) fits into a similar exact sequence.

MW 5.1. Hilbert symbols. In order to obtain a Moore reciprocity sequence for K2 , we first need to define Hilbert symbols in the setting of Milnor–Witt K-theory. These should be particular instances of maps of the following type: Definition 5.1. Let A be a GW(F )-module. A Milnor–Witt symbol on F with values in A is a GW(F )-bilinear map ( , ):KMW(F ) KMW(F ) A − − 1 × 1 → satisfying ([a], [1 a]) = 0 for all a F × 1. − ∈ \ Remark 5.2. Note that any abelian group A is also a GW(F )-module via the rank map rk: GW(F ) Z. Thus we should think of the definition of a Milnor–Witt symbol as a lift of the classical no- → tion of a symbol, i.e., a Z-bilinear map ( , ): F × F × A satisfying the Steinberg relation − − × → [Tat71]. Just as K2(F ) is the universal object with respect to symbol maps, it follows from MW [Mor12, Remark 3.2] that K2 (F ) is the universal object with respect to Milnor–Witt symbols.

Definition 5.3. For any place v of F , let Bv be the group

µ(Fv), v Pl0 ∈ r Bv := Z, v Pl  ∈ ∞ 0, v Plc . ∈ ∞ Furthermore, define a map qv : Bv µ(Fv) by letting qv be the identity if v Pl0; reduction r → c ∈ modulo 2 if v Pl∞; or the trivial homomorphism if v Pl∞. Finally, write q := v∈Plnc qv. ∈ ∈ F L NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 11

MW nc 5.1.1. By Proposition 2.2 we have K2 (Fv) ∼= Bv Av for each v PlF . Thus we can define, for any noncomplex place v of F , the homomorphism⊕b :KMW(F ) ∈ B as the composition of v 2 v → v the isomorphism KMW(F ) = B A followed by the projection B A B . 2 v ∼ v ⊕ v v ⊕ v → v Definition 5.4. For each v Plnc, let hMW :KMW(F ) B denote the composite ∈ F v 2 → v i b KMW(F ) v KMW(F ) v B , 2 −→ 2 v −→ v where the first map is induced by the inclusion i : F ֒ F . Moreover, let v → v hMW :KMW(F ) B 2 → v v∈Plnc MF MW MW be the map h := nc h . v∈PlF v MW 5.1.2. For any placeLv of F , we can define a Milnor–Witt Hilbert symbol ( , )v on F similarly as the classical Hilbert symbols as follows: − − nc Definition 5.5. Let v Plv be a noncomplex place of F . We define the local Milnor–Witt Hilbert symbol at v, denoted∈ ( , )MW, as the composite − − v hMW ( , )MW :KMW(F ) KMW(F ) KMW(F ) v B . − − v 1 × 1 → 2 −−−→ v Here the first map is multiplication on Milnor–Witt K-theory. Example 5.6. It follows from Proposition 2.4 that given any A-valued Milnor–Witt symbol MW MW ( , ):K1 (Q) K1 (Q) A on Q, there exist unique homomorphisms ψp : Bp A such that− − × → → MW (α, β)= ψp((α, β)p ) 2 0 = v v (1, iv(a),iv(b) < 0 (a,b) (mod 2). ≡ v The result follows.  NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 12

Corollary 5.8. With notations as above, the diagram

(−,−)MW KMW(F ) KMW(F ) v B 1 × 1 v p×p qv

× × F F µ(Fv) × (−,−)v commutes for any v Plnc. ∈ F Proof. For finite places this holds trivially; for real places the claim follows from Lemma 5.7. 

5.1.4. Wild kernels. Using the map hMW we can define wild kernels for Milnor–Witt K-theory similarly as the classical case:

MW MW MW Definition 5.9. Let WK (F ) denote the kernel of the map h = nc h . 2 v∈PlF v 5.1.5. We are now ready to formulate a Moore reciprocity sequence forL Milnor–Witt K-theory: Proposition 5.10. MW For any number field F , there is an isomorphism WK2 (F ) ∼= WK2(F ). Moreover, there is an exact sequence

MW 0 WK (F ) KMW(F ) h B µ(F ) 0, → 2 → 2 −−−→ v → → v∈Plnc MF where v runs over all noncomplex places of F , and where the Bv’s are the groups defined in Definition 5.3.

Proof. Consider the following commutative diagram with exact rows:

ker p′ I3(F ) ι I3(F ) ker q′ =∼ v v∈Plnc MF

MW MW MW h MW 0 WK2 (F )K2 (F ) Bv coker h 0 nc v∈PlF M ′ p′ p q q

h π 0 WK2(F )K2(F ) µ(Fv) µ(F ) 0. v∈Plnc MF Here p′ and q′ are induced from p and q, respectively. According to Lemma 4.9, the map 3 3 ′ ι: I (F ) v∈Plnc I (Fv) is an isomorphism. It follows from this and a diagram chase that p → F and q′ are isomorphisms.  L Example 5.11. We see from Proposition 2.4 that in the case F = Q, the exact sequence of Proposition 5.10 reads

0 Z F× Z Z/2 F× Z/2 0. → ⊕ p → ⊕ ⊕ p → → p>2 p>2 M M Here we have used that WK2(Q) = 0. NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 13

6. Regular kernels and Milnor–Witt K-theory of rings of integers We will now consider various kernels of the Hilbert symbols and the tame symbols. Recall that in classical K-theory, there are three subgroups of K2(F ) of particular interest:

× K2( F ) = ker ∂ :K2(F ) k(v) O → ! vM∈Pl0 ⊆

+ K2 ( F ) := ker hv :K2( F ) µ(Fv) O  K2(OF ) O →  v∈Plr v∈Plr M∞ M∞   ⊆

WK (F ) := ker h:K (F ) µ(F ) . 2  2 → v  v∈Plnc MF   We have already encountered the wild kernel WK2(F ), and it is a classical result of Quillen that K ( ) is the kernel of the tame symbols ∂. The group K+( ) was introduced by Gras in 2 OF 2 OF [Gra86] and is referred to as the regular kernel. It is a modification of K2( F ) that takes into account also the real places of F . O Definition 6.1. For any n 1, let KMW( ) denote the kernel of ≥ n OF ∂ := ∂πv :KMW(F ) KMW(k(v)). v n → n−1 vM∈Pl0 vM∈Pl0 Here πv is any choice of uniformizer for the discrete valuation v. Moreover, we let KMW,+( ) := ker KMW( ) Zr1 ;KMW,+(F ) := ker KMW(F ) Zr1 , n OF n OF → n n → where the maps are given by the signatures with respenct to the orderings on F . 

πv Remark 6.2. By [Mor12, Proposition 3.17 (3)], the kernel of ∂v is independent of the choice of uniformizer πv, hence Definition 6.1 makes sense. Remark 6.3. If S is a set of places of F containing the infinite places, note that we can also define Milnor–Witt K-theory of the ring of S-integers in F as OF,S

KMW( ) := ker ∂πv :KMW(F ) KMW(k(v)) . n OF,S  v n → n−1  Mv6∈S Mv6∈S   Here π is a uniformizer for the place v. For n = 2, the groups KMW( ) were also considered v 2 OF,S in [Hut16]. More precisely, Hutchinson defines a subgroup K (2, ) of the second unstable 2 OF,S K-group K2(2, F ) by e K (2, ) := ker K (2, F ) k(v)× . 2 OF,S  2 →  Mv6∈S ∼ e   = MW By [Hut16, Proposition 3.12] there is a natural isomorphism K2(2, F ) K2 (F ), hence MW −→ K2(2, F,S) ∼= K2 ( F,S). The main theorem of [Hut16] states that if S is a set of places Q O O MW Z Z Z Z of containing 2 and 3, then K2 ( S ) ∼= H2(SL2( S ), ) (where S denotes the localization e NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 14

Z MW Z Z Z of at the primes of S). In particular, K2 ( [1/m]) ∼= H2(SL2( [1/m]), ) provided 6 m. It is a conjecture that this isomorphism holds for any even m. | Example 6.4. By Proposition 2.4 we have KMW(Z) = Z for all n 1. n ∼ ≥ Proposition 6.5. We have a short exact sequence q 0 Zr1 KMW( ) K ( ) 0, → → 2 OF −→ 2 OF → where the homomorphism q is induced by the forgetful map from Milnor–Witt K-theory to Milnor K-theory. Proof. Consider the commutative diagram with exact rows

0 KMW( )KMW(F ) ∂ KMW(k(v)) 0 2 OF 2 1 vM∈Pl0 q p =∼

0 K ( )K (F ) ∂ k(v)× 0. 2 OF 2 vM∈Pl0

3 r1 Since the right hand vertical map is an isomorphism, it follows that ker(q) ∼= ker p =I (F ) ∼= Z , while coker q = 0.  Proposition 6.6. MW,+ + MW,+ + (1) We have K2 ( F ) ∼= K2 ( F ) and K2 (F ) ∼= K2 (F ). (2) The groups KMW(O ) and KOMW(F ) decompose as 2 OF 2 KMW( ) = K+( ) Zr1 ;KMW(F ) = K+(F ) Zr1 . 2 OF ∼ 2 OF ⊕ 2 ∼ 2 ⊕ In particular, by Garlands theorem on the finiteness of K ( ), the group KMW( ) is a finitely 2 OF 2 OF generated abelian group of rank r1. Proof. The snake lemma applied to the diagram

MW,+ 0 K ( )KMW( ) Zr1 0 2 OF 2 OF

0 K+( )K ( ) (Z/2)r1 0 2 OF 2 OF yields the first claim of (1); the second claim follows similarly. For (2), we use (1) along with the observation that the short exact sequence 0 K+( ) KMW( ) Zr1 0 (4) → 2 OF → 2 OF → → MW  is split, and similarly for K2 (F ). Remark 6.7. In contrast, the corresponding short exact sequence for K ( ), 2 OF 0 K+( ) K ( ) (Z/2)r1 0, → 2 OF → 2 OF → → + need not split. For example, [Keu89, Example 3.10] shows that if F = Q(√14), then rk2(K2 ( F )) = 1 while rk (K ( )) = 2. On the other hand, the sequence O 2 2 OF 0 K+(F ) K (F ) (Z/2)r1 0 → 2 → 2 → → is always split (see [Keu89, §2.1]). NUMBER THEORETIC ASPECTS OF MILNOR–WITT K-THEORY 15

MW 6.1. Sample computations of K2 ( F ). Using Proposition 6.6 along with similar results for O MW K2( F ) we deduce some computations of K2 ( F ) for various fields F . Below we make use O MW O the calculations of [BG04] to determine K2 ( F ). In the following table, we consider the numberO fields F = Q[x]/(P ) defined by the polynomial P . We let ∆F denote the of F , and (r1, r2) the signature. P ∆ (r , r )K ( )KMW( ) F 1 2 2 OF 2 OF x3 + x2 2x 1 49 (3, 0) (Z/2)3 Z3 x3 + x2 +2− x +1− 23 (1, 1) Z/2 Z x3 + x2 +3 −255 (1, 1) Z/6 Z/3 Z x4 x 1 −283 (2, 1) (Z/2)2 Z2 ⊕ x5 − x3− x2 + x +1 −1609 (1, 2) Z/2 Z − −

MW 7. Hasse’s norm theorem for K2 The classical norm theorem of Hasse states that if L is a cyclic extension of the number field F , then a nonzero element of F is a local norm at every place if and only if it is a global norm. We can think of this result as a norm theorem for K1. In [BR84], Bak and Rehmann extended Hasse’s norm theorem to K2. More precisely, they showed that if L/F is any finite extension of number fields, then an element of K2(F ) lies in the image of the transfer map τL/F :K2(L) K2(F ) if and only if its image in each K2(Fv) lies in the image of the map → τ : K (L ) K (F ); Lw/Fv 2 w → 2 v Mw|v Mw|v see [BR84, Theorem 1]. By [BR84, p. 4], this result can be reformulated as the exactness of the sequence

τ Lv∈Σ hv K (L) L/F K (F ) L/F µ(F ) 0. (5) 2 −−−→ 2 −−−−−−−−→ v → v∈MΣL/F

Here ΣL/F denotes the set of infinite real places of F that are complexified in the extension L/F , and hv denotes the classical local Hilbert symbol at v.

MW 7.0.1. The aim of this section is to show that a similar result as (5) holds also for K2 : MW Proposition 7.1. Let L/F be an extension of number fields. Then an element of K2 (F ) lies in the image of the transfer map τ :KMW(L) KMW(F ) if and only if its image in KMW(F ) L/F 2 → 2 2 v lies in the image of τ for all v Pl . w|v Lw/Fv ∈ F 7.0.2. We note thatL Proposition 7.1 is equivalent to the following assertion:

Proposition 7.2. Let L/F be an extension of number fields. Denote by ΣL/F the set of infinite real places of F that are complexified in the extension L/F . Then there is an exact sequence

MW τ Lv∈Σ hv KMW(L) L/F KMW(F ) L/F Z 0, 2 −−−→ 2 −−−−−−−−−−→ → v∈MΣL/F where τL/F denotes the transfer map on Milnor–Witt K-theory as defined in (2.1.2), and the right hand map is given by the local Milnor–Witt Hilbert symbols. REFERENCES 16

Remark 7.3. To show the equivalence of Propositions 7.1 and 7.2, notice first that Proposition 7.1 is equivalent to the assertion that the map coker(τL/F ) coker τL /F is injec- → v∈PlF w|v w v tive. It is therefore enough to show that coker(τ ) is trivial except for v Σ , in which Lw/Fv Q L L/F  case it is Z. But this follows from Proposition 2.2. ∈ 7.0.3. Let us proceed with the proof of Proposition 7.2. We follow closely the strategy of Bak and Rehmann [BR84]. Recall from Definition 5.3 the definition of the groups B for v Pl . v ∈ F Definition 7.4. If v is any place of F and w v is a place of L above v, we define a homomorphism | b : B B w|v w → v by 0, v Plr and w Plc ∈ F,∞ ∈ L,∞ b := id, v Plr and w Plr w|v  ∈ F,∞ ∈ L,∞ nw/mv, v PlF,0 . ∈ Here n := #µ(L ) and m := #µ(F ). w w v  v Lemma 7.5. For any place v of F , the diagram

L hMW MW w|v w K2 (L) Bw Mw|v τL/F Lw|v bw|v MW MW hv K2 (F ) Bv is commutative. MW r Proof. Since K2 (Fv) ∼= K2(Fv) for each v Pl∞ it suffices by [BR84, Proof of Proposition 2] to note that τ = id: KMW(R) KMW(6∈R) whenever v and w both are infinite real places. Lw/Fv 2 → 2 Indeed, this follows since Lw/Fv is then the trivial extension.  MW Lemma 7.6. Let x WK2 (F ), and let p be a rational prime. Then there is an element z KMW(F ) such that∈ pz = x. ∈ 2 Proof. By Proposition 5.10 we have WKMW(F ) = WK (F ). Moreover, KMW(F ) = K+(F ) Zr1 2 ∼ 2 2 ∼ 2 ⊕ by Proposition 6.6. Then, since WKMW(F ) K+(F ), the statement follows from the fact that 2 ⊆ 2 any element in the classical wild kernel has a p-th root in K2(F ) [Tat71]. 

Lemma 7.7. Suppose that x ker hMW . If nz = x for some z KMW(F ) and ∈ v∈ΣL/F v ∈ 2 some integer n 1, then there is an element y KMW (L) such that x + τ (y) nWKMW(F ). ≥ L ∈ 2 L/F ∈ 2 Proof. Using the Moore reciprocity sequence 5.10 along with the fact that the homomorphism KMW(F ) Z is split, the same proof as that of [BR84, Lemma 1] goes through.  2 → v∈ΣL/F Proof of PropositionL 7.2. The desired result now follows from Lemma 7.7 in an identical manner as the proof of [BR84, Theorem 2].  References

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H˚akon Kolderup, University of Oslo, Postboks 1053, Blindern, 0316 Oslo, Norway E-mail address: [email protected]