EUCLIDEAN QUADRATIC FORMS AND ADC-FORMS: I

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1 EUCLIDEAN QUADRATIC FORMS AND ADC-FORMS: I PETE L. CLARK Abstract. A classical result of Aubry, Davenport and Cassels gives conditions for an integral quadratic form to integrally represent every integer that it rationally represents. We give a generalization which allows one to pass from rational to integral representations for suitable quadratic forms over a normed ring. The Cassels-Pfister theorem follows as another special case. This motivates a closer study of the classes of forms which satisfy the hypothesis of our theorem ( Euclidean forms ) and its conclusion ( ADC forms ) Normed rings. Let R be a commutative, unital ring. We write R for R \ {0}. A norm on R is a function : R N such that (N0) x = 0 x = 0, (N1) x R, x = 1 x R, and (N2) x, y R, xy = x y. A norm is non-archimedean if for all x, y R, x + y max( x, y ). A normed rins a pair (R, ) where is a norm on R. A ring admitting a norm is necessarily an integral domain. We denote the fraction field by K. The norm extends uniquely to a homomorphism of groups (K, ) (Q >0, ). Example 1: The usual absolute value (inherited from R) is a norm on Z. Example 2: Let k be a field, R = k[t], and let a 2 be an integer. Then the map f k[t] a deg f is a non-archimedean norm a on R. When k is finite, it is most natural to take a = #k (see below). Otherwise, we may as well take a = 2. Example 3: An abstract number rins an infinite integral domain R such that for every nonzero ideal I of R, R/I is finite. (In particular, R may be an order in a number field, the ring of regular functions on an irreducible affine algebraic curve over a finite field, or any localization or completion thereof.) The map x R #R/(x) gives a norm on R [Cl10, Prop. 5], which we will call the canonical norm. The standard norm on Z is canonical, as is the norm q on the polynomial ring F q [t]. Example 4: Let R be a discrete valuation ring (DVR) with valuation v : K Z Partially supported by National Science Foundation grant DMS c Pete L. Clark,

2 2 PETE L. CLARK and residue field k. Choosing an integer a 2 we can define a norm a : R Z + by x a = a v(x). R is an abstract number rinff the residue field k is finite, and in this case the norm #k is the canonical norm. Note well that the norm a is not quite the usual norm x a v(x) associated to a DVR: rather, it is the reciprocal of the usual norm. Especially, beware: the norm a is not non-archimedean. A norm on a ring R is Euclidean if for all x K, there exists y R such that x y < 1. A domain which admits a Euclidean norm is a principal ideal domain (PID). It is known that the converse is not true, both in the sense that a given norm on a PID need not be Euclidean and in the stronger sense that there are PIDs which do not admit any Euclidean norm. However, for our purposes we wish to consider the norm as part of the given structure on R, so when we say R is Euclidean, we really mean the given norm on R is a Euclidean norm. Example 5: The canonical norm on Z is a Euclidean norm. For any field k and any a 2, the norm a on k[t] (c.f. Example 2) is a Euclidean norm. If (R, v) is discrete valuation ring, then the norm a (c.f. Example 4) is Euclidean: indeed, for x K, x K \ R v(x) < 0 x a = a v(x) < 1, so we may take y = Euclidean quadratic forms. Let (R, ) be a normed ring of characteristic not 2. A quadratic form over R, is a polynomial q R[x] = R[x 1,..., x n ] which is homogeneous of degree 2. Throughout this note we only consider quadratic forms which are non-degenerate over the fraction field K of R. A nondegenerate quadratic form q /R is isotropic if there exists a = (a 1,..., a n ) R n \ {(0,..., 0)} such that q(a) = 0; otherwise q is anisotropic. A form q is anisotropic over R iff it is anisotropic over K. A quadratic form q /R is universal if for all d R, there exists x R n such that q(x) = d. A quadratic form q on a normed ring (R, ) is Euclidean if for all x K n \ R n, there exists y R n such 0 < q(x y) < 1. Remark 1: An anisotropic quadratic form q is Euclidean iff for all x K n there exists y R n such that q(x y) < 1. Proposition 1. The norm on R is a Euclidean norm iff the quadratic form q(x) = x 2 is a Euclidean quadratic form. Proof. Noting that q is an anisotropic quadratic form, this comes down to: x, y K, x y < 1 q(x y) = (x y) 2 = x y 2 < 1. Remark 2: Also (R, ) is Euclidean iff the hyperbolic plane q = x 1 x 2 is Euclidean. Example 6: Let n, a 1,..., a n Z +. Then the integral quadratic form q(x) = a 1 x a n x 2 n is Euclidean iff i a i < ADC-forms. A quadratic form q(x) = q(x 1,..., x n ) over R is an ADC-form if for all d R, if

3 EUCLIDEAN QUADRATIC FORMS AND ADC-FORMS: I 3 there exists x K n such that q(x) = d, then there exists y R n such that q(y) = d. Example 7: Any universal quadratic form is an ADC-form. If R = Z and q is positive definite and positive universal i.e., represents all positive integers then q is an ADC-form. Thus for each n 5 there are infinitely many positive definite ADC-forms, e.g. x x 2 n 1 + dx 2 n for d Z +. Example 8: Let R be the integral closure of R in K. Then q(x) = x 2 is not an ADC-form iff there exists a R \ R such that a 2 R. In particular x 2 is an ADC-form if R is integrally closed. Example 9: Let R be a UFD and a R. iff a is squarefree. Then q(x) = ax 2 is an ADC-form Example 10: Suppose R is an algebra over a field k, and let q /k be isotropic. Then the base extension of q to R is universal. Indeed, since q is isotropic over k, it contains the hyperbolic plane as a subform. That is, after a k-linear change of variables, we may assume q = x 1 x 2 +q (x 3,..., x n ), and the conclusion is now clear. Example 11: The isotropic form q(x, y) = x 2 y 2 is not an ADC-form over Z: indeed it is universal over Q but not over Z. Example 12: In 1912, L. Aubry showed that q = x x x 2 3 is an ADC-form over Z [Aub12]. This leads to an elegant and conceptual proof of the Legendre- Gauss Three Squares Theorem, via reduction to the Hasse-Minkowski theory. By Example 6, q is Euclidean. Indeed, Aubry s proof exploits the Euclidean property. However, his argument seems to have been forgotten for many years, and circa 1960 Davenport and Cassels (unpublished) essentially rediscovered it. Theorem 2. (Aubry-Davenport-Cassels-Serre-Weil) Every Euclidean quadratic form q over Z is an ADC-form. Remark 3: In his widely read text [Se73, Lemma B, p. 46], Serre states and proves this theorem with the additional hypotheses that q be positive-definite and classically integral: i.e., that (x, y) 1 2 (q(x + y) q(x) q(y)) be Z-valued. The first to state Theorem 2 in full seems to be A. Weil in [We84, p. 294]. In 2009, Serre gave a new argument [SeMO] proving a slightly more general result. ADC-forms also appear (although heretofore not by name) in the following result. Theorem 3. (Cassels-Pfister [Ca64] [Pf65]) Let k be a field of characteristic different from 2. Let q /k be a quadratic form, and consider q as a quadratic form over the polynomial ring R = k[t]. Then q /R is an ADC-form: that is, every polynomial p k[t] which is represented by q over the field k(t) of rational functions is also represented by q over the ring k[t] of polynomial functions The Main Theorem. Theorem 4. Let (R, ) be a normed ring not of characteristic 2 and q /R a Euclidean quadratic form. Then q is an ADC-form.

4 4 PETE L. CLARK Proof. For x, y K n, put x y := 1 2 (q(x + y) q(x) q(x)). Then (x, y) x y is bilinear and x x = q(x). Note that for x, y R n, we need not have x y R, but certainly we have 2(x y) R. Let d R, and suppose there exists x K n such that q(x) = d. Equivalently, there exists t R and x R n such that t 2 d = x x. Choose x and t such that t is minimal. It is enough to show that t = 1, for then by (N1) t R. Apply the Euclidean hypothesis with x = x d : there is y R such that if z = x y, 0 < q(z) < 1. Now put a = y y d, b = 2dt 2(x y), T = at + b, X = ax + by. Then a, b, T R, and X R n. Claim: X X = T 2 d. Indeed, X X = a 2 (x x ) + ab(2x y) = b 2 (y y) = a 2 t 2 d + ab(2dt b) + b 2 (d + a) = d(a 2 t 2 + 2abt + b 2 ) = T 2 d. Claim: T = t(z z). Indeed, tt = at 2 + bt = t 2 (y y) dt 2 + 2dt 2 t(2x y) = t 2 (y y) t(2x y) + x x = (ty x ) (ty x ) = ( tz) ( tz) = t 2 (z z). Since 0 < z z < 1, we have 0 < T < t, contradicting the minimality of t. Remark 4: This proof is modelled on that of [Se73, pp ] Deducing the (Generalized) Cassels-Pfister Theorem. Lemma 5. Let q be an anisotropic quadratic form over a field k. Then q remains anisotropic over the rational function field k(t). Proof. If there exists a nonzero vector x k(t) n such that q(x) = 0, then (since k[t] is a UFD) there exists y = (y 1,..., y n ) such that y R n, gcd(y 1,..., y n ) = 1 and q(y) = 0. The polynomials y 1,..., y n do not all vanish at 0, so (y 1 (0),..., y n (0)) k n \ (0,..., 0) is such that q(y 1 (0),..., y n (0)) = 0, i.e., q is isotropic over k. Remark 5: The argument of Lemma 5 actually shows that a projective variety V /k has a k-rational point iff it has a k(t)-rational point. Proof of Theorem 3: let q = i,j a ijx i x j be a quadratic form over k. We view q as a quadratic form over R = k[t] via base extension. If q is isotropic over k, then by Example 10, q /R is universal. So suppose q is anisotropic over k, hence also, by Lemma 5, over k(t). By Theorem 4, it suffices to show that as a quadratic form over R = k[t] endowed with the norm = 2 of Example 2, q is Euclidean. Given an element x = ( f1(t) g,..., fn(t) 1(t) g ) n(t) Kn, by polynomial division we may write fi = y i + ri with y i, r i k[t] and deg(r i ) < deg( ). Putting y = (y 1,..., y n ) and using the non-archimedean property of, we find (1) q(x y) = i,j a i,j ( r i )( r j g j ) ( max a i,j i,j ) ( max r i i ) 2 < 1.

5 EUCLIDEAN QUADRATIC FORMS AND ADC-FORMS: I 5 Remark 6: Suppose q = i,j a i,jx i x j is an anisotropic quadratic form over k[t] such that each a i,j (t) has degree at most one. Then max i,j a i,j 2 while max i ri 2 1 4, so (1) still holds and shows that q is Euclidean and hence ADC. This is sometimes called the Generalized Cassels-Pfister Theorem: c.f. [Pf95, Remark 1.2.3]. By Example 9, extension to forms with coefficients of degree at most 2 is not possible Another class of anisotropic Euclidean forms. Let R be a complete discrete valuation ring with fraction field K. Let q : R n R be a quadratic form on R. By tensoring to K, we may view q as being a quadratic form on K n and R n as an R-lattice Λ K n. The quadratic form q satisfies the integrality property q(λ) R. We say that Λ is maximal if there does not exist an R-lattice Λ strictly containing Λ such that q(λ ) R. Theorem 6. (Eichler s Maximal Lattice Theorem) Let q be an anisotropic quadratic form over a complete discrete valuation field K with valuation ring R. Then there is a unique maximal R-lattice for q, namely Proof. See [Ei52] or [Ge08, Thm. 8.8]. Λ = {x K n q(x) R}. Corollary 7. Let R be a complete, discretely valued ring (not of characteristic 2), with fraction field K. Then, for each anisotropic quadratic form q /K, restricting to the unique maximal R-lattice Λ defines a Euclidean quadratic form over R. Proof. The proof is immediate from the fact that R n = {x K n q(x) R}. Indeed, this is equivalent to R n = {x K n q(x) a 1}, where x a = a v(x) is the norm of Example 4. Therefore, x K n \ R n q(x) a = q(x 0) a < Future work. In this note we have attempted to set the stage for a systematic study of Euclidean forms and ADC-forms. Work is in progress on the following topics: Classification of Euclidean quadratic forms over Z: In [Co07], H. Cohen reports on work of J. Houriet classifying all positive definite classically integral Euclidean quadratic forms. However, the form 2x 2 + 2xy + 3y 2 is missing from this list. We wish to verify that Houriet s list is otherwise complete and extend the list to non-classically integral positive definite Euclidean forms. The corresponding problem for indefinite anisotropic forms is also of interest, but seems more difficult. Quadratic Rings: There is a correspondence between primitive anisotropic binary quadratic forms over Z and invertible ideal classes in quadratic orders (see e.g. [Co93, 5.2]) under which Euclidean forms correspond to H.W. Lenstra s Euclidean ideal classes [Le79]. Combining Lenstra s work with the classification of Euclidean quadratic rings gives a complete list of anisotropic Euclidean binary forms over Z. This correspondence can be generalized to other normed domains. Quaternion rings: Let B be a free quaternion ring over a domain R in the sense of [Vo11]. Then the reduced norm N on B defines a quaternary quadratic form

6 6 PETE L. CLARK (q B ) /R. The forms which arise in this way are all principal of square discriminant. If is a norm on R, then q B is Euclidean iff the ring B is left-euclidean for the induced norm x B N(x). The definite norm-euclidean quaternion rings over Z have been classified by Fitzgerald [Fi10], who noted the relation to the classical quaternionic proof of Lagrange s Four Squares Theorem. Moreover, the notion of a Euclidean ideal class for B makes sense in this setting. Boundary-Euclidean forms: For an anisotropic quadratic form q over a normed ring (R, ) with fraction field K, define the Euclideanity of q as E(q) = sup x K n y R inf q(x y). n Thus q is Euclidean if E(q) < 1 and is certainly not Euclidean if E(q) > 1. The case E(q) = 1 is ambiguous: q is Euclidean iff the supremum in the definition of E(q) is not attained. However, certain non-euclidean forms q /Z with E(q) = 1 can nevertheless be shown to be ADC-forms via the method of Theorem 4 [We84, pp ], e.g. x 2 + 3y 2. (There are also non-adc forms with E(q) = 1, e.g. 2x 2 + 2y 2.) So the forms q with E(q) = 1 seem to deserve further study. References [Aub12] L. Aubry, Sphinx-Œdipe 7 (1912), [Cl10] P.L. Clark, Factorization in Integral Domains, pete/factorization2010.pdf [Ca64] J.W.S. Cassels, On the representation of rational functions as sums of squares. Acta Arith. 9 (1964), [Co93] H. Cohen, A course in computational algebraic number theory. Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, [Co07] H. Cohen, Number theory. Vol. I. Tools and Diophantine equations. Graduate Texts in Mathematics, 239. Springer, New York, [Ei52] M. Eichler, Martin Quadratische Formen und orthogonale Gruppen. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Band LXIII. Springer-Verlag, [Fi10] R.W. Fitzgerald, Norm Euclidean Quaternionic Orders, preprint, [Ge08] L.J. Gerstein, Basic quadratic forms. Graduate Studies in Mathematics, 90. American Mathematical Society, Providence, RI, [Le79] H.W. Lenstra, Jr., Euclidean ideal classes. Journées Arithmétiques de Luminy (Colloq. Internat. CNRS, Centre Univ. Luminy, Luminy, 1978), pp , [Pf65] A. Pfister, Multiplikative quadratische Formen. Arch. Math. (Basel) 16 (1965), [Pf95] A. Pfister, Quadratic forms with applications to algebraic geometry and topology. London Math. Society Lecture Note Series, 217. Cambridge University Press, Cambridge, [Se73] J.-P. Serre, A course in arithmetic. Translated from the French. Graduate Texts in Mathematics, No. 7. Springer-Verlag, New York-Heidelberg, [SeMO] J.-.P. Serre communicated to B. Poonen communicated to MathOverflow.net: mathoverflow.net/questions/3269 [Vo11] J. Voight, Characterizing quaternion rings over an arbitrary base, to appear. [We84] A. Weil, Number theory. An approach through history from Hammurapi to Legendre. Reprint of the 1984 edition. Modern Birkhäuser Classics, Boston, MA, Department of Mathematics, Boyd Graduate Studies Research Center, University of Georgia, Athens, GA , USA address: pete@math.uga.edu