LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS

VÍTĚZSLAV KALA AND PAVLO YATSYNA

Abstract. We study totally real number fields that admit a universal quadratic form whose coeffi-cients are rational integers. We show that Q(

√5) is the only such real quadratic field, and that among

fields of degrees 3, 4, 5, and 7 which have principal codifferent ideal, the only one is Q(ζ7 + ζ−17 ), overwhich the form x2 + y2 + z2 +w2 + xy+ xz+ xw is universal. Moreover, we prove an upper bound forPythagoras numbers of orders in number fields that depends only on the degree of the number field.

1. Introduction

The question which integers can be represented by a given quadratic form has long played a centralrole in number theory, involving works of mathematicians such as Diophantus, Brahmagupta, Fermat,Euler, and Gauss. Of particular interest have been universal quadratic forms, i.e., positive definiteforms that represent all natural numbers. The first example of the sum of four squares x2 + y2 + z2 +w2 was followed by many others, including classification of quaternary diagonal universal forms byRamanujan and Dickson, and culminating in the 15- and 290- theorems of Conway-Schneeberger andBhargava-Hanke [Bh, BH].

A natural generalization has been the study of universal quadratic forms over number fields Kand their rings of algebraic integers OK . When the field has a complex embedding, every quadraticform over K is indefinite, and so it is comparatively easy to understand which algebraic integers itrepresents. For example, Siegel [Si3] and Estes-Hsia [EH] considered complex fields with universalsums of 5 and 3 squares (respectively) and characterized them. Hence of particular interest are totallyreal number fields where one expects to have a rich and hard theory of representations by totallypositive quadratic forms.

In 1941 Maaß [Ma] used theta series to show that the sum of three squares is universal over thering of integers of Q(

√5). Siegel [Si3] then in 1945 proved that the sum of any number of squares

is universal only over the number fields K = Q,Q(√

5). However, universal forms exist over everytotally real number field [HKK], and there have been numerous recent results concerning them, see,e.g., [CKR, EK, Ea, Ki1, Ki2, CI, Sa2, De1, BK1, BK2, Ka1, Ya, KS, CL+, KTZ] and the referencestherein.

Almost all of these results involve quadratic forms which do not have rational integers as all oftheir coefficients. This is not an accident, as indeed Siegel’s result immediately implies that a diagonalpositive definite quadratic form with Z-coefficients can be universal only over K = Q,Q(

√5). This

suggests the following natural generalization: when is it possible for a positive definite quadratic formwith Z-coefficients to be universal over the ring of integers OK of a number field K? Or more generally,one can consider two (totally real) number fields K ⊂ L and ask whether there is a quadratic formwith OK coefficients that is universal over OL. This is sometimes known as the lifting problem foruniversal quadratic forms over number fields; the main goal of the present article is to consider it forZ-forms, i.e., positive definite quadratic form with Z-coefficients.

We completely solve this problem for real quadratic fields by proving

Theorem 1.1. There does not exist a Z-form that is universal over a real quadratic number field K,unless K = Q(

√5).

Date: April 15, 2020.2010 Mathematics Subject Classification. 11E12, 11E25, 11R11, 11R18, 11H06, 11H55.Key words and phrases. universal quadratic form, totally real number field, trace form, lattice of E-type, Dedekind

zeta function, additively indecomposable integer.Both authors were supported by the project PRIMUS/20/SCI/002 from Charles University. The first author was

supported by Czech Science Foundation GAČR, grant 17-04703Y, and by Charles University Research Centre programUNCE/SCI/022. Most of the work on this article was done when the second author was visiting Charles University, alsosupported by Czech Science Foundation GAČR, grant 17-04703Y.

1

arX

iv:1

808.

0226

2v2

[m

ath.

NT

] 1

4 A

pr 2

020

2 VÍTĚZSLAV KALA AND PAVLO YATSYNA

Over Q(√

5), there are indeed quite a few universal Z-forms, such as x2 +y2 +z2 [Ma], x2 +y2 +2z2[CKR], x2 + xy+ y2 + z2 + zw+w2 [De1], x2 + y2 + z2 +w2 + xy+ xz+ xw [De2]. Lee [Le] classifiedall quaternary classical universal forms (recall that a form is classical if all its off-diagonal coefficientsare divisible by 2), but his list does not give any other examples that are Z-forms. We are not awareof any classification of universal Z-forms over Q(

√5) (or any other number field); this is another very

interesting open problem.We then turn our attention to number fields of higher degree, where the situation is much more

convoluted.In the spirit of the study of the minimal number of variables required by a universal form, we first

show in Corollary 3.4 that there are no classical universal Z-forms of rank strictly less than 6 over anytotally real number field (of arbitrary degree), except for Q,Q(

√5).

We finally focus on the existence of universal Z-forms over certain number fields of small degree.

Theorem 1.2. There does not exist a totally real number field K of degree 1, 2, 3, 4, 5 or 7 which hasprincipal codifferent ideal and a universal Z-form defined over it, unless K = Q,Q(

√5) or Q(ζ7 +ζ−17 ).

The Z-form x2 + y2 + z2 + w2 + xy + xz + xw is universal over Q(ζ7 + ζ−17 ).

We prove this result as Theorems 5.1 and 6.2. Note that the codifferent is principal for examplewhen OK = Z[α] for some α or when K has class number one.

The limiting assumptions in the theorem come from the tools that we use. First of all, the compo-sition of a Z-form with (twisted) trace form decomposes as a tensor product, and so we study tensorproducts of positive definite Z-lattices and their minimal vectors. In particular, lattices of E-type(which were first introduced by Kitaoka [Kt2]) play a prominent role in Section 4. We use them toshow that if certain additively indecomposable algebraic integers are represented by a Z-form, thenthey have to be squares. This in turn for example implies that if a number field possesses a universalZ-form, then it has units of all signatures. Not every lattice is of E-type, but in the small degreesconsidered in Theorem 1.2 this poses no restrictions.

Our second main tool is Siegel’s formula for the value of Dedekind zeta function at s = −1 [Si1, Za],which expresses this value in terms of elements of the codifferent of small trace. In particular, weare interested in elements of trace 1, and these are the only ones that appear in the formula fordegrees 2, 3, 4, 5 or 7. When the codifferent is principal, the resulting bound on their number gives(together with the results of Section 4) an estimate on the number of minimal vectors of the traceform. However, this estimate can hold only for very few number fields, which in turn implies thetheorem. It is tempting to try to apply Siegel’s formula also for higher degrees by (for example) usingelements of trace 2 to deduce the existence of elements of trace 1. Unfortunately, the resulting boundsseem to be too weak to be of much use.

In several of the proofs we use computer calculations to deal with specific number fields and qua-dratic forms. All of these computations were done in Magma [BCP] and are straightforward.

Note that Theorem 1.1 is a special case of Theorem 1.2. However, in the quadratic case we havemore explicit control of elements of small trace, and so the proof of Theorem 1.1 is more elementaryand does not require the use of Dedekind zeta function.

The question whether there exists a universal Z-form over a number field not covered by Theorem1.2 remains open and may be very hard. Our results provide some clues towards conjecturing thatthere are perhaps no number fields with Z-forms except for Q,Q(

√5), and Q(ζ7 + ζ−17 ), but the

evidence is of course quite weak. Even more broadly, the following general lifting problem questionremains completely open.

Question. Is there a totally real number field K such that there are infinitely many totally realnumber fields L ⊃ K that admit a universal quadratic form with OK-coefficients?

The auxiliary results that we obtain are also useful for the study of Pythagoras numbers of ordersO in totally real fields. While we know that typically not all totally positive integers are sums ofsquares, we can ask what is the smallest integer m such that if an element is the sum of squares,then it is the sum of at most m squares. This integer m is called the Pythagoras number of theorder O and is known to be always finite, but can be arbitrarily large [Sc2] (cf. also [Po]). In theaforementioned article, Scharlau asked whether Pythagoras numbers of orders are bounded by thedegree of the corresponding number field. We answer this question affirmatively as Corollary 3.3.

LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS 3

Let us note that while we state most of our results only for the maximal order OK (as it is, arguably,the most interesting case), many of them can probably be quite straightforwardly extended to generalorders, at least the material in Sections 2, 3, 4, and 6. However, the references that we use typicallyalso deal only with the full ring of integers OK , and so this extension would require reproving all thesereferenced results in the more general setting.

Finally, let us note that besides from studying representations of integers by quadratic forms, therehave been numerous works considering representations of quadratic forms by quadratic forms and, inparticular, by the sum of squares, e.g., [BI, Ic, KO1, Sa1, Oh, KO2, KO3, JKO, BC+] (in fact, we usesome of them in the proof of Corollary 3.3). Most of them deal with forms over Z, but it is anotherexciting direction of future research to consider the situation over number fields in detail and to studythe lifting problem for representations of quadratic forms of a given rank > 1.

Acknowledgments

We thank Tomáš Hejda for our interesting conversations and for his help with computations relatedto this research, and to the anonymous referees for numerous useful comments that have helped toimprove the paper and its exposition.

2. Preliminaries

Throughout the article, K will denote a totally real number field of degree d over Q with the ringof integers OK . For an order O ⊂ OK we fix an integral basis ω1, . . . , ωd and denote its group of unitsby O×.

Let σ1 = id, σ2, . . . , σd : K ↪→ R be the (distinct) real embeddings of K. The norm of α ∈ K isthen N(α) = σ1(α) · · ·σd(α) and its trace is Tr(α) = σ1(α) + · · ·+ σd(α).

We write α � β to mean σi(α) > σi(β) for all 1 ≤ i ≤ d; moreover, α � β denotes α � β or α = β.An algebraic integer α ∈ OK is totally positive if α � 0; the semiring of totally positive integers thatlie in the order O will be denoted O+; moreover we let O×,+ = O× ∩O+. By the signature of α ∈ Kwe mean the d-tuple of signs of σi(α).

We say that α ∈ O+ is indecomposable if it can not be decomposed as the sum of two totallypositive elements of O, or equivalently if there is no β ∈ O+ such that α � β. Indecomposableintegers and their norms are quite well studied, especially over real quadratic fields [DS, JK, Ka2].

The following lemma shows that when an element has sufficiently small norm, then it has to beindecomposable. In particular, every totally positive unit is indecomposable.

Lemma 2.1. a) For all α1, α2 ∈ O+ we have

N(α1 + α2)1/d ≥ N(α1)1/d + N(α2)1/d.

b) If β ∈ O+ has norm N(β) < 2d, then β is indecomposable.

Proof. Both parts are easy to show and quite well-known, but let us include the proofs for reader’sconvenience:

a) follows by a simple use of Hölder’s inequality [OM2, 3.1]: By (a version of) generalized Hölder’sinequality [HLP, Theorem 12] we have

r∏i=1

s∑j=1

arij

1/r ≥ s∑j=1

r∏i=1

aij , where r, s ∈ Z+ and aij ∈ R+.

Setting r = d, s = 2, and aij = σi(αj)1/d we get

N(α1 + α2)1/d =

d∏i=1

(σi(α1 + α2))1/d ≥

d∏i=1

σi(α1)1/d +

d∏i=1

σi(α2)1/d = N(α1)

1/d + N(α2)1/d.

b) Assume that β is decomposable as α1 + α2. Then 2 > N(β)1/d = N(α1 + α2)

1/d ≥ N(α1)1/d +N(α2)

1/d ≥ 1 + 1, which is not possible. �

We denote O∨ := {β ∈ K : Tr(βO) ⊂ Z} the codifferent of O; O∨,+ is the semiring of all totallypositive elements of O∨. Recall that if OK = Z[ω] for some ω with minimal polynomial f(x) ∈ Z[x],then the codifferent is the principal fractional ideal O∨K =

1f ′(ω)OK [Nar, Proposition 4.17].

4 VÍTĚZSLAV KALA AND PAVLO YATSYNA

We will often work with positive definite quadratic formsQ ∈ Z[x1, . . . , xr], i.e., Q(x) =∑

i≥j aijxixjwith aij ∈ Z, and refer to such quadratic forms as Z-forms from now on; r is the arity (or rank) ofQ. Recall that Q is positive semidefinite if Q(x) ≥ 0 for all x ∈ Rr, and positive definite if moreoverQ(x) = 0 implies x = 0. If aij ∈ 2Z for all i 6= j, then we say that Q is classical Z-form (andnon-classical, otherwise).

For a given quadratic form Q we can define the bilinear form BQ ∈ Z[x1, . . . , xr] by BQ(x, y) =(Q(x+y)−Q(x)−Q(y))/2. Note that if e(1) = (1, 0, . . . , 0), e(2) = (0, 1, 0, . . . , 0), . . . , e(r) = (0, . . . , 0, 1),then BQ(e

(i), e(i)) = aii and BQ(e(i), e(j)) = aij/2 for i 6= j; the r × r matrix MQ := (BQ(e(i), e(j)))

is called the Gram matrix of Q. We have Q(x) = xTMQx and BQ(x, y) = xTMQy for x, y ∈ Zr (AT

denotes the transpose of a matrix A).Unless specified otherwise, throughout the paper Q will denote a Z-form of rank r.

We will sometimes also need to work with quadratic forms over O, i.e., Q(x) =∑

i≥j aijxixj withaij ∈ O. The corresponding bilinear form BQ and Gram matrix MQ are defined in the same wayas over Z; we have Q(x) = xTMQx and BQ(x, y) = xTMQy for x, y ∈ Or. Such a form Q is totallypositive if Q(a) � 0 for all a ∈ Or, a 6= 0.

A Z-form Q can be naturally viewed also as a totally positive quadratic form Q(O) over O with thesame Gram matrix MQ(O) = MQ; although we will not explicitly distinguish between the two, as it

should lead to no confusion. Thus we will say that a Z-form Q represents an element α ∈ O+ over theorder O if Q(v) = α for some v ∈ Or. We say that Q is universal over O if it represents every elementα ∈ O+ over O. When dealing with the maximal order OK , we often just say that Q is universal (oruniversal over K to specify the number field).

Sometimes it is useful to work in lattice-theoretic language, which we now briefly introduce. Atotally positive quadratic space over K is an r-dimensional vector space V over the field K equippedwith a symmetric bilinear form B : V ×V → V such that the associated quadratic form Q(v) = B(v, v)attains totally positive values at all non-zero v ∈ V . A quadratic O-lattice L ⊂ V is an O-submodulesuch that KL = V ; L is equipped with the restricted quadratic form Q, and so we often talk about aquadratic O-lattice as the pair (L,Q). A sublattice of the lattice (L,Q) is an O-submodule equippedwith the restriction of the quadratic form Q. Note that to a totally positive quadratic form Q over Owe can naturally associate the quadratic O-lattice (Or, Q), and so we will interchangeably talk of Qas a quadratic form and as an O-lattice.

We will use Siegel’s result on the non-universality of sums of squares in the paper, so let us state ithere formally as a theorem.

Theorem 2.2. [Si3, Theorem 1] Let K be a totally real number field different from Q and Q(√

5).Then some totally positive integer α ∈ O+K is not the sum of squares of elements of OK . In particular,the quadratic form x21 + · · ·+ x2r is not universal over K for any r ∈ Z+.

We say that a quadratic form Q with the Gram matrix MQ ∈ Zr×r is represented (over Z) bya quadratic form Q′ with the Gram matrix MQ′ ∈ Zs×s if there exists X ∈ Zs×r such that MQ =XTMQ′X (see [OM1, §41B]). Note that if Q is unary, that is r = 1 and Q(x) = αx2, say, then anyquadratic form Q′ that represents α represents Q. In particular, this definition generalizes the notionof a quadratic form representing an integer to a quadratic form representing another quadratic form.(One can analogously define representation of forms by forms over O.)

Two quadratic forms Q and Q′ of the same rank are equivalent if they represent each other, in otherwords, if MQ = X

TMQ′X for an invertible matrix X.

Let Q1(x1, . . . , xr), Q2(y1, . . . , ys) be two quadratic forms (over Z or O). Their orthogonal sum isdefined as the (r+ s)-ary form (Q1 ⊥ Q2)(x1, . . . , xr, y1, . . . , ys) := Q1(x1, . . . , xr) +Q2(y1, . . . , ys). Inthe language of lattices, if (L1, Q1) and (L2, Q2) are the corresponding quadratic lattices, then theirorthogonal sum is the lattice L1 ⊕ L2 equipped with the quadratic form Q1 ⊥ Q2. Similarly when Qis a quadratic form of rank r and m ∈ N, then Q⊥m := Q ⊥ Q ⊥ · · · ⊥ Q (m-times) is a quadraticform of rank rm.

Let Q be a Z-form. If it cannot decomposed as the orthogonal sum Q = Q1 ⊥ Q2 of Z-formsQ1, Q2, we say that Q is an indecomposable form, otherwise that it is decomposable. Each Z-form Qcan be uniquely decomposed as the orthogonal sum of indecomposable Z-forms, its indecomposableconstituents; we also have analogous notions for totally positive quadratic forms over O.

LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS 5

3. Pythagoras number

Let us start with a preliminary consideration of the rank of a Z-form that represents a given totallypositive element α of an order O ⊂ K.Proposition 3.1. If α ∈ O+ is represented by some Z-form over O, then there exists a Z-form Q ofrank at most d that represents α over O. Moreover, there exists a d-ary positive semidefinite quadraticform Q0 with Z-coefficients that represents α over O.Proof. Let ω1, . . . , ωd be an integral basis of the order O and let Q′ be a Z-form of rank r thatrepresents α, i.e., Q′(v) = α for some v ∈ Or. We can write v =

∑di=1 v

(i)ωi, where v(i) ∈ Zr. In

particular, we have that

Q′(v) = Q′

(d∑i=1

v(i)ωi

)=

d∑i=1

ω2iQ′(v(i)) + 2

∑i>j

ωiωjBQ′(v(i), v(j)).

Consider the quadratic form Q0 with the Gram matrix (BQ′(v(i), v(j))) ∈ Zd×d, i.e.,

Q0(x1, . . . , xd) =∑

1≤i,j≤dBQ′(v

(i), v(j))xixj = Q′

(d∑i=1

v(i)xi

).

This quadratic form is positive semidefinite, as Q′ is a positive definite quadratic form. The form Q0is d-ary and we see that Q0(ω1, . . . , ωd) = α.

Note that if MQ′ is the r× r Gram matrix of Q′ and V = (v(1), . . . , v(d)) ∈ Zr×d is the r× d matrixcontaining the vectors v(i) as columns, then for the d× d Gram matrix MQ0 of Q0 we have

(3.1) MQ0 = VTMQ′V,

i.e., the quadratic form Q′ represents the form Q0.To finish the proof by proving the existence of a Z-form Q that represents α, let us prove the

following lemma (that is essentially well-known, e.g., see the discussion at the beginning of [Mo2]).

Lemma. Let Q0(x) =∑d

i=1 aijxixj be a d-ary positive semidefinite quadratic form with Z-coefficients.Then there is a Z-form Q of rank ≤ d such that Q represents Q0.Proof. The radical Rad(L) := {v ∈ L | BQ0(v, w) = 0 for all w ∈ L} of the lattice (L = Zd, Q0) isclearly a sublattice of L. Further, it makes sense to define the quotient lattice (Zd/Rad(L), Q1) byQ1(v mod Rad(L)) := Q0(v). By the classification of finitely generated abelian groups, we have thatZd/Rad(L) is isomorphic to Zm for some m ≤ d.

The quadratic lattice (Zm, Q) ' (Zd/Rad(L), Q1) is then positive definite (by the Cauchy-Schwarzinequality, if Q(v) = 0, then B(v, w) = 0 for all w), i.e., Q is a Z-form of rank m ≤ d which representsQ0 by definition. �

As the form Q from the lemma represents Q0 (over Z), it clearly represents all the elements thatare represented by Q0, not only over Z, but also over O ⊃ Z. In particular, Q represents α, whichfinishes the proof of Proposition 3.1. �

This proposition in particular gives an algorithm for deciding whether a given element α ∈ O isrepresented by some Z-form. Namely, it allows us to restrict our attention only to Z-forms Q of rankat most d. Now we can use the following theorem of Conway-Sloane.

Theorem 3.2. [CS, Theorem 1] For each r ∈ Z+ there is k = k(r) ∈ Z+ with the following property:If Q is a classical Z-form of rank r, then the form kQ is the sum of squares of linear forms, in otherwords, it is represented by the quadratic form x21 + · · ·+ x2s for some s. For r ≤ 5 we have k(r) = 1.

Returning to the situation when a given element α is represented by a Z-form Q of rank at most d,we see that 2Q is classical and so 2kQ is represented by the sum of squares (where k = k(d)). Thus weneed only to check whether 2kα is the sum of squares in O (which we can do, e.g., using Lemma 2.1a))and then whether these squares are of the correct shape corresponding to the decomposition of 2kQ.

For a ring R, let Sq(R) denote the set of elements that are sums of squares in R and Sqm(R) denotethe set of elements that are sums of m squares in R. Then

P(R) := min{m : Sqm(R) = Sq(R)}

6 VÍTĚZSLAV KALA AND PAVLO YATSYNA

is the Pythagoras number of R (if no such m exists, then P(R) :=∞). For a totally real number fieldK, we have that P(K) ≤ 4 [Si2, Ho]. For orders O in K, P(O) is finite, but can grow arbitrary large[Sc2]. In this work, Scharlau also asked whether the Pythagoras number of orders is bounded in termsof the degree of the field K. Let us now show that this is indeed so.

Corollary 3.3. Let O be an order in a totally real number field of degree d. Then P(O) ≤ f(d), wheref is some function which depends only on d. If d = 2, 3, 4, or 5, then P(O) ≤ d+ 3.Proof. Assume that α ∈ Sq(O), i.e., α is represented by the Z-form x21 + · · ·+ x2r over O for some r.From Proposition 3.1 it follows that α is represented by some d-ary semidefinite form Q0. Moreover,from its construction (3.1) in the proof of Proposition 3.1, we see that this form Q0 is represented bythe original form x21 + · · ·+ x2r .

Icaza [Ic, Proposition 3] (cf. also [BI, Theorem 1.3]) showed that there exists a function f(d), whichdepends only on d, such that any quadratic form of rank d that is represented by the sum of squaresquadratic form is represented by the sum of f(d) squares. Therefore, Q0 is represented by f(d) squaresy21 + · · · + y2f(d). Since Q0 represents α over O, we see that α is the sum of f(d) squares of elementsof O. In other words, P(O) ≤ f(d).

Finally, when 2 ≤ d ≤ 5, then we are dealing with Q0 of rank d ≤ 5. For such forms, Ko andMordell [Ko, Mo1] proved that they are represented by at most d+ 3 squares. Therefore we can takef(d) = d+ 3 for 2 ≤ d ≤ 5. �

The bound for real quadratic number fields K = Q(√n) is sharp [Pe]. In fact, one can show that

P(OK) = 3 for n = 2, 3, 5 [Co, Sc1] and determine all n for which P(OK) = 4 (as in [CP]).Let us now consider Z-forms again. If we restrict to classical ones, we get the following result:

Corollary 3.4. Let K 6= Q(√

5) be a totally real number field of degree d > 1 and OK the ring ofintegers in K. Then there does not exist a classical Z-form of rank 3, 4, or 5 that is universal overOK .Proof. For contradiction assume that Q is a classical Z-form of rank strictly less than 6 that is universalover OK . By Theorem 3.2 of Conway-Sloane, Q is represented by the sum of s squares x21 + · · ·+ x2sfor some s. Since Q is universal over OK , it follows that the sum of squares x21 + · · ·+ x2s is universalas well. But this is impossible if K 6= Q(

√5) by Siegel’s Theorem 2.2. �

The previous corollary answers a (very) special case of Kitaoka’s conjecture [Km] that there existonly finitely many totally real number fields which admit a universal ternary quadratic form. Notethat the use of Theorem 2.2 was the only place in the proof of Corollary 3.4 where we used theassumption that OK is the maximal order. This probably can be avoided by generalizing Siegel’stheorem to general orders (most likely using essentially the same proof).

4. Forms of E-type

Given two (positive definite) Z-lattices (L1, Q1) and (L2, Q2), we define their tensor product overZ as (L1 ⊗ L2, Q1 ⊗Q2), so that

(Q1 ⊗Q2)(v ⊗ w) = Q1(v)Q2(w) and BQ1⊗Q2(v ⊗ v′, w ⊗ w′) = BQ1(v, w)BQ2(v′, w′)

for all v, v′ ∈ L1 and all w,w′ ∈ L2. Given bases {v(1), . . . , v(b)} of L1 and {w(1), . . . , w(c)} of L2,then {v(i) ⊗ w(j)} is the canonical basis of the tensor product L1 ⊗ L2. The Gram matrix MQ1⊗Q2of L1 ⊗ L2 with respect to this canonical basis equals MQ1 ⊗MQ2 , i.e., the Kronecker product of theGram matrices of Q1 and Q2 (see [Kt1, Chapter 7] for more details on tensor products of lattices).

In general (L1⊗L2, Q1⊗Q2) is not a Z-lattice, as the quadratic form Q1⊗Q2 need not be integervalued: for example, the tensor product of the lattice corresponding to the non-classical quadraticform x2 + xy + y2 with itself will have quadratic form

X2 + Y 2 + Z2 +W 2 +XY +XZ + Y Z + YW +1

2(XW + Y Z).

However, this happens if and only if we tensor two non-classical forms; as long as one of the forms isclassical, the tensor product will be a Z-lattice. This will always be the case in our paper.

For a Z-lattice (L,Q), let min(L) := min06=v∈LQ(v) be the minimum of L andM(L) := {v ∈ L : Q(v) = min(L)}

LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS 7

be the set of minimal vectors of L. For two Z-lattices (L1, Q1) and (L2, Q2) (one of which is classical)we clearly have

min(L1 ⊗ L2) ≤ min(L1) min(L2).There are examples of this inequality being strict (see [MH, page 47]), but there are important classesof lattices for which one has equality:

Definition 4.1. We say that a Z-lattice L is of E-type ifM(L⊗M) ⊂ {v⊗w : v ∈M(L), w ∈M(M)}for every classical Z-lattice M .

Note that although lattices of E-type are usually defined only for classical Z-lattices in the literature(e.g., [Kt1]), we are extending the definition also to non-classical lattices. Nevertheless, all the resultsconcerning lattices of E-type, such as Kitaoka’s Theorem 4.2 below, still hold since a non-classicallattice L is of E-type if and only if the classical lattice 2L is of E-type.

Given two Z-lattices L1 and L2, then of course not all elements of L1⊗L2 are split, i.e., of the formv1⊗ v2, vi ∈ Li. However, if either of Li is of E-type, then all the minimal vectors of L1⊗L2 are split[Kt1, Lemma 7.1.1].

Although certainly not every lattice is of E-type, this is true in several important cases, as thefollowing theorem of Kitaoka shows.

Theorem 4.2. [Kt1, Theorems 7.1.1, 7.1.2, 7.1.3] Let Q be a Z-form of rank r. Then Q is of E-typeif at least one of the following conditions holds:

• r ≤ 43,• min(Q) ≤ 3,• Q(x1, . . . , xr) = TrK/Q((

∑xiωi)

2), where K is an abelian number field of degree r with integralbasis ω1, . . . , ωr.

We will only use the first criterion in this paper; it is an open question what is the smallest rank ofa Z-form not of E-type.

From now on, we will work only with the maximal order OK in a totally real number field K ofdegree d, although as we discussed in the Introduction, probably many of our results generalize to thecase of general orders.

For δ ∈ O∨,+K , we can consider the “twisted trace form”, i.e., the quadratic form Tδ(x) = Tr(δx2)for x ∈ OK . Note that Tδ is not a quadratic form over OK (as Tδ(αx) 6= α2Tδ(x) for general α ∈ OK),but (OK , Tδ) is a quadratic Z-lattice of rank d. Fixing an integral basis ω1, . . . , ωd for OK , we identifyOK with Zd. Then we can denote tδ the Z-form of rank d such that

tδ(x1, . . . , xd) := Tr

(δ(∑

xiωi

)2),

i.e., (OK , Tδ) = (Zd, tδ) under the identification of OK with Zd.Since δ lies in the codifferent, the form Tδ is Z-valued. Because δ is totally positive, all the values

of Tδ are positive (except for Tδ(0) = 0), and so tδ is positive definite. Moreover, we see thatBTδ(β, β

′) = Tr(δββ′) for all β, β′ ∈ OK , and so the Gram matrix of the form tδ is (Tr(δωiωj))ij ,hence all its entries are (rational) integers. In other words, we have verified that tδ is a classicalZ-form, and so it makes sense to consider the tensor product tδ ⊗ Q with any Z-form Q. Finally,although tδ of course depends on the choice of the integral basis, we will not need to worry about this,as the basis will be considered fixed throughout the paper. We now have the following classical resulton tensor products.

Lemma 4.3. For a Z-form Q of rank r and δ ∈ O∨,+K , we have that

(OrK ,Tr(δQ)) = (OK ⊗ Zr, Tδ ⊗Q) = (Zd ⊗ Zr, tδ ⊗Q).

Hence we will freely switch between these three Z-lattices (and the corresponding quadratic forms)in the following. In particular, note that they have the same minimum.

Proof. Given that OrK , OK ⊗ Zr, and Zd ⊗ Zr are isomorphic as Z-modules and that Tδ and tδ areequivalent under this isomorphism by definition, it suffices to show that Tr(δQ) is equivalent to Tδ⊗Q.It suffices to show that the corresponding bilinear forms are equal on all split vectors.

8 VÍTĚZSLAV KALA AND PAVLO YATSYNA

Let β, β′ ∈ OK , w,w′ ∈ Zr, and BQ(x, y) =∑

i,j aijxiyj . Then

BTδ⊗Q(β ⊗ w, β′ ⊗ w′) = BTδ(β, β

′)BQ(w,w′) = Tr(δββ′)

∑i,j

aijwiw′j

= Tr

δ∑i,j

ββ′aijwiw′j

= Tr(δBQ(βw, β′w′))= BTr(δQ)(βw, β

′w′). �

The most important case for us will be when tδ is of E-type, which we will assume from now; letus summarize all our assumptions for the rest of the paper:

• K is a totally real number field of degree d over Q,• OK is the ring of integers in K,• the quadratic form tδ is of E-type for every δ ∈ O∨,+K ; this is true if d ≤ 43 by Theorem 4.2,• Q(x) is a Z-form of rank r, i.e., a positive definite quadratic form with Z-coefficients.

Let us now prove a series of auxiliary results that restrict possible number fields K over which theremay exist a universal Z-form.

Proposition 4.4. Assume that an indecomposable element α ∈ O+K is represented by Q over OK andsatisfies Tr(δα) = min(tδ ⊗Q) for some δ ∈ O∨,+K . Then α is a square in OK and min(Q) = 1.

Proof. From the assumption that Tr(δα) = min(tδ ⊗ Q), we conclude that the element v of OrKrepresenting α ∈ OK is a minimal vector of the quadratic form tδ ⊗Q (which we identify with Tr(δQ)by Lemma 4.3). Since tδ is of E-type, the minimal vector v ∈ OrK = OK⊗Zr is split, that is, v = β⊗w,where β ∈ OK and w ∈ Zr. We then have

α = Q(v) = β2Q(w).

Given that α is indecomposable and Q(w) ∈ Z+, we conclude that Q(w) = 1 (otherwise α = β2 +β2(Q(w)− 1)) and that α is a square. �

Corollary 4.5. If Q is universal over OK , then every totally positive unit is a square in OK . Hencethere is a unit of every signature in OK .

Proof. By Lemma 2.1b) we know that totally positive units are indecomposable. Hence in order tobe able to use Proposition 4.4, it suffices to show that for a given totally positive unit ε there existsδ ∈ O∨,+K such that Tr(δε) = min(tδ ⊗Q).

We are assuming that Q is universal, and so let w ∈ OrK be such that Q(w) = ε. Clearly ε−1 ∈ O∨,+K

and the minimum of tε−1 is greater or equal than the minimal value of Tr on totally positive integers.This minimal value is d = Tr(1), as by the inequality between arithmetic and geometric means we

have Tr(α) ≥ dN(α)1/d ≥ d.Our assumption that tε−1 is of E-type then implies

min(tε−1 ⊗Q) = min(tε−1) min(Q) ≥ dmin(Q) ≥ d.On the other hand,

Tr(ε−1Q(w)) = Tr(1) = d,

and so ε satisfies the assumptions of Proposition 4.4 for δ = ε−1, hence it is a square.Finally, recall the well-known fact [Nar, p. 111, Corollary 3] that every totally positive unit is a

square in OK if and only if there is a unit of every signature in OK . �

Lemma 4.6. Assume that Q is universal over OK and let α ∈ O+K . If there exists δ ∈ O∨,+K such that

Tr(δα) ≤ Tr(δβ) for all β ∈ O+K , then α is a unit in OK .

Of course, when Q is universal, the condition Tr(δα) ≤ Tr(δβ) (for all β ∈ O+K) in the lemma isequivalent to our earlier assumption from Proposition 4.4 that Tr(δα) = min(tδ ⊗Q).

Proof. Every element α that satisfies the assumption of the lemma is indecomposable (if α = α1 +α2with α1, α2 ∈ O+K , then Tr(δα) > Tr(δα1)), and Proposition 4.4 implies that α is a square, say α = γ2for some γ ∈ OK . Suppose that α satisfies the assumption, but is not a unit. We can also assumewithout loss of generality that α is such an element with the smallest possible norm.

LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS 9

Let ε ∈ O×K be such that εγ � 0; such a unit exists by Corollary 4.5. Then Q represents εγ, sothere is v ∈ OrK such that Q(v) = εγ. Denote δ′ := δγε−1 ∈ O

∨,+K . Then

min(tδ′ ⊗Q) ≥ min(tδ ⊗Q).The equality here is attained: Let a be the element of Zd ⊗ Zr that corresponds to v ∈ OrK by

Lemma 4.3. Then

(tδ′ ⊗Q)(a) = Tr(δ′Q(v)) = Tr(δ′εγ) = Tr(δα) = min(tδ ⊗Q).Therefore the element εγ ∈ O+K has the property that Tr(δ′εγ) ≤ Tr(δ′β) for all β ∈ O

+K , but has

norm N(εγ) = |N(γ)| =√

N(α) < N(α). This is a contradiction with our assumption that α is theelement of smallest possible norm. �

Clearly, if for α ∈ O+K there exists δ ∈ O∨,+K such that Tr(δα) = min(tδ), then α is indecomposable.

Unfortunately, the converse implication does not hold, one counterexample being the indecomposableζ27 + ζ

−27 − 2 in K = Q(ζ7 + ζ

−17 ). Nevertheless, it holds in real quadratic fields and there are similar

characterizations in higher degrees as well [KT].

Let us now record for further use the following well-known facts.

Lemma 4.7. Let (L,Q) be a classical Z-lattice of rank r such that min(Q) = 1. Then:a) The number #M(L) of minimal vectors v ∈ L is at most 2r.b) If Q is moreover indecomposable quadratic form, then r = 1 and Q is equivalent to the quadratic

form x2.

Proof. We shall prove the lemma as an easy application of the following classical Theorem (e.g., [Mar,Proposition 4.10.7]):

Let (L,Q) be a classical Z-lattice of rank r such that min(Q) = 1. Let L1 be the sublattice of Lgenerated by the minimal vectorsM(L) of L (i.e., by v ∈ L such that Q(v) = 1), and Q1 the restrictionof Q to L1. Then L1 is equivalent to Zs equipped with the usual inner product (for some s ≤ r), andL = L1 ⊥ L2 for some lattice L2.

a) By definition, each v ∈ L with Q(v) = 1 corresponds to precisely one v′ ∈ L1 with Q1(v′) = 1.The number of such vectors in L1 is the same as the number of these vectors in Zs, which is clearly2s ≤ 2r.

b) Since L = L1 ⊥ L2, the indecomposability assumption implies that L = L1, which is equivalentto Zs. But this is indecomposable only when s = r = 1. �

We can now turn our attention to the (non-)existence of universal Z-forms using the results es-tablished above. For specific fields, one can often use the following proposition to deal with classicalforms, although in general there of course need not exist any non-unit with norm smaller than 2d.

Proposition 4.8. If there exists α ∈ OK such that 1 < |N(α)| < 2d, then there does not exist aclassical Z-form that is universal over OK .

Proof. Let α ∈ OK be such that 1 < |N(α)| < 2d. If a Z-form Q is universal over OK , then byCorollary 4.5 there are units of all signatures in OK . Thus, after multiplying by a suitable unit, wecan assume that α � 0, and furthermore take α to be such element of the smallest possible norm. ByLemma 2.1b), α is indecomposable.

Let us consider the orthogonal decomposition of Q (over Z), say Q = Q1 ⊥ · · · ⊥ Qk. If v =(v(1), . . . , v(k)) is the corresponding decomposition of a vector v representing α, then α = Q(v) =

Q1(v(1)) + · · · + Qk(v(k)). We have Qi(v(i)) � 0 for all i, and so all but one of these values must be

zero, as α is indecomposable. Hence α is represented by one of the indecomposable constituents, whichwe will denote Q′. Note that Q′ is an indecomposable classical Z-form (because Q was classical).

Let m ∈ N be such that mTr(βα

)∈ Z for all β ∈ OK . We denote δ := mα ∈ O

∨,+K . Let us now

prove that α is a square by distinguishing two possible cases:1. If Tr(δα) = min(tδ ⊗Q′), then by Proposition 4.4 it follows that α is a square.2. Otherwise there exists β ∈ O+K such that Tr(δα) > Tr(δβ) and Q′(w) = β with w ∈M(tδ ⊗Q′).

Therefore

dm = Tr(m) = Tr(δα) > Tr(δβ) = mTr

(β

α

),

10 VÍTĚZSLAV KALA AND PAVLO YATSYNA

and so d > Tr(βα

).

The inequality between arithmetic and geometric means then gives

1 >1

dTr

(β

α

)≥ N

(β

α

)1/d,

and so 2d > N(α) > N(β). Thus β is indecomposable by Lemma 2.1b).Applying Proposition 4.4 for the element β we obtain that min(Q′) = 1, and given that Q′ is an

indecomposable classical Z-form, this implies that Q′ is just a form of one variable, Q′(x) = x2, byLemma 4.7b). This form represents α, which therefore is a square.

In both cases we proved that α is a square, say α = γ2, and therefore there exists a non-unit elementγ with a smaller norm than α, contradicting the assumption of minimality of norm of α. �

Theorem 1.1. There does not exist a Z-form that is universal over a real quadratic number field K,unless K = Q(

√5).

Proof. Let K = Q(√D), where D ≥ 2 is squarefree integer. Let OK = Z[ω] be the ring of integers in

K, where ω =√D or 1+

√D

2 , depending on whether D ≡ 2, 3, or 1 (mod 4). Let f(x) be the minimalpolynomial of ω. From Corollary 4.5 it follows that if there exists a universal Z-form over OK , thenOK has units of all signatures; fix a unit ε > 0 such that its conjugate ε < 0.

We know that O∨K =1

f ′(ω)OK . Thus we can let δ := ε ·1

f ′(ω) so that we have δ � 0 and O∨K = δOK .

The form tδ has rank 2, and so from Kitaoka’s Theorem 4.2 it follows that tδ is of E-type.The element α := ωε−1 ∈ O+K satisfies Tr(δα) = 1: Let us show the easy computation only in the

case ω =√D, when we have δ = εf ′(ω) =

ε2√D

, and so Tr(δα) = Tr( √

D2√D

)= 1. (In fact, all elements

with Tr(δα) = 1 form an arithmetic progression [Ya, Example 1].)By Lemma 4.6 we therefore have that α ∈ O+K is a unit, and so N(ω) = −N(α) = −1. If D ≡ 2, 3

(mod 4), then N(ω) = N(√D) = −D 6= −1. When D ≡ 1 (mod 4), we have −1 = N(ω) = 1−D4 , and

so the only possibility is D = 5. �

5. Dedekind zeta function

Let K be a totally real number field of degree d and let ∆K denote the discriminant of K. Resultsof the previous section suggest that elements of the codifferent which have small trace play a keyrole in the study of universal Z-forms; we shall use Siegel’s formula [Si1] to estimate the number ofthese elements in terms of Dedekind zeta function ζK(s). We start by reviewing its basic propertiesfollowing [Za, §1] as reference for all the facts that we mention.

Dedekind zeta function ζK(s) of K for s ∈ C is the meromorphic function that for Re(s) > 1 satisfies

ζK(s) =∞∑n=1

F (n)

ns,

where F (n) is the number of ideals in OK of norm n (and the norm of an ideal I is N(I) = #OK/I).

We will be interested in the values at the points s = 2 and s = −1; clearly ζK(2) >F (1)

12= 1.

From the functional equation we see that

(5.1) ζK(−1) = (−1)d|∆K |3/2(

1

2π2

)dζK(2).

Assume from now on that the degree d = 2, 3, 4, 5, 7, and let bd =1

240,−1504

,1

480,−1264

,−124, respec-

tively. Then we have

(5.2) ζK(−1) = 2dbd∑

α∈O∨,+KTr(α)=1

σ((α)(O∨K)−1),

whereσ(I) =

∑J |I

N(J)

and (α) denotes the fractional ideal αOK .

LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS 11

Putting together (5.1) and (5.2), we get

(5.3)∑

α∈O∨,+KTr(α)=1

σ((α)(O∨K)−1) =(−1)d

bd|∆K |3/2

(1

4π2

)dζK(2).

We can now use this formula to prove our main result that greatly restricts possible number fieldsof small degrees with a universal Z-form.

Theorem 5.1. There does not exist a totally real number field K of degree 2, 3, 4, 5, or 7, with aprincipal codifferent ideal and a universal Z-form defined over it, unless K = Q(

√5) or Q(ζ7 + ζ−17 ).

Proof. Since d ≤ 7 < 43, by Kitaoka’s Theorem 4.2 the form tδ is of E-type. Hence we can use theresults of Section 4. Let us assume that there exists a universal Z-form over OK . By Corollary 4.5there are units of all signatures in OK . By the assumption that O∨K is a principal ideal, there existssome δ ∈ K such that O∨K = (δ). Without loss of generality, let δ � 0.

By Lemma 4.6, if α = α′δ ∈ O∨,+K is such that Tr(α) = 1, then α′ ∈ O×,+K . As then (α) = (δ) = O∨K ,

we deduce that

σ((α)(O∨K)−1) = σ(O∨K(O∨K)−1) = σ(OK) = 1.Therefore, the left-hand side of (5.3) is equal to the number of α ∈ O∨,+K that have Tr(α) = 1. Theright-hand side of (5.3) is non-zero as ζK(2) 6= 0, which in particular implies that there is at least onesuch α ∈ O∨,+K and that min(tδ) = 1.

Let α = α′δ ∈ O∨,+K be such that Tr(α) = 1. We have already seen that by Lemma 4.6 we haveα′ ∈ O×,+K , and moreover by Corollary 4.5, α′ is a square, so let α′ = β2 for some β ∈ OK . Furtherlet v ∈ Zd be such that β =

∑viωi for an integral basis ωi of OK . We then have

tδ(±v) = Tr(δ(±∑

viωi

)2)= Tr(δβ2) = Tr(δα′) = 1.

Thus the two vectors ±v that were attached to α are minimal vectors of the Z-lattice (Zd, tδ), whoseminimum is 1. In the discussion just before Lemma 4.3 we saw that tδ(x) is classical, and so it has atmost 2d minimal vectors by Lemma 4.7a). Therefore,

2d ≥ 2 ·#{α ∈ O∨,+K |Tr(α) = 1} = 2∑

α∈O∨,+KTr(α)=1

σ((α)(O∨K)−1).

Using this inequality and the fact that ζK(2) > 1 in equation (5.3), we get

|∆K | <∣∣∣(4π2)d dbd∣∣∣2/3 .

In the table below we summarize the resulting bounds:

d |∆K | <2 5.6...3 51.2...4 742.8...5 14886.9...7 12386158.6...

From online database of number fields (described in [JR]; cf. also [Vo]) we find that there are only afew totally real number fields K = Q(ω) that satisfy the above bounds:

d Minimal polynomial of ω |∆K | Narrow class number2 x2 − x− 1 5 13 x3 − x2 − 2x+ 1 49 14 x4 − x3 − 3x2 + x+ 1 725 15 x5 − x4 − 4x3 + 3x2 + 3x− 1 14641 1

First, note that there does not exist a number field satisfying the above bound for degree 7. Quadratic,cubic, and quintic number fields correspond to the maximal real subfields of cyclotomic fields. In

12 VÍTĚZSLAV KALA AND PAVLO YATSYNA

particular, we have Q(ζ5 + ζ−15 ),Q(ζ7 + ζ−17 ), and Q(ζ11 + ζ

−111 ), respectively. We need to exclude the

cases in degrees d = 4 and 5.

d = 4. Let K = Q(ω), where ω is a root of f(x) = x4−x3−3x2 +x+1. Given that ∆f = ∆K (where∆f is the discriminant of the polynomial f), we have OK = Z[ω]. The totally positive integer ω + 2has norm 11 < 24, and so it is indecomposable by Lemma 2.1b), and from Proposition 4.8 it followsthat there is no universal classical Z-form. However, we also need to exclude non-classical forms.

Assume that there is a universal (non-classical) Z-form over OK . In particular it represents ω + 2,and so by Proposition 3.1, this element is represented by a Z-form Q′ of rank ≤ 4. But then 2(ω + 2)is represented by the classical Z-form 2Q′, which in turn is represented as sum of squares by Theorem3.2.

Hence it suffices to show that 2(ω + 2) cannot be represented as a sum of squares; assume that2(ω + 2) � α2 for some α 6= 0, without loss of generality α � 0 (as there are units of all signatures).2(ω + 2) is not a square, and so 2(ω + 2) = α2 + γ with γ � 0. Then by Lemma 2.1a) we have

N(α)1/2 + 1 ≤ N(α)1/2 + N(γ)1/4 ≤ N(2(ω + 2))14 = 2 · 111/4,

and so N(α) < 7. We can easily check in Magma that the only elements of norm less than 11 in OKare units, and so if 2(ω+ 2) is a sum of squares, then it is a sum of squares of units. From the boundof Lemma 2.1a) it further follows that there are at most three summands. The trace of 2(ω+ 2) is 18,thus by checking all the combinations of totally positive units of small trace, we confirm that 2(ω+ 2)cannot be represented as a sum of squares.

d = 5. Let ω = ζ11 + ζ−111 and K = Q(ω). There (again) exists a totally positive integer of norm 11

in OK , i.e., β = ω + 2. Given that N(β) = 11 < 25, Lemma 2.1b) implies that β is indecomposable.By the different theorem [Nar, Theorem 4.24], every ramified prime ideal of OK divides the different

(O∨K)−1. In our case, the norm of β is prime and β | 11 | ∆K , and so the principal ideal (β) is aramified prime ideal. It therefore divides the different, and so inversely, β−1 lies in the codifferent,i.e., β−1 ∈ O∨,+K .

If there exists a universal Z-form Q over K, thenmin(tβ−1) min(Q) = min(tβ−1 ⊗Q) ≤ Tr(β−1β) = Tr(1) = 5,

as Q represents β. Since Z[ω] = OK , we compute in Magma the Gram matrix of tβ−1 correspondingto the integral basis {1, ω, ω2, ω3, ω4}:

5 −5 11 −13 30−5 11 −13 30 −3511 −13 30 −35 86−13 30 −35 86 −9430 −35 86 −94 252

.Using this, we check (in Magma again) that min(tβ−1) = 5. Thus by Proposition 4.4 it follows thatβ is a square, which is impossible given that the norm of β is 11. Therefore, there does not existuniversal Z-form over OK . �

We note that the argument in the second part of the proof fails for Q(ζ7 + ζ−17 ). Even though wecan find an indecomposable element β of norm 7, we have that min(tβ−1) = 2, and so β does notcorrespond to a minimal vector.

6. Existence of universal forms

In the previous sections we have proved that there does not exist a universal Z-form in a numberof cases. Let us now turn our attention to the converse problem, namely, of proving the existence ofa universal Z-form over Q(ζ7 + ζ−17 ).

Lemma 6.1. Let K = Q(ζ7 + ζ−17 ). Then the binary quadratic form Q(x, y) = x2 +xy+y2 representsall indecomposable integers of K.

Proof. All totally positive units are squares in OK , and so Q represents all of them, as Q represents 1.Let us now find all the indecomposable elements in O+K . Brunotte [Br2, Korollar 2] proved that

the norm of every indecomposable integer in a general real cyclotomic field Q(ζn + ζ−1n ) is at most

LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS 13

cϕ(n)n , where ϕ(n) is Euler’s totient function and cn is certain constant defined using the unit group ofQ(ζn) (this is the constant denoted cQ(n) in [Br1] and c(En) in [Br2]). When n = 7, Brunotte [Br1,Table 2] also showed that c7 < 1.6, and so each indecomposable has norm at most 16. We then checkin Magma all these candidate elements to show that the only non-unit indecomposable integer of K(up to multiplication by totally positive units) is an element of norm 7, which can be written as

(ζ27 + ζ−27 + 2) + (ζ7 + ζ

−17 ) + 1.

This element is represented by Q, as we have that

(ζ27 + ζ−27 + 2) + (ζ7 + ζ

−17 ) + 1 = (ζ7 + ζ

−17 )

2 + (ζ7 + ζ−17 ) + 1 = x

2 + xy + y2,

where x = ζ7 + ζ−17 and y = 1. �

Strictly speaking, we do not need this lemma for the following theorem, but it gives us someimportant clues concerning the possible shape of a universal Z-form over Q(ζ7 + ζ−17 ). In particular,it may be advantageous for it to contain x2 + xy + y2 as a subform. This is indeed the case for theform x2 + y2 + z2 + w2 + xy + xz + xw, whose universality over Q(ζ7 + ζ−17 ) we will now show. Thisform is the norm form for a subring of the quaternions, which was used by Deutsch [De2] to prove itsuniversality over Q(

√5); this form is also universal over Z.

Theorem 6.2. Let K = Q(ζ7 + ζ−17 ). Then the quadratic form Q(x, y, z, w) = x2 + y2 + z2 + w2 +xy + xz + xw is universal over OK .

Proof. We use genus theory (and the mass formula) in the proof, following similar lines as the proofof Theorem 3.1 in [CKR]. Let us recall that two quadratic forms over a number field K are in thesame genus if they are equivalent at every completion of K. Therefore, if the genus of a quadraticform Q consist of only one class, then the local-global principle holds for Q. Specifically, in thiscase Q represents all the elements α ∈ OK that it represents locally everywhere, i.e., all α thatare represented over all the archimedean completions and over the rings of integers of all the non-archimedean completions.

All the one-class genera of quadratic forms (of rank ≥ 3) over all totally real fields K 6= Q weredetermined by Kirschmer [Kr]; there are 471 such genera. Our form Q = x2+y2+z2+w2+xy+xz+xwover K = Q(ζ7+ζ−17 ) is one of them, and so we know that the local-global principle holds for Q. Let usnote that the classification uses Siegel’s mass formula [Si4] and that one can directly check in Magmathat the class number of Q over K is 1 (the mass of Q is 11152 and the order of the automorphismgroup of Q is 1152).

Thanks to the local-global principle, it now suffices to show that Q is universal locally.There are three archimedean places, all of them real, and Q clearly represents all positive elements

over each of them, as Q is positive definite.Over all the non-dyadic places, 2 is a unit, and so Q is unimodular (i.e., the Gram matrix of Q is

invertible). A well-known theorem [OM1, 92:1b] says that if q is a unimodular quadratic form of rank≥ 3 over a non-dyadic local field F , then q represents all the elements of the ring of integers OF . Asthe rank of Q is 4, it follows that Q is universal there too.

Finally, 2 is inert in K. We directly check that Q represents all the square classes over the corre-sponding 2-adic completion of K, which is the degree 3 extension of Q2 (and so contains 32 squareclasses as we directly compute).

Hence Q is indeed universal over OK . �

From Hilbert reciprocity law it follows that there cannot exist a universal quadratic form overQ(ζ7 + ζ−17 ) of smaller rank than 4 [EK, Lemma 3].

Let us remark that another quaternary quadratic form, Q′ = x2 + xy + y2 + z2 + zw + w2, thatwas also considered by Deutsch [De1], appears to be universal over Q(ζ7 + ζ−17 ) as well. However, thisform has class number 2, and so we have not proved its universality. Deutsch proved the universalityof Q′ over Q(

√5) and also showed that it is not universal over several real quadratic fields of small

discriminant. By our Theorem 1.1 it now follows that it is not universal over any other real quadraticfield.

Finally, let us show a general proposition that provides a way of constructing a universal form froma quadratic form that represents all indecomposable integers.

14 VÍTĚZSLAV KALA AND PAVLO YATSYNA

Proposition 6.3. Let K be a totally real number field of degree d and Q a totally positive quadraticform over OK of rank r that represents all indecomposable integers. Then there is m ∈ N such thatthe orthogonal sum Q⊥m is universal over OK . In fact, m is the product of the Pythagoras numberP(OK) with the number of indecomposable integers in OK up to multiplication by squares of units.

In particular, if there is a Z-form that represents all indecomposables such as in Lemma 6.1, thenthere is a universal Z-form over OK . So if we were interested only in the existence of a universalZ-form, we could have used this proposition instead of the specific (and much stronger!) constructionof Theorem 6.2.

Proof. Let B be a set containing certain indecomposable integers so that every indecomposable is ofthe form ε2β for some β ∈ B and ε ∈ O×K . Brunotte [Br2] proved that every indecomposable has normless than a constant cOK (depending only on K). Thus the set B is finite.

Every element of α ∈ O+K can be expressed as the sum of indecomposables. By combining indecom-posables βε2i with the same representative β ∈ B, we see that α can be written as a sum

∑β∈B βσβ,

where each σβ is a sum of squares of units. The Pythagoras number of OK is finite (cf. Corollary3.3), and so there is a constant c = f(d) such that each σβ is a sum of c squares, i.e., it is representedby the form x21 + · · ·+ x2c .

Each β ∈ B is represented by the form Q, and so each of the elements βσβ = β(s21 + · · ·+ s2c) (forsome s1, . . . , sc ∈ OK) is represented by Q⊥c. The element α is the sum of #B such elements βσβ,and so it is represented by the form

(Q⊥c

)⊥(#B)= Q⊥(#B·c). �

Let us conclude the article by the example of the form Q = x2 + xy + y2 over K = Q(ζ7 + ζ−17 ).In the proof of Lemma 6.1, we have seen that, up to multiplication by units, there are exactly twoindecomposables, 1 and β = ζ27 + ζ

−27 + ζ7 + ζ

−17 + 3. As every totally positive unit in OK is a square,

we see that #B = 2. We are in a number field of degree d = 3, and so the Pythagoras number is ≤ 6by Corollary 3.3. Thus the Z-form Q⊥12 of rank 24 is universal over K.

Further, as pointed out by the anonymous referee, observing that 2β and 3β are sums of squares,this can be improved to show that the Z-form Q⊥6 + x21 + · · ·+ x26 of rank 18 is universal over K.

References

[BI] R. Baeza, M.I. Icaza, Decomposition of positive definite integral quadratic forms as sums of positive definitequadratic forms. In: K-theory and algebraic geometry: connections with quadratic forms and division algebras(Santa Barbara, CA, 1992), Proc. Sympos. Pure Math. 58, Part 2, Amer. Math. Soc. (1995), 63–72

[BC+] C. N. Beli, W. K. Chan, M. I. Icaza, J. Liu, On a Waring’s problem for integral quadratic and Hermitian forms,Trans. Amer. Math. Soc. 371 (2019), 5505–5527

[Bh] M. Bhargava, On the Conway-Schneeberger Fifteen Theorem, Contemp. Math. 272 (1999), 27–37[BH] M. Bhargava, J. Hanke, Universal quadratic forms and the 290-theorem, Invent. Math., to appear[BK1] V. Blomer, V. Kala, Number fields without universal n-ary quadratic forms, Math. Proc. Cambridge Philos. Soc.

159 (2015), 239–252[BK2] V. Blomer, V. Kala, On the rank of universal quadratic forms over real quadratic fields, Doc. Math. 23 (2018),

15–34[BCP] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24

(1997), 235–265[Br1] H. Brunotte, The Computation of a Certain Metric Invariant of an Algebraic Number Field, Math. Comput 38

(1982), 627–632[Br2] H. Brunotte, Zur Zerlegung totalpositiver Zahlen in Ordnungen totalreeller algebraischer Zahlkörper, Arch. Math.

(Basel) 41 (1983), 502–503[CL+] M. Čech, D. Lachman, J. Svoboda, M. Tinková, K. Zemková, Universal Quadratic Forms and Indecomposables

over Biquadratic Fields, Math. Nachr. 292 (2019), 540–555[CI] W. K. Chan, M. I. Icaza, Positive definite almost regular ternary quadratic forms over totally real number fields,

Bull. Lond. Math. Soc. 40 (2008), 1025–1037[CKR] W. K. Chan, M.-H. Kim, S. Raghavan, Ternary universal integral quadratic forms, Japan. J. Math. 22 (1996),

263–273[Co] H. Cohn, Decomposition into four integral squares in the fields of 21/2 and 31/2, Amer. J. Math. 82 (1960),

301–322[CP] H. Cohn, G. Pall, Sums of four squares in a quadratic ring, Trans. Amer. Math. Soc. 105 (1962), 536–556[CS] J. H. Conway, N. J. A. Sloane, Low-dimensional lattices. V. Integral coordinates for integral lattices, Proc. Roy.

Soc. London Ser. A 426 (1989), 211–232

[De1] J. I. Deutsch, Universality of a non-classical integral quadratic form over Q(√

5), Acta Arith. 136 (2009), 229–242

[De2] J. I. Deutsch, A non-classical quadratic form of Hessian discriminant 4 is universal over Q(√

5), Integers 16(2016), # A19

LIFTING PROBLEM FOR UNIVERSAL QUADRATIC FORMS 15

[DS] A. Dress, R. Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14(1982), 292–306

[Ea] A. G. Earnest, Universal and regular positive quadratic lattices over totally real number fields, Integral quadraticforms and lattices (Seoul, 1998), Contemp. Math. 249 (1999), Amer. Math. Soc., 17–27

[EK] A. G. Earnest, A. Khosravani, Universal positive quaternary quadratic lattices over totally real number fields,Mathematika 44 (1997), 342–347

[EH] D. R. Estes, J. S. Hsia, Sums of three integer squares in complex quadratic fields, Proc. Amer. Math. Soc. 89(1983), 211–214

[HLP] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge, at the University Press, 1934[Ho] D. Hoffmann, Pythagoras numbers of fields, J. Amer. Math. Soc. 12 (1999), 839–848[HKK] J. S. Hsia, Y. Kitaoka, M. Kneser, Representations of positive definite quadratic forms, J. Reine Angew. Math.

301 (1978), 132–141[Ic] M. I. Icaza, Sums of squares of integral linear forms, Acta Arith. 74 (1996), 231–240[JK] S. W. Jang, B. M. Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234–243[JKO] Y.-S. Ji, M.-H. Kim, B.-K. Oh, Binary quadratic forms represented by a sum of nonzero squares, J. Number

Theory 148 (2015), 257–271[JR] J. W. Jones, D. P. Roberts, A database of number fields, LMS J. Comput. Math. 17 (2014), 595–618[Ka1] V. Kala, Universal quadratic forms and elements of small norm in real quadratic fields, Bull. Aust. Math. Soc.

94 (2016), 7–14[Ka2] V. Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193–207[KS] V. Kala, J. Svoboda, Universal quadratic forms over multiquadratic fields, Ramanujan J. 48 (2019), 151–157[KT] V. Kala, M. Tinková, Universal quadratic forms over families of cubic fields, in preparation[Ki1] B. M. Kim, Finiteness of real quadratic fields which admit positive integral diagonal septenary universal forms,

Manuscr. Math. 99 (1999), 181–184[Ki2] B. M. Kim, Universal octonary diagonal forms over some real quadratic fields, Commentarii Math. Helv. 75

(2000), 410–414[Km] M.-H. Kim, Recent developments on universal forms. In Algebraic and arithmetic theory of quadratic forms, In

Algebraic and arithmetic theory of quadratic forms, 344 of Contemp. Math., pages 215228. Amer. Math. Soc.,Providence, RI, 2004

[KO1] M.-H. Kim, B.-K. Oh, Representations of positive definite senary integral quadratic forms by a sum of squares,J. Number Theory 63 (1997), 89–100

[KO2] M.-H. Kim, B.-K. Oh, Bounds for quadratic Waring’s problem, Acta Arith. 104 (2002), 155–164[KO3] M.-H. Kim, B.-K. Oh, Representations of integral quadratic forms by sums of squares, Math. Z. 250 (2005),

427–442[Kr] M. Kirschmer, One-class genera of maximal integral quadratic forms, J. Number Theory 136 (2014), 375–393[Kt1] Y. Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Mathematics 106, Cambridge University Press

1993[Kt2] Y. Kitaoka, Tensor products of positive definite quadratic forms, Proc. Japan Acad. 52 (1976), 498–500[KTZ] J. Krásenský, M. Tinková, K. Zemková, There are no universal ternary quadratic forms over biquadratic fields,

arXiv:1909.05422[Ko] C. Ko, On the representation of a quadratic form as a sum of squares of linear forms, Q. J. Math. 1 (1937),

81–98[Le] Y.M. Lee, Universal forms over Q(

√5), Ramanujan J. 16 (2008), 97–104

[Ma] H. Maaß, Über die Darstellung total positiver Zahlen des Körpers R(√

5) als Summe von drei Quadraten, Abh.Math. Sem. Univ. Hamburg 14 (1941), 185–191

[Mar] J. Martinet, Perfect lattices in Euclidean spaces, Springer-Verlag, Berlin, 2003[MH] J. Milnor, D. Husemoller, Symmetric Bilinear Forms, Springer-Verlag, New York, 1973[Mo1] L. J. Mordell, On the representation of a binary quadratic form as a sum of squares of linear forms, Math. Z. 35

(1932), 1–15[Mo2] L. J. Mordell, The Representation of a Definite Quadratic Form as a Sum of Two Others, Ann. Math. 38 (1937),

751–757[Nar] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, 3rd Edition, Springer-Verlag, Berlin, 2004[Oh] B.-K. Oh, Universal Z-lattices of minimal rank, Proc. Amer. Math. Soc. 128 (2000), 683–689[OM1] O. T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag, Berlin, 1973[OM2] O. T. O’Meara, On indecomposable quadratic forms, J. Reine Angew. Math. 317, (1980), 120–156[Pe] M. Peters, Quadratische Formen über Zahlringen, Acta Arith. 24 (1973), 157–165[Po] P. Pollack, A remark on the number field analogue of Waring’s constant g(k), Math. Nachr. 291 (2018), 1893–1898[Sa1] H. Sasaki, Sums of squares of integral linear forms, J. Austral. Math. Soc. Ser. A 69 (2000), 298–302.

[Sa2] H. Sasaki, Quaternary universal forms over Q[√

13], Ramanujan J. 18 (2009), 73–80[Sc1] R. Scharlau, Darstellbarkeit von ganzen Zahlen durch Quadratsummen in einigen totalreellen Zahlkörpern, Math.

Ann. 249 (1980), 49–54[Sc2] R. Scharlau, On the Pythagoras number of orders in totally real number fields, J. Reine Angew. Math. 316 (1980),

208–210[Si1] C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys.

Kl. II (1969), 87–102[Si2] C. L. Siegel, Darstellung total positiver Zahlen durch Quadrate, Math. Z. 11 (1921), 246–275

http://arxiv.org/abs/1909.05422

16 VÍTĚZSLAV KALA AND PAVLO YATSYNA

[Si3] C. L. Siegel, Sums of m-th powers of algebraic integers, Ann. of Math. 46 (1945), 313–339

[Si4] C. L. Siegel, Über die analytische Theorie der quadratischen Formen, Ann. of Math. 36 (1935), 527–606[Vo] J. Voight, Enumeration of totally real number fields of bounded root discriminant, in Algorithmic number theory

5011, Springer, Berlin 2008, 268–281[Ya] P. Yatsyna, A lower bound for the rank of a universal quadratic form with integer coefficients in a totally real

field, Comment. Math. Helvet. 94 2019, 221–239[Za] D. Zagier, On the values at negative integers of the zeta-function of a real quadratic field, Enseignement Math.

(2), 22 (1976), 55–95

Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83,18600 Praha 8, Czech Republic

E-mail address: vita.kala@gmail.com

Charles University, Faculty of Mathematics and Physics, Department of Algebra, Sokolovská 83,18600 Praha 8, Czech Republic

E-mail address: pvyatsyna@gmail.com

1. IntroductionAcknowledgments2. Preliminaries3. Pythagoras number4. Forms of E-type5. Dedekind zeta function6. Existence of universal formsReferences