Failure of the Local-Global Principle for Isotropy of Quadratic Forms Over Surfaces, Oberwolfach Reports 10 (2013), Issue 2, Report 31, 1823–1825

arXiv:1709.03707v1 [math.AG] 12 Sep 2017 om al ohl ndimension in hold to fails forms otiilzr.If for zero. quadratic nontrivial a that a Recall nondegenerate. forms quadratic ISOTROPY FOR PRINCIPLE LOCAL-GLOBAL THE OF FAILURE edo hrceitc0 e [ see 0, ove characteristic field of function 2 field degree transcendence any over geometry, character of be to assumed be will fields our All transcendenc fields. higher closed of n fields real function over the forms over quadratic variables two in field function rational a over over[ forms for over quadratic principle 4 of local-global dimension the of of forms dratic failure [ of Lind principle examples 4). local-global also found (and the 3 of dimension failure in forms the of examples found the of instance famous, most led o ainlfnto edi n aibeoe nu a over variable one in field function rational a for Already hoe 1. Theorem following. the is result main Our lsapaigi [ in appearing ples edi stoi vreeycmlto,then completion, every over isotropic is field udai om,smtmsrfre oa h togHseP Hasse Strong the as to referred dimension sometimes forms, quadratic auto igof ring valuation K FQARTCFRSOE AINLFNTO FIELDS FUNCTION RATIONAL OVER FORMS QUADRATIC OF 13 v Date pr local-global the of failure the in interested are we Here, form quadratic a if that states theorem Hasse–Minkowski The rvosy nytecs of case the only Previously, h.1.4.Cses lio,adPse [ Pfister and Ellison, Cassels, 17.14]. Thm. , h rcinfil ftecmlto wt epc othe to respect (with completion the of field fraction the o vr iceevaluation discrete every for If etme 3 2017. 13, September : eeaie umrvreiscniee yBre [ Borcea by considered varieties Kummer generalized ucinfil ftasednedegree transcendence of field function iceevlain,friorp fqartcfrso dim of forms quadratic of isotropy for valuations, discrete Abstract. q saqartcfr in form quadratic a is d vragvnfield given a over i any Fix v epoetefiueo h oa-lblpicpe ihres with principle, local-global the of failure the prove We 15 hnw pa ftelclgoa rnil o stoyof isotropy for principle local-global the of speak we When . ,adltri [ in later and ], K n safil and field a is Q ≥ ( SE ULADV SURESH V. AND AUEL ASHER t ya plcto fteAe–rmrterm[ theorem Amer–Brumer the of application an by ) 3 2 ,[ ], h oa-lblpicpefriorp fquadratic of isotropy for principle local-global The . oa-lblprinciple local-global n K a nw,wt h rtkonepii exam- explicit known first the with known, was 2 = 4 d ema h olwn statement: following the mean we , 2 Q , Introduction n v aibe over variables § se[ (see ] hsltrrsl oiae h following. the motivates result later This 6]. vrtertoa ucinfield function rational the over 5 on n [ and ] v K sadsrt auto on valuation discrete a is 17 n 1 1 then , o ealdacut,gvn examples giving account), detailed a for ] n ecad [ Reichardt and ] over 14 .Fracntuto,uigalgebraic using construction, a For ]. q C q u osrcinivle the involves construction Our . siorpc hsi h rt and first, the is This isotropic. is K 8 on xmlso ieso 4 dimension of examples found ] siorpcover isotropic is and o stoyo udai forms. quadratic of isotropy for 6 n ykadHlk[ Hulek and Cynk and ] nin2 ension q ereoe algebraically over degree e siorpcover isotropic is siorpci tadmits it if isotropic is m o stoyo quadratic of isotropy for stoyo ar fqua- of pairs of isotropy 18 nagbaclyclosed algebraically an r v umbers. ,adltrCses[ Cassels later and ], ai oooy fthe of topology) -adic nil o stoyof isotropy for inciple n brfil,Wt [ Witt field, mber istic vrarational a over icpe nagiven a in rinciple, K q K ett all to pect C n l our all and 2 6= ednt by denote we , vranumber a over . ( x 1 x , . . . , 11 K ]. v n ) 20 . 2 7 ], ], ] 2 AUEL AND SURESH

Conjecture 2. Let K be a finitely generated field of transcendence degree n 2 over an algebraically closed field k of characteristic = 2. Then the local-global principle≥ for isotropy of quadratic forms fails to hold in dimension6 2n over K. We recall that by Tsen–Lang theory [16, Theorem 6], such a function field is a n n Cn-field, hence has u-invariant 2 , and thus all quadratic forms of dimension > 2 are already isotropic. An approach to Conjecture 2, along the lines of the proof in the n = 2 case given in [4, Cor. 6.5], is outlined in Section 4. Finally, we point out that in the n = 1 case, with K = k(X) for a smooth projective curve X over an algebraically closed field k, the local-global principle for isotropy of binary quadratic forms (equivalent to the “global square theorem”) holds when X has genus 0 and fails for X of positive genus. We would like to thank the organizers of the summer school ALGAR: Quadratic forms and local-global principles, at the University of Antwerp, Belgium, July 3–7, 2017, where the authors obtained this result. We would also like to thank Jean-Louis Colliot-Thélène, David Leep, and Parimala for helpful discussions. The first author received partial support from NSA Young Investigator grant H98230-16-1-0321; the second author from National Science Foundation grant DMS-1463882.

1. Hyperbolicity over a quadratic extension Let K be a field of characteristic = 2. We will need the following result about isotropy of quadratic forms, generalizing6 a well-known result in the dimension 4 case, see [19, Ch. 2, Lemma 14.2]. Proposition 1.1. Let n> 0 be divisible by four, q a quadratic form of dimension n and discriminant d defined over K, and L = K(√d). If q is hyperbolic over L then q is isotropic over K. Proof. If d K×2, then K = K(√d) and hence q is hyperbolic over K. Suppose ×2 ∈ d K and q is anisotropic. Since qL is hyperbolic, q < 1, d > q1 for some 6∈ ≃ − ⊗ quadratic form q1 over K, see [19, Ch. 2, Theorem 5.3]. Since the dimension of q is divisible by four, the dimension of q1 is divisible by two, and a computation of the discriminant shows that d K×2, which is a contradiction. ∈ × For n 1 and a1, . . . , an K , recall the n-fold Pfister form ≥ ∈ a1, . . . , an = <1, a1 > <1, an > ≪ ≫ − ⊗···⊗ − n ⊗n and the associated symbol (a1) (an) in the Galois cohomology group H (K,µ ). · · · 2 Then a1, . . . , an is hyperbolic if and only if a1, . . . , an is isotropic if and ≪ ≫ ≪ ≫ only if (a1) (an) is trivial. For d K· ·× · and n 2, we will consider quadratic forms of discriminant d related ∈ ≥ n to n-fold Pfister forms, as follows. Write a1, . . . , an as q0 <( 1) a1 . . . an >, ≪ n ≫ ⊥ − then define a1, . . . , an; d = q0 <( 1) a1 . . . and>. For example: ≪ ≫ ⊥ − a, b; d = <1, a, b, abd> ≪ ≫ − − a, b, c; d = <1, a, b, c, ab, ac, bc, abcd> ≪ ≫ − − − − for n = 2and n = 3, respectively. We remark that every quadratic form of dimension 4 is similar to one of this type. If q = a1, . . . , an; d , we note that, in view of ≪ ≫ Proposition 1.1 and the fact that qL is a Pfister form over L = K(√d), we have that q is isotropic if and only if qL is isotropic, generalizing a well-known result about quadratic forms of dimension 4, see [19, Ch. 2, Lemma 14.2]. FAILURE OF THE LOCAL-GLOBAL PRINCIPLE FOR ISOTROPY 3

2. Generalized Kummer varieties In this section, we review a construction, considered in the context of modular Calabi–Yau varieties [11, 2] and [12], of a generalized Kummer variety attached to a product of elliptic curves.§ This recovers, in dimension 2, the Kummer K3 surface associated to a decomposable abelian surface, and in dimension 3, a class of Calabi–Yau threefolds of CM type considered by Borcea [6, 3]. § Let E1,...,En be elliptic curves over an algebraically closed field k of characteristic = 2 and let Y = E1 En. Let σi denote the negation automorphism on 6 1 ×···× Ei and Ei P the associated quotient branched double cover. We lift each σi to an automorphism→ of Y ; the subgroup G Aut(Y ) they generate is an elementary abelian 2-group. Consider the exact sequence⊂ of abelian groups Π 1 H G Z/2 0, → → −→ → where Π is defined by sending each σi to 1. Then the product of the double covers Y P1 P1 is the quotient by G and we denote by Y X the quotient by the→ subgroup×···×H. Then the intermediate quotient X P1 → P1 is a double cover, branched over a reducible divisor of type (4,...,→4). ×···× We point out that X is a singular degeneration of smooth Calabi–Yau varieties that also admits a smooth Calabi–Yau model, see [11, Cor. 2.3]. For n = 2, the minimal resolution of X is isomorphic to the Kummer K3 surface Kum(E1 E2). 1 × Given nontrivial classes γi H (Ei,µ2), we consider the cup product ∈ ét n ⊗n γ = γ1 γn H (Y,µ ) · · · ∈ ét 2 and its restriction to the generic point of Y , which is a class in the unramified part n ⊗n n ⊗n Hnr(k(Y )/k,µ2 ) of the Galois cohomology group H (k(Y ),µ2 ) of the function field k(Y ) of Y (see [9] for background on the unramified cohomology groups). These classes have been studied in [10]. We remark that γ is in the image of the restriction n 1 1 ⊗n n ⊗n map H (k(P P ),µ2 ) H (k(Y ),µ2 ) in Galois cohomology since each ×···× → 1 1 1 γi is in the image of the restriction map H (k(P ),µ2) H (k(Ei),µ2). → 1 We make this more explicit as follows. For each double cover Ei P , we choose a Weierstrass equation in Legendre form → 2 (1) yi = xi(xi 1)(xi λi) 1 − − where xi is a coordinate on P and λi k r 0, 1 . Then the branched double cover X P1 P1 is birationally defined∈ by{ the} equation → ×···× n 2 (2) y = Y xi(xi 1)(xi λi)= f(x1,...,xn) i=1 − −

where y = y1 yn in C(Y ). Up to an automorphism, we can even choose the · · · 1 1 Legendre forms so that the image of γi under Hét(E,µ2) H (k(E),µ2) coincides → 1 with the square class (xi) of the function xi, which clearly comes from k(P ). The main result of this section is that the class γ considered above is already unramified over k(X). We prove a more general result. Proposition 2.1. Let k be an algebraically closed field of characteristic = 2 and 6 K = k(x1,...,xn) a rational function field over k. For 1 i n, let fi(xi) k[xi] ≤ ≤ n ∈ be polynomials of even degree satisfying fi(0) = 0, and let f = i=1 xif(xi). Then 6 n ⊗n Qn ⊗n the restriction of the class ξ = (x1) (xn) H (K,µ2 ) to H (K(√f),µ2 ) is unramified with respect to all discrete· · valuations. · ∈ 4 AUEL AND SURESH

Proof. Let L = K(√f) and v a discrete valuation on L with valuation ring v, O maximal ideal mv, and residue field k(v). Suppose v(xi) < 0 for some i. Let di be the degree of fi and consider the reciprocal ∗ di 1 d +2 1 ∗ 1 i i i i i i i polynomial fi (x )= xi f ( xi ), so that x f (x )= x xi fi ( xi ). Since d is even, 1 ∗ 1 · × ×2 i i i we have that the polynomials x f (x ) and xi fi ( xi ) have the same class in K /K . ∗ Thus, up to replacing, for all i with v(xi) < 0, the polynomial fi by fi in the 1 definition of f and replacing xi by , we can assume that v(xi) 0 for all i without xi ≥ changing the extension L/K. Hence k[x1,...,xn] v. ⊂ O Let p = k[x1,...,xn] mv. Then p is a prime ideal of k[x1,...,xn] whose residue ∩ field is a subfield of k(v). Let Kp be the completion of K at p and Lv the completion of L at v. Then Kp is a subfield of Lv. If v(xi) = 0 for all i, then ξ is unramified at v. Suppose that v(xi) = 0 for some i. 6 By reindexing x1,...,xn, we assume that there exists m 1 such that v(xi) > 0 for ≥ 1 i m and v(xi)=0for m+1 i n, i.e., x1,...,xm p and xm+1,...,xn p. In≤ particular,≤ the transcendence degree≤ ≤ of k(p) over k is at∈ most n m. 6∈ − First, suppose fi(xi) p for some m + 1 i n. Since fi(xi) is a product of ∈ ≤ ≤ linear factors in k[xi], we have that xi ai p for some ai k, with ai = 0 since − ∈ ∈ 6 fi(0) = 0. Thus the image of xi in k(p) is equal to ai and hence a square in Kp. In 6 particular, xi is a square in Lv, thus ξ is trivial (hence unramified) at v. Now, suppose that fi(xi) p for all m + 1 i n. Then for each 1 i m, 6∈ ≤ ≤ ≤ ≤ we see that since xi p and fi(0) = 0, we have fi(xi) p. Consequently, we can ∈ 6 6∈ assume that f = x1 xmu for some u k[x1,...,xn] r p. Computing with symbols,· · · we have ∈ m ⊗m (x1) (xm) = (x1) (xm−1) (x1 xm) H (K,µ ). · · · · · · · · · · ∈ 2 By definition, f = x1 xmu is a square in L, and thus we have that · · · m (x1) (xm) = (x1) (xm−1) (u) H (L,µ2). · · · · · · · ∈ Thus it is enough to show that (x1) (xm−1) (u) (xm+1) (xn) is unramified at v. n ⊗n n−1 · · · ⊗n−1 · · · · · Let ∂v : H (L,µ ) H (k(v),µ ) be the residue homomorphism at v. 2 → 2 Since xi, for all m + 1 i n, and u are units at v, we have ≤ ≤ ∂v((x1) (xm−1) (u) (xm+1) (xn)) = α (u) (xm+1) (xn) · · · · · · · · · · · · · m−2 m−2 for some α H (k(v),µ2 ), where for any h k[x1,...,xn], we write h for the image of h ∈in k(p). Since the transcendence degree∈ of κ(p) over k is at most n m n−m+1 ⊗n−m+1 − and k is algebraically closed, we have that H (κ(p),µ2 ) = 0. Since u, xi κ(p), we have (u) (xm+1) (xn) = 0. In particular ∂v(ξ) = 0. Finally, the class ∈ξ is unramified at all· discrete· · valuations · on L. As an immediate consequence, we have the following.

Proposition 2.2. Let E1,...,En be elliptic curves over an algebraically closed ﬁeld k of characteristic = 2, given in the Legendre form (1), with K = k(x1,...,xn). 6 n ⊗n n ⊗n Then the restriction of the class γ = (x1) (xn) in H (K,µ2 ) to H (k(X),µ2 ) is unramiﬁed at all discrete valuations. · · · Finally, we will need the fact, proved in the appendix by Gabber to the article [10], that if k = C and the j-invariants j(E1),...,j(En) are algebraically independent, n ⊗n 1 then any cup product class γ = γ1 γn H (C(Y ),µ2 ), with γi H (C(Ei),µ2) nontrivial as considered above, is itself· · · nontrivial.∈ ∈ FAILURE OF THE LOCAL-GLOBAL PRINCIPLE FOR ISOTROPY 5

3. Failure of the local global principle

Given elliptic curves E1,...,En deﬁned over C with algebraically independent j-invariants, presented in Legendre form (1), and X P1 P1 the double 2 → ×···× cover deﬁned by y = f(x1,...,xn) in (2), we consider the quadratic form

q = x1,...,xn; f ≪ ≫ 1 1 over C(P P )= C(x1,...,xn), as in Section 1. Our main×···× result is that q shows the failure of the local-global principle for isotropy, with respect to all discrete valuations, for quadratic forms of dimension 2n over C(x1,...,xn), thereby proving Theorem 1.

Theorem 3.1. The quadratic form q = x1,...,xn; f is anisotropic over ≪ ≫ C(x1,...,xn) yet is isotropic over the completion at every discrete valuation.

Proof. Let K = C(x1,...,xn) and L = K(√f) = C(X). Let v be a discrete valuation of K and w an extension to L, with completions Kv and Lw and residue fields κ(v) and κ(w), respectively. We note that κ(v) and κ(w) have transcendence degree 0 i n 1 over C. By Proposition 1.1, we have that q K Kv is isotropic ≤ ≤ − ⊗ if and only if q K Lw is isotropic. ⊗ n ⊗n By Proposition 2.2, the restriction (x1) (xn) H (L,µ ) is unramified at w, · · · ∈ 2 hence q K L = x1,...,xn is an n-fold Pfister form over L unramified at w. ⊗ ≪ ≫ Thus the first residue form for q K L, with respect to the valuation w, is isotropic ⊗ n i since the residue field κ(w) is a Ci-field and q has dimension 2 > 2 . Consequently, by a theorem of Springer [19, Ch. 6, Cor. 2.6], q K L, and thus q, is isotropic. ⊗ Finally, q is anisotropic since the symbol (x1) (xn) is nontrivial when restricted to C(Y ) by [10, Appendice], hence is nontrivial· when · · restricted to C(X). To give an explicit example, let λ, κ, ν Cr 0, 1 be algebraically independent complex numbers. Then over the function∈ field K{ =}C(x,y,z), the quadratic form q = <1,x,y,z,xy,xz,yz, (x 1)(y 1)(z 1)(x λ)(y κ)(z ν)> − − − − − − is isotropic over every completion Kv associated to a discrete valuation v of K, and yet q is anisotropic over K.

4. Over general function fields We have exhibited locally isotropic but globally anisotropic quadratic forms of n dimension 2 over the rational function field C(x1,...,xn). In [4, Cor. 6.5], we proved that locally isotropic but anisotropic quadratic forms of dimension 4 exist over any function field of transcendence degree 2 over an algebraically closed field of characteristic zero. Taking these as motivation, we recall Conjecture 2, that over any function field of transcendence degree n 2 over an algebraically closed field of characteristic = 2, there exist locally isotropic≥ but anisotropic quadratic forms of dimension 2n.6 In this section, we provide a possible approach to Conjecture 2, motivated by the geometric realization result in [4, Proposition 6.4]. Proposition 4.1. Let K = k(X) be the function field of a smooth projective variety X of dimension n 2 over an algebraically closed field k of characteristic = 2. n ≥⊗n n ⊗n 6 If either Hnr(K/k,µ2 ) = 0 or Hnr(L/k,µ2 ) = 0 for some separable quadratic extension L/K, then there6 exists an anisotropic6 quadratic form of dimension 2n over K that is isotropic over the completion at every discrete valuation. 6 AUEL AND SURESH

Proof. First, by a standard application of the Milnor conjectures, every element in n ⊗n n ⊗n H (K,µ ) is a symbol since K is a Cn-field. If Hnr(K/k,µ ) = 0, then taking 2 2 6 a nontrivial element (a1) (an), the n-fold Pfister form a1, . . . , an is locally isotropic (by the same argument· · · as in the proof of Theorem≪ 3.1) but is≫ anisotropic, n ⊗n giving an example. So we can assume that Hnr(K/k,µ2 ) = 0. n ⊗n Now assume that Hnr(L/k,µ ) = 0 for some separable quadratic extension 2 6 L = K(√d) of K. Then taking a nontrivial element (a1) (an), the corestric- n n tion map Hn (L/k,µ⊗ ) Hn (K/k,µ⊗ ) = 0 is trivial,· so · · by the restriction- nr 2 → nr 2 corestriction sequence for Galois cohomology, we have that (a1) (an) is in the n ⊗n n ⊗n · · · image of the restriction map Hnr(K/k,µ2 ) Hnr(L/k,µ2 ) = 0, in which case × → we can assume that a1, . . . , an K . Then the quadratic form a1, . . . , an; d is locally isotropic over K (by the∈ same argument as in the proof of≪ Theorem 3.1)≫ but globally anisotropic. Hence we are naturally led to the following geometric realization conjecture for unramified cohomology classes. Conjecture 4.2. Let K be a finitely generated field of transcendence degree n over n ⊗n an algebraically closed field k of characteristic = 2. Then either Hnr(K/k,µ2 ) = 0 6 n ⊗n 6 or their exists a quadratic extension L/K such that H (L/k,µ ) = 0. nr 2 6 Proposition 4.1 says that the geometric realization Conjecture 4.2 implies Con- jecture 2 on the failure of the local-global principle for isotropy of quadratic forms. Proposition 2.2 establishes the conjecture in the case when K is purely transcen- dental over k; in [4, Proposition 6.4], we established the conjecture in dimension 2 and characteristic 0, specifically, that given any smooth projective surface S over an algebraically closed field of characteristic zero, there exists a double cover T S 2 ⊗2 → with T smooth and Hnr(k(T )/k,µ2 ) = Br(T )[2] = 0. In this latter case, Propo- sition 4.1 gives a different proof, than the one presented6 in [4, 6], that there exist locally isotropic but anisotropic quadratic forms of dimension 4§ over K = k(S).

References [1] W. Aitken and F. Lemmermeyer, Counterexamples to the Hasse principle, Amer. Math. Monthly 118 (2011), no. 7, 610–628. 1 [2] D. Leep, The Amer–Brumer theorem over arbitrary fields, preprint, 2007. 1 [3] A. Auel, Failure of the local-global principle for isotropy of quadratic forms over surfaces, Oberwolfach Reports 10 (2013), issue 2, report 31, 1823–1825. 1 [4] A. Auel, R. Parimala, and V. Suresh, Quadric surface bundles over surfaces, Doc. Math., Extra Volume: Alexander S. Merkurjev’s Sixtieth Birthday (2015), 31–70. 1, 2, 5, 6 [5] A. Bevelacqua, Four dimensional quadratic forms over F (X) where ItF (X) = 0 and a failure of the strong Hasse principle, Commun. Algebra 32 (2004), no. 3, 855–877. 1 [6] C. Borcea, Calabi–Yau threefolds and complex multiplication, Essays on Mirror Manifolds, Int. Press, Hong Kong, 1992, 489–502 1, 3 [7] J. W. S. Cassels, Arithmetic on Curves of Genus 1 (V). Two Counter-Examples, J. London Math. Soc. 38 (1963), 244–248. 1 [8] J. W. S. Cassels, W. J. Ellison, and A. Pfister, On sums of squares and on elliptic curves over function fields, J. Number Theory 3 (1971), 125–149. 1 [9] J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), Amer. Math. Soc., Proc. Sympos. Pure Math. 58 (1995), 1–64. 3 [10] J.-L. Colliot-Thélène, Exposant et indice d’algèbres simples centrales non ramifiées, with an appendix by O. Gabber, Enseign. Math. (2) 48 (2002), no. 1–2, 127–146. 3, 4, 5 FAILURE OF THE LOCAL-GLOBAL PRINCIPLE FOR ISOTROPY 7

[11] S. Cynk and K. Hulek, Higher-dimensional modular CalabiYau manifolds, Canad. Math. Bull. 50 (2007), no. 4, 486–503. 1, 3 [12] S. Cynk and M. Schütt, Generalised Kummer constructions and Weil restrictions, J. Number Theor. 129 (2009), no. 8, 1965–1975. 3 [13] R. Elman, N. Karpenko, and A. Merkurjev, The algebraic and geometric theory of quadratic forms, American Mathematical Society Colloquium Publications, vol. 56, American Mathe- matical Society, Providence, RI, 2008. 1 [14] P. Jaworski, On the strong Hasse principle for fields of quotients of power series rings in two variables, Math. Z. 236 (2001), no. 3, 531–566. 1 [15] K. H. Kim and W. Roush, Quadratic forms over C[t1,t2], J. Algebra 140 (1991), no. 1, 65–82. 1 [16] S. Lang, On quasi algebraic closure, Ann. of Math. (2) 55 (1952), 373–390. 2 [17] C.-E. Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, Dissertation, University Uppsala, 1940. 1 [18] H. Reichardt, Einige im Kleinen überall lösbare, im Großen unlösbare diophantische Gleichun- gen, J. Reine Angew. Math. 184 (1942), 12–18. 1 [19] W. Scharlau, Quadratic and Hermitian forms, Spring-Verlag, Berlin, 1985. 2, 5 [20] E. Witt, Uber¨ ein Gegenbeispiel zum Normensatz, Math. Zeitsch. 39 (1934), 462–467. 1

Asher Auel, Department of Mathematics, Yale University, New Haven, Connecticut E-mail address: [email protected]

V. Suresh, Department of Mathematics & Computer Science, Emory University, Atlanta, Georgia E-mail address: [email protected]