In this chapter, we shortly introduce quadratic forms over fields and their connection to the study of sums of squares, and present Springer’s Theorem for discrete valued fields. Then we present a new local-global principle that we derive from a similar on by Harbater, Harmann, and Krashen. Finally, we present a closely related local-global principle by Colliot-Th´el`ene, Parimala, and Suresh, and include a short proof.
1. Quadratic forms
Let K be a field and n ∈ N. A quadratic form of dimension n over K is a homogeneous quadratic polynomial inK[x1, . . . , xn]. We call a graded K-algebra automorphism on K[x1, . . . , xn], where the grading is given by the total degree, alinear change of variables. Two quadratic forms q1 and q2 of dimension n over K are called isometric, if there exists a linear change of variables L : K[x1, . . . , xn] → K[x1, . . . , xn] such thatq1=L(q2), and we then writeq1∼=q2. We callqisotropic, if the projective quadricV+(q)( Pn−1K overK has a rational point.
Let a1, . . . , an ∈ K. we call the n-dimensional form a1x21 +· · ·+ anx2n a diagonal quadratic form, denoted ha1, . . . , ani. Note that for c1, . . . , cn ∈ K×, we have that ha1, . . . , ani ∼= hc21a1, . . . , c2nani. If char(K) 6= 2, then for every n-dimensional quadratic form q, there exists a linear change of variablesdsuch thatd(q) =ha1, . . . , aniand we call it adiagonalization ofq.
Form≤n, a gradedK-algebra homorphisms of degree zero K[x1, . . . , xm],→K[x1, . . . , xn],
given by the obvious inclusion followed by a linear change of variables, is called adimension extension.
We call ann-dimensional quadratic formregular, if it is not a dimension extension of anm-dimensional quadratic form for anym < n.
Conversely, form≤n, a gradedK-algebra homorphisms of degree zero K[x1, . . . , xn] → K[x1, . . . , xm], given by a linear change of variables
53
54 5. LOCAL-GLOBAL PRINCIPLES
followed by the partial evaluations xj 7→ 0 for j > m, is called a dimension restriction. We call anm-dimensional quadratic form q1 a subform of an n-dimensional quadratic form q2, if m ≤ n and q1 is obtained by a dimension restriction ofq2.
For an n-dimensional quadratic form q1 and an m-dimensional qua-dratic formq2, we denote byq1⊥q2 then+m-dimensional quadratic formq1(x1, . . . , xn)+q2(xn+1, . . . , xn+m)∈K[x1, . . . , xn+m], called the orthogonal sum of q1 and q2. For a quadratic form q and ` ∈ N, we write `×qfor the`-fold orthogonal sumq⊥q⊥ · · · ⊥q.
Note that, fora∈K×andn∈N, the fact thata∈DK(n) is equivalent to then+ 1 dimensional formn× h1i ⊥ h−aibeing isotropic.
LetL/K be a field extension, andqann-dimensional quadratic form.
We denote by qL thescalar extension of q toL, considered now as a polynomial over L.
2. Isotropy over complete discrete valued fields
Let K be a field and v a nondyadic discrete valuation on K. Let a1, . . . , an ∈ K× and consider the regular diagonal quadratic form ha1, . . . , anioverK. Fixing a uniformizerπforv, we writeai=biπv(ai) forbi∈K×ifv(ai)∈2Z, orai=ciπv(ai)forci∈K×ifv(ai)∈/ 2Z, for every i ∈ {1, . . . n}. After renumerating the variables, we can assume that there exists k ∈ {1, . . . , n} such that v(ai) ∈ 2Z for i ≤ k and v(ai)∈/ 2Zfori > k.
The diagonal formshb1, . . . , bkiandhck+1, . . . , cnioverκv are regular, and their isometry classes are called the first and the second residue form of ha1, . . . , aniwith respect to(v, π).
Note that the first residue form does not depend on the chosen uni-formizerπ, while the second residue form is only independent of πup to a scalar inκ×v. However, whether the second residue form is isotropic or not does not depend on the chosen uniformizer π, thus we sloppily omit to specify the uniformizer in the following (5.1).
Theorem 5.1 (Springer). Let (K, v)be a nondyadic complete discrete valued field, and a1, . . . , an∈K×. The regularn-dimensional diagonal quadratic formha1, . . . , anioverK is isotropic if and only if one of its two residue forms overκv is isotropic.
Reference:. [Lam05, VI.1.9.] or [Sch85, 6.2.6].
3. GEOMETRIC LOCAL-GLOBAL PRINCIPLE 55
3. Geometric local-global principle
LetT denote a complete discrete valuation ring,kits residue field,K its field of fractions, andF/K an algebraic function field. Recall that by (3.41), there exists a regular projective integral curveCoverKwith function field F. By (3.51), there exists a regular model Cover T for F/K. Note thatC is not unique. LetX (Cdenote the special fiber ofC. For any P ∈X, we denote by RbP the completion of the regular local domainOC,P with respect to its maximal ideal mC,P. Note that the natural morphismOC,P →RbP is an embedding by (2.9), and that RbP is a regular local domain by (2.10). Its field of fractions FP is a field extension of F. As opposed to [HHK09], the notation RbP and FP in our case is not reserved for closed pointsP ∈X only. The rings and fields just defined will remain the same throughout the rest of this chapter.
The following local global principle is the main result of this section and will be deduced from [HHK09, 4.2]. In [HHK11, 9.3], the authors of [HHK09] also mention this consequence1of their previous work. The result holds more generally whenT-modelsCforF/Kis only considered to be normal. The proof is identical, except for a more sophisticated reasoning for why the ringsRbP are domains also in the case whereCis not regular but only normal.
Theorem 5.2. Let qbe a regular quadratic form over F of dimension at least3. Then qis isotropic if and only if qFP is isotropic for every P ∈X.
LetU (X be a nonempty open subset of the special fiber ofC. LetRU
denote the direct limit lim−→OC(W) of the direct system of ringsOC(W) whereW⊂Care open neighbourhoods ofU. Lett∈T be a generator of the maximal ideal in T. We denote by RbU the t-adic completion lim←−RU/(tn).
Lemma 5.3. The natural map RU →RbU is an embedding. Moreover, ifU is irreducible, then RbU is a domain.
Proof. Let η ∈ U denote the generic point of a component of X that has nonempty intersection withU. By (3.45 & 3.46), the point η∈Cis regular of codimension one, whereby the ringOC,ηis a discrete valuation ring. Hence T
n∈NtnOC,η = {0} by (2.1). As η ∈ U, the
1which they obtained independently.
56 5. LOCAL-GLOBAL PRINCIPLES
obvious embeddingRU ,→OC,ηyields thatT
n∈NtnRU ={0}, whereby the natural map RU →RbU is an embedding by (2.9).
Now assume that U is irreducible, and let η denote the generic fiber of the irreducible component of X that contains U. The completion ObC,ηofOC,ηwith respect tomC,ηis a discrete valuation ring and thus a domain. We are going show thatRbU ,→ObC,η, whereby it is a domain.
First, we claim that the homomorphisms RU/(tn) → OC,η/(tn) are injective for all n∈N, whereby it then follows that RbU embedds into lim←−OC,η/(tn). So let f ∈ RU. Suppose there exists g ∈ OC,η such thatf =tng. It is sufficient to show that then necessarilyg∈RU. Let W⊂Cbe an irreducible open neighbourhood ofUsuch thatf ∈OC(W) andW∩X=U.
We claim that theng∈OC(W). SinceCis regular and thus in particular normal, it is sufficient by [Eis95, 11.4] to show thatg∈OC,P for every codimension one point P ∈ W. So letP ∈W be of codimension one.
If P ∈ X, then P = η by the choice of W, and thus g ∈ OC,P by assumption. Otherwise, we have thatt ∈O×C,P, thus g= t1nf ∈OC,P. Hence,g∈OC(W).
Thus RbU ,→ lim←−OC,η/(tn) and we have that lim←−OC,η/(tn)∼= ObC,η by (2.7), astOC,η=mmC,η for somem∈N. For U (X irreducible, we denote by FU the field of fractions of the domain RbU. It is easy to see that for ∅ 6=U0⊆U, we have FU ⊆FU0. Note that if U has an openaffine neighborhoodW⊆C, thenF is the field of fractions ofRU, wherebyFU is a field extension ofF.
We are now prepared to formulate the geometric local-global principle for quadratic forms by Harbater, Hartmann, and Krashen. Let S(X denote the finite subset of points of the special fiber that lie on at least two distinct components of X. Let f : C → P1T denote a finite T-morphism as in (3.49) such that S ⊂f−1({∞}), where ∞ ∈ P1k = proj(k[x0, x1]) denotes an arbitrary rational point. DenoteA1k =P1k\ {∞}. There exists a homogeneous linear polynomial`∈k[x0, x1] such that A1k = D+(`). Let L ∈ T[x0, x1] be a homogeneous lift of ` and consider the affine open neighbourhood D+(L) of A1k in P1T. Since f is finite, f−1(D+(L)) is an affine neighbourhood of f−1(A1k). In particular, the field FU is an extension fields of F for any irreducible component U of f−1(A1k)(X.
Theorem5.4 (Harbater, Hartmann, Krashen). Letqbe a regular qua-dratic form over F of dimension at least 3. Then q is isotropic if
3. GEOMETRIC LOCAL-GLOBAL PRINCIPLE 57
and only if for allP ∈f−1({∞})and all irreducible componentsU of f−1(A1k), the formsqFP andqFU are isotropic.
Reference:. [HHK09, 4.2].
Remark 5.5. In [HHK09, 4.2], the T-curve C was only assumed to be normal, not necessarily regular. It is more difficult to justify the existence ofFP for a closed pointP ∈X in this case. As we apply the theorem only in the situation where C is regular, we just stated it in that special case.
Note that RbP is a complete ring with respect to mb = mC,PRbP and RbP/mb =OC,P/mC,P. As a first consequence of (5.4) & (2.11) we obtain the following.
Corollary5.6. Let qbe a regular quadratic form overF. LetP∈X be a closed point and assume there exist a1, . . . , an ∈ O×C,P such that q∼=F ha1, . . . , ani. Then qFP is isotropic if ha1, . . . , ani is isotropic as a form overOC,P/mC,P.
Proof. Let x1, . . . , xn ∈ OC,P such that a1x21+· · ·+anx2n ∈ mC,P and, without loss of generality, x1 ∈ O×C,P. The monic polynomial f(X) =a1X2+a2x22+· · ·+anx2n considered as a polynomial overRbP
has a zerox1modulomb =mC,PRbP. Moreover dXd f(x1) = 2a1x1∈Rb×P. By (2.11) there existsz∈Rb×P such thata1z2+a2x22+· · ·+anx2n = 0.
Thusha1, . . . , aniFP is isotropic.
Proposition 5.7. Let Y be an irreducible component of X. Let η denote the generic point forY inC. Let qbe a quadratic form over F such that qFη is isotropic. Then there exists a nonempty open subset U ⊆Y such thatqFU is isotropic.
Proof. The pointη∈Chas codimension one, by (3.45 & 3.46). In par-ticular,OC,η is a discrete valuation ring. The completion ofF with re-spect to the corresponding valuation isFη. Lets∈F such thatmC,η= sOC,η. Then there exists somen∈Nandu∈O×C,η such thatsn =ut.
One can find a diagonalization q ∼= ha1, . . . , am, sb1, . . . , sb`i, where a1, . . . , am, b1, . . . , b` ∈ O×C,η for somem, ` ∈ N. By (5.1), one of the two residue formsϕ=ha1, . . . , ami, ψ=hb1, . . . , b`ioverκvis isotropic.
Let us assume, without loss of generality, that ϕ is isotropic. Let x1, . . . , xm∈OC,η, such thata1x21+· · ·+amx2m∈mη =sOC,ηand, with-out loss of generality,x1 ∈O×C,η. Let w∈OC,η such that a1x21+· · ·+ amx2m=ws. AsOC,ηis the direct limit overOC(W), whereW⊂Cruns
58 5. LOCAL-GLOBAL PRINCIPLES
over the open neighbourhoods ofη inC, we can find a neighbourhood U of η, such that a1, a−11 , . . . , am, a−1m, u, u−1, x1, . . . , xm, x−11 , w, s ∈ OC(U).
Set U = Y ∩U. The polynomial a1T2+ (a2x22+· · ·+amx2m) has in (RU/(s))× the simple zero x1+sRU. By (2.11) this zero lifts to a solution in RbU of a1T2+ (a2x22+· · ·+anx2n) = 0, asRbU is complete with respect to sbRU by (2.7) andRbU/sRbU ∼=RU/sRU by (2.8). Thus ha1, . . . , aniFU is isotropic and so isqFU. Proof of (5.2). Suppose a quadratic formqof dimension at least 3 is isotropic at FP for everyP ∈X. By (3.45), each nonclosed point η ∈X is the generic point of an irreducible componentYη of X. By (5.7), there exists a nonempty openUη⊆Yηsuch thatqFUη is isotropic.
SetS= (X\S
η∈X(0)Uη)∪S
η6=ρ∈X(0)(Yη∩Yρ).ThenSis a finite set of closed points. There exists a finite T-morphism f :C→P1T such that S⊂f−1({∞}), by (3.49). Every irreducible componentU of f−1(A1k) is contained inUηfor some non closed pointη, wherebyqFU is isotropic.
The assertion now follows with (5.4).
4. Valuation theoretic local-global principle
The following local global principle will appear in [CTPS, 3.1]. We use our variant (5.2) of (5.4) to give a short presentation of its proof. Recall that Ω(F) denotes the set of discrete valuations onF up to equivalence.
Theorem 5.8 (Colliot-Th´el`ene, Parimala, Suresh). Let q be a regu-lar quadratic form defined over F of dimension at least 3. Then q is isotropic if and only if qFv is isotropic for everyv∈Ω(F)onF. Proof. Let a1, . . . , an ∈ F such that q ∼= ha1, . . . , ani. Let S = Sn
i=1supp(div(ai))⊂C(1). Set D=X
x∈S
[x],
i.e. the effective divisor onCconsisting of the supports of the div(ai) for each 1≤i≤n. There exists a birationalT-morphismg:C0→Csuch that the divisor g∗D on C0 is effective (3.27) and has normal cross-ings, by (3.52). This birational morphism induces a K-isomorphism gη# : OC,η → OC0,η0, where η = g(η0) and η0 are the generic points of C and C0 respectively. Since F ∼= OC,η ∼= OC0,η0, the map gη# is a K-automorphism on F. The supports of the principal divisors div(g#η(ai)) onC0are contained in the support (3.27) ofg∗D. Certainly
4. VALUATION THEORETIC LOCAL-GLOBAL PRINCIPLE 59
qis anisotropic if and only if q0 =hgη#(a1), . . . , gη#(an)iis anisotropic.
Moreover, the residue forms of q with respect to a discrete valuation v onF are anisotropic if and only if the residue forms of q0 with re-spect to the discrete valuation v◦(gη#)−1 are anisotropic. Thus q0Fv
is anisotropic for everyv ∈Ω(F) if and only if qFv is anisotropic for everyv∈Ω(F).
We can thus assume that the divisorD onC has normal crossings to begin with. LetX (Cdenote the special fiber. Supposeqis anisotropic overF. By (5.2) the form qFP is anisotropic for someP ∈X. IfP is a nonclosed point inC, then OC,P is a discrete valuation ring andFP
is the completion of the corresponding discrete valuation, by (2.18).
IfP is a closed point, there exist x, y∈mC,P such thatmC,P = (x, y) andai=uix`iysi for some`i, si∈Z, by (3.28). For 1≤i≤4 there are quadratic formsq(i) of dimension ni over F with q(i) =hu(i)1 , . . . u(i)nii for someu(i)j ∈O×C,P for 1≤j≤ni, such that
q∼=q(1)⊥yq(2) ⊥xq(3)⊥xyq(4).
For 1≤i≤4 we denote the quadratic formhu(i)1 +m, . . . , u(i)ni+miover O/m by ˜q(i). Note that the forms ˜q(i) for i= 1, . . . ,4 are anisotropic, as otherwise the multiple of some subform ofqwould become isotropic overFP by 5.6.
The ring O(x) is a discrete valuation ring by (2.5) with residue field isomorphic to Quot(O/(x)). The two residue forms of q with respect to its corresponding valuation and the uniformizer x are ¯q(1) ⊥ y¯q¯(2) and ¯q(3) ⊥ y¯q¯(4), where we write ¯a for the residue a+xO(x) for any a ∈ F, and we write ¯qi for the quadratic forms h¯u(i)1 , . . . ,u¯(i)nii over Quot(O/(x)) for i= 1, . . . ,4.
By (2.5), the ringO/(x) is a discrete valuation ring with uniformizer y+ (x) and residue field O/m. The first and second residue form of
¯
q(1) ⊥y¯q¯(2) are the anisotropic forms ˜q(1) and ˜q(2), whereby the first residue form ¯q(1) ⊥y¯q¯(2) of q with respect to the discrete valuationv corresponding toO(x) is anisotropic.
Analogous, the second residue form ¯q(3)⊥y¯q¯(4) ofqwith respect tov is anisotropic. This yields thatqFv is anisotropic.
CHAPTER 6