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vv† between lines Cv, Cw, where Πv v,v is the rank-1 hermitian projector onto Cv. = ( ) It is easily shown [11] that Cd can contain at most d2 equiangular lines. Following established terminology in quantum information theory [25], we say that a complete set of d2 equiangular lines is Symmetric and Informationally Complete (SIC ), or a SIC. The 1 1 common angle of a SIC is necessarily cos− [11]. ‰ √d 1 Ž At the time of writing, explicit constructions+ of SICs have been published in dimen- sions d 1–21, 23, 24, 28, 30, 35, 39, 48, 124, 195, 323 while published numerical evidence = suggests that they exist for all d 151 as well as d 168, 172, 199, 228, 259, and 844. ≤ = SICs are conjectured to exist in every finite dimension, forming a crucial part of what is known as Zauner’s conjecture [33, 25]. With one exception [16], every known SIC is unitarily equivalent to an orbit of the group Z dZ 2 under the projective unitary j1 j2 ( ~ ) representation taking j to the image of Xd Zd in the projective unitary group PU d , ( ) where

0 0 0 . . . 0 1 10 0 0 . . . 0 ⎛ 1 0 0 . . . 0 0 ⎞ ⎛ 0 ζd 0 0 . . . 0 ⎞ ⎜ ⎟ ⎜ 0 0 ζ2 0 . . . 0 ⎟ ⎜ 0 1 0 . . . 0 0 ⎟ ⎜ d ⎟ (1) Xd ⎜ ⎟ ,Zd ⎜ 3 ⎟ = ⎜ 0 0 1 . . . 0 0 ⎟ = ⎜ 0 0 0 ζd . . . 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⋮⋮⋮ ⋱⋱⋮ ⎟ ⎜ d 1⎟ 0 0 0 . . . 1 0 ⎜ 00⋮⋮⋮ 0 0 ⋮⋱... ζ ⋮ − ⎟ ⎝ ⎠ ⎝ d ⎠ 2πi d and where ζd e ~ is a primitive dth root of unity. All SICs considered in this = paper will be orbits of Z dZ 2 under the action of this representation. Following the ( ~ ) terminology usual in the physics literature we say they are Weyl-Heisenberg covariant. If v Cd is such that Cv is contained in some SIC, we call v and its corresponding ∈ projector Πv fiducial. d Given a nonzero v C , let Q v Q vi vj vj 0 be the minimal field of definition ∈ ([ d])1= ( ~ ∶ ≠ ) of the corresponding point v P − in projective space. For a given SIC E, let Q E [ ] ∈ d 1 ( ) be the minimal field of definition of the corresponding subset of P − , i.e. the smallest subfield of C containing the fields Q v for every Cv E. We refer to Q E as the ([ ]) ∈ ( ) SIC-field. Then we have the following result and subsequent natural conjecture:

Proposition 1. Let d be any one of the following dimensions:

(2) 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 28, 30, 35, 39, 48.

Then Cd contains at least one SIC for which the corresponding SIC field is equal to R, the ray class field of K Q √D with conductor d′ 1 2. Moreover, for every other = ( ) ∞ ∞ known SIC in dimension d the SIC field is a finite Galois extension of Q containing R.

Proof. The proof involves lengthy computations using Magma [7]. The main steps are indicated in Section 7. 

Conjecture 2. The statement of Proposition 1 holds for every dimension d 4. ≥ GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 3

These statements are a substantial strengthening of results and conjectures in [4]. As a further strengthening, at the end of Section 2 we give a putatively complete classifi- cation of the SICs in the dimensions of Eq. (2) for which the SIC field equals R. The completeness of the classification hinges on the numerical work of Scott and Grassl [26], who have with high probability identified all the (Weyl-Heisenberg covariant) SICs in these dimensions. After introducing the necessary geometric and number-theoretical apparatus in the next two sections, we informally introduce ray class fields and establish some properties of the relevent fields in §4. In §5 and §6 we relate invariants of the geometric objects to arithmetic quantities in the ray class fields and to associated canonical units. Section 7 outlines the proofs of Proposition 1 above, and Propositions 3, 16, 21 below.

1.1. Subsequent developments and marginalia. Since the first version of this paper appeared, explicit solutions have been published [6, 15, 20] in the further dimensions 23, 124, 195 and 323. These new solutions were in whole or in part directly inspired by the first version of this paper. Our original analysis has not yet been extended to include these, hence the list of dimensions in Eq. (2) is not exhaustive with respect to the known solutions. To see the actual solutions themselves, the reader should first consult Scott and Grassl [26]. In addition to many solutions found by them, they give a comprehen- sive listing of solutions, both explicit and numerical, published by previous authors. For solutions discovered subsequently see Refs. [2, 3, 6, 13, 15, 20, 27]. Many solutions are available in conveniently downloadable form on the websites [12, 34]. Finally, it is worth elaborating our previous claim that the only known SICs (with the exception of the Hoggar lines [16]) are Weyl-Heisenberg covariant SICs. Histori- cally, at first it appeared that there exist other examples of non-Weyl-Heisenberg co- variant SICs [14, 25]. However, it was then discovered [36] that these other ostensible exceptions are covariant with respect to more than one group, one of them being the Weyl-Heisenberg group. It is, in general, an open question whether there exist exam- ples of non-Weyl-Heisenberg covariant SICs yet to be discovered. However, it can be proved [35, 36] that if d is prime, and if a SIC has a group covariance property at all, then it must be Weyl-Heisenberg covariant.

2. SIC orbits under the extended Weil representation

A subgroup H U d sitting in a central extension ⊂ ( ) (3) 1 → U 1 → H → Z dZ 2 → 0 ( ) ( ~ ) canonically determines a projective unitary representation Z dZ 2 → PU d via j ↦ ( ~ ) ( ) ∆j , where ∆j ↤ j is any choice of section, and where U PU d denotes the image [ ] [ ] ∈ ( ) of U U d in the projective unitary group. The ∆j determine a 2-cocycle j, k ↦ cj,k ∈ ( ) ( ) ∈ U 1 via ∆j∆k cj,k∆j k, by definition satisfying ck,ℓcj,k ℓ cj k,ℓcj,k. In turn, this ( ) = + + = + GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 4 cocycle determines a skew-multiplicative pairing j, k ↦ ej,k cj,k ck,j, which satisfies 1 1 ( ) = ~ ∆j∆k∆j− ∆k− ej,kI, where ej,k ζd for every j, k. It is independent of the choice of = ∈ ⟨ ⟩ section, hence an invariant of the extension. The extension (3) is a Heisenberg group if e is nondegenerate (i.e. induces an isomorphism between Z dZ 2 and its dual, whence 2 2 ( ~ ) e Z dZ , Z dZ ζd . (( ~ ) ( ~ ) ) = ⟨ ⟩ j1 j2 2 From now on let H φXd Zd φ U 1 , j Z dZ . Then H U d is a = { ∶ ∈ ( ) ∈ ( ~ ) } ⊂ ( ) Heisenberg group and the Stone-von Neumann theorem [28, 30] asserts that every unitary 1 representation ρ H → U d acting as the identity on scalars has the form ρ h UρhUρ− , ∶ ( ) ( ) = for some Uρ U d that is uniquely determined up to scalars. Thus follows the existence ∈ ( ) of the Weil representation [31], a canonical projective unitary representation of the group 0 0 Aut H of automorphisms of H fixing its center, taking t Aut H to Ut for any Ut ( ) † ∈ ( ) 0 [ ] satisfying t h UthUt for every h H. In quantum computing, Aut H is known as ( ) = ∈ ( ) the projective generalized Clifford group and the normalizer of H in U d is known as ( ) the generalized Clifford group. This group sits in an extension [24]

2 0 (4) 0 → Z dZ → Aut H → Sp2 Z dZ → 1 ( ~ ) ( ) ( ~ ) of the subgroup Sp2 Z dZ SL2 Z dZ of GL2 Z dZ fixing e. The action of t 0 ( ~ ) = ( ~ ) ( ~ ) ∈ Aut H on a section ∆j ↤ j is given by [24] ( )

(5) t ∆j ϕt j ∆Ftj, ( ) = ( ) 2 where ϕt Z dZ → U 1 is a function and t ↦ Ft is a surjective homomorphism. The ∶ ( ~ ) ( ) 2 injection in (4) is defined by j ↦ h ↦ ∆jh∆ j and gives an action of Z dZ that ( − ) ( ~ ) depends only on the pairing:

1 j ∆k ∆j∆k∆j− ej,k∆k. ( ) = = ↦ j1 j2 For the section j ∆j φjXd Zd H determined by phases φj U 1 , the cocycle 1= j2k1 ∈ ∈ ( ) is given by cj,k φjφkφj− kζd , while the pairing is independent of the phases: ej,k + k1j2 k=2j1 j1j2 = cj,k ck,j ζd − . The specific choice φj ζ2d of phases gives the section ~ = = (− ) j1j2 j1 j2 (6) ∆j ζ2d Xd Zd = (− ) 1 j2k1 j1k2 ′ used in [4, 26] and in §5 and §6, which satisfies ∆j− ∆ j and cj,k ζ2d − ζd . − 2−1 mod d = = (− ) ∈ ⟨ ⟩ When d is odd, then ζ2d ζd , in which case the following hold [24]: e and c have − = 2 the same isometry group, the phase in (5) is a homomorphism ϕt Z dZ → ζd and 0 2∶ ( ~ ) ⟨ ⟩ the extension (4) splits as a semidirect product Aut H Z dZ Sp2 Z dZ via the ( ) ≃ ( ~ ) ⋊ ( ~ ) injective homomorphism taking F Sp2 Z dZ to the automorphism of H defined by ∈ ( ~ ) 0 F ∆j ∆Fj. This splitting identifies each automorphism t Aut H with a pair t ( ) = 2 ∈ ( ) = jt, Ft . We henceforth identify Z dZ and Sp2 Z dZ with their images j ↦ j, 1 and ( ) ( ~ ) ( ~ ) ( ) 0, F ↤ F in Aut0 H under this splitting. The extension (4) does not split when 4 d, ( ) ( ) 2 0 S but for general d, there is a surjective homomorphism Z dZ Sp2 Z d′Z → Aut H ( ~ ) ⋊ 3 ( ~ ) ( ) that is an isomorphism when d is odd, but has kernel Z 2Z ker SL2 Z d′Z → ≃ ( ~ ) ≃ ( ( ~ ) SL2 Z dZ when d is even [1]. ( ~ )) GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 5

Let EAut0 H be the group of automorphisms of H acting either as the identity or ( ) by complex conjugation on the center. It may be viewed as the projectivization of the extended Clifford group, defined in [1] as the normalizer of H in the group EU d of ( ) all unitary and anti-unitary operators in dimension d [32, Chap. 26]. It is generated by Aut0 H together with complex conjugation in the standard basis. It lies in an extension ( ) 2 0 0 → Z dZ → EAut H → ESp2 Z dZ → 1 ( ~ ) ( ) ( ~ ) of the determinant 1 subgroup ESp2 Z dZ ⊂ GL2 Z dZ , with complex conjuga- 1± 0 ( ~ ) ( ~ ) 2 tion mapping to 0 1 . There is similarly a surjective homomorphism Z dZ ‰ −0 Ž ( ~ ) 2⋊ ESp2 Z d′Z → EAut H . The SICs we consider are orbits under the action of Z dZ . ( ~ ) ( ) ( ~ ) It follows that in a given dimension they decompose into a set of orbits of ESp2 Z d′Z . ( ~ ) Aside2 from dimension 3, there are only finitely many such orbits in every known case.

Proposition 3. : The SIC field is constant on each orbit of EAut0 H . In particular, ( ) if Proposition 1 holds for one fiducial on a given orbit of EAut0 H then it holds for all. ( ) Proof. See Section 7. 

2 We define the stabilizer group of a fiducial as the subgroup of Z dZ ESp2 Z d′Z ( ~ ) ⋊ ( ~ ) leaving it invariant. Every known orbit contains a fiducial such that S The stabilizer group 0 is contained in ESp2 Z d′Z . ● S ( ~ ) 0 contains an order-3 element of Sp2 Z d′Z with trace ≡ 1 mod d. ● ( ~ ) − We refer to such a fiducial as centred (in [4] such fiducials were called “simple”). As observed in [26], the order-3 element can always be chosen to be conjugate to either Fz 0 d 1 1 d 3 = d 1 d−1 or Fa 4d 3 1 d+2 , the latter possibility only occurring when d 3 mod 9. ‰ + − Ž = ‰ ~ − − Ž ≡ We note that S0 is in fact always cyclic in the known instances for d 4. ≥ In the enumeration scheme of [26] the orbits for d 3 are specified by the dimension ≠ followed by a letter, and for future reference we also list here the type of stabilizer (Fz or Fa). For dimensions in the list (2), the orbits that are known to generate R over the rationals are as follows:

(7) Fz-type 4a, 5a, 6a, 7b, 8b, 9ab, 10a, 11c, 13ab, 14ab, 15d, 16ab, 17c, 18ab, 19e, [ ] 20ab, 24c, 28c, 35j,

(8) Fa-type 12b, 21e, 30d, 39ij, 48g. [ ] Here, two or more letters indicates several orbits—for instance 13ab is shorthand for 13a, 13b and 39a-f means 39a, . . . , 39f. Scott and Grassl [26] have, with high probability, identified every orbit for d 48 (although in many cases the solutions are only known ≤ numerically). For the dimensions in list (2) the orbits they identify additional to those

2In d 3 there is an uncountably infinite pencil of inequivalent SICs with (necessarily) transcendental = fields of definition apart from special points [4, §10],[35]. For an in-depth look at the case d 3, see = also [17]. GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 6 in lists (7) and (8) are

(9) Fz-type 7a, 8a, 11ab, 12a, 15abc, 17ab, 19abcd, 21abcd, 24ab, 28ab, 30abc, [ ] 35a-i, 39a-f, 48abcdf,

(10) Fa-type 39gh, 48e. [ ] Explicit solutions have been calculated for all of these orbits. In every case it is found that the field generated contains, and is strictly larger than R. For a detailed analysis of these fields, and those generated by the SICs in lists (7) and (8), see refs. [4, 5] . Finally let us note that for lists (7) and (8) the number of orbits in each dimension is the same as the class number of K for that dimension. This is likely not an accident, as we discuss in §5.

3. Pell’s equation and towers of ray class fields over K

We now prove the assertions made in the introduction about infinite sequences of dimensions d1, d2,... all of which give rise to the same value of D. As we shall now show, the dimensions dk are indexed precisely by the solutions of the generalised Pell’s equation X2 DY 2 4, − = which form a rank-1 Z-module by the theory of binary quadratic forms [10]. Conjec- turally these give us ‘towers’ of ray class fields lying above each quadratic field K.

Fix D, K Q √D as above and write 1 for the place associated to the embedding = ( ) ∞ of K into R under which √D is sent to the positive square root of D. Let uf ZK× be the ∈ 2 fundamental unit of K which is 1 under 1. If the norm of uf is 1 then set uD uf ; > ∞ − = otherwise set uD uf . In other words, uD is the first power of uf of norm 1. = Lemma 4. The following are equivalent. (i) d is a positive integer such that the square-free part of d 1 2 4 is equal to D d 1 r ( − ) − (ii) −2 is the rational part of uD for some r N. ∈ d 1 2 y 2 Proof. Statement (i) is equivalent to saying there exists y Z such that −2 D 2 d 1 y ∈ ‰ Ž − ( ) = 1, which in turn is equivalent to −2 2 √D being an element of K of norm 1. But it + is also a root of the monic integral polynomial X2 d 1 X 1, hence a unit. There- − ( − ) + fore [10, §11B] it is a power of uD, proving (ii). The converse is just a restatement of the definitions. 

Corollary 5. For every square-free value of D 2 there exist infinitely many values of d ≥ such that the square-free part of d 1 2 4 d 1 d 3 is equal to D. ( − ) − = ( + )( − ) Definition 6. All d yielding the same value of D are indexed by the appropriate power of uD:

r r (11) dr 1 uD uD− . = + + GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 7

In particular, d1 is the smallest dimension corresponding to D. We shall generally omit reference to the underlying D but the notation dr always applies with reference ur u−r r D+ D to a specific fixed value of D. Note that the rational part of uD is given by 2 = cosh r log uD . The Chebyshev polynomials of the first kind Tn tell us how to go from ( ) r rs the rational part of uD to that of uD :

dr 1 drs 1 ds 1 (12) Ts 2− Ts cosh r log uD cosh sr log uD 2− Tr 2− . ‰ Ž = ( ( )) = ( ) = = ‰ Ž Let us define a shifted version of the functions Tn by

x 1 Tn∗ x 1 2Tn −2 , ( ) = + ‰ Ž and for convenience extend it to negative n by defining T ∗n Tn∗. Equation (12) may be − = rephrased in terms of the new functions Tn∗ and rearranged to read:

(13) Tr∗ Ts∗ d1 Tr∗ ds drs dsr Ts∗ dr Ts∗ Tr∗ d1 . ( ( )) = ( ) = = = ( ) = ( ( )) It is also easy to show that d1, d2, d3,... is a strictly increasing sequence of positive integers.

Proposition 7. Within each sequence d1, d2,... of dimensions corresponding to a fixed D, there are infinitely many distinct infinite subsequences dk1 , dk2 ,... with the property that dk1 dk2 ⋯. S S Proof. Fix D. The defining recursion

T0 x 1, T1 x x, Tn x 2xTn 1 x − Tn 2 x ( ) = ( ) = ( ) = − ( ) − ( ) for the Chebyshev polynomials yields the following recursion for the shifted version:

T0∗ x 3, T ∗1 x x, Tn∗ x xTn∗ 1 x − xTn∗ 2 x + Tn∗ 3 x . ( ) = ± ( ) = ( ) = − ( ) − ( ) − ( ) If x Z then all terms are in Z, so for the particular case of a positive integer d, ∈ (14) Tn∗ d Tn∗ 3 d mod d, ( ) ≡ − ( ) and in fact for any positive integer r we have dnr d n 3 r mod dr. We now have three ≡ ( − ) cases according to the congruence class of n modulo 3:

● C0: n 0 mod 3: T0∗ dr 3, implying dnr − 3 is a multiple of dr. ≡ ( ) = ● C1: n 1 mod 3: T1∗ dr dr, implying dnr is a multiple of dr. ≡ ( ) = ● C2: n 2 mod 3: T2∗ dr dr dr − 2 , implying dnr is a multiple of dr. ≡ ( ) = ( ) So if ij j 1 is any increasing sequence of integers coprime to 3 such that i1 i2 i3 ⋯, it ( ) ≥ S S S follows that di1 di2 di3 ⋯. There are infinitely many such dimension towers because S S S ∀ Ô⇒ m n  ds dr 3 s r, so by C0, n m d3 ik d3 ij for any j, k. > > > > ∤ Corollary 8. If Conjecture 2 is true, then for every square-free D 2, there exist infin- ≥ itely many infinite ray class field towers above each Q √D whose successive generators d ( ) may be found by constructing SICs in the C ki . GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 8

4. Properties of the ray class fields

4.1. Definition of ray class fields. By the Kronecker-Weber theorem [9], any abelian extension of Q is a subfield of some cyclotomic extension Q ζm , where ζm is a primi- ( ) tive m-th root of unity for some minimal positive integer m. By the fundamental theorem of Galois theory, therefore, its Galois group is isomorphic under the Artin map to a quo- tient of Z mZ ×, the multiplicative group of the quotient ring Z mZ. The ideal mZ is ( ~ ) ~ called the conductor of the extension: and indeed Q ζm is the narrow ray class field ( ) of Q modulo mZ. The unique infinite prime of Q is allowed to ramify here and we obtain a totally complex field. If on the other hand we were to restrict ramification just to the finite places dividing m then the ray class field modulo the same conductor mZ would 1 be the maximal totally real subfield Q ζm + ζm− of Q ζm . ( ) ( ) The classical ideal-theoretic version of Artin’s global reciprocity law for finite abelian extensions of any finite extension K of Q then leads naturally to the more general notion of the ray class fields of K modulo any conductor. Indeed, let F be any finite abelian extension of K. Define S as the set of finite and infinite primes of K which ramify in F ; let IS be the free abelian group on the finite primes of K which do not lie in S.

Then the classical Artin map is the homomorphism φF K ∶ IS → Gal F K which sends ~ ( ~ ) an unramified finite prime v of K to the Frobenius automorphism of v in Gal F K . ( ~ ) Artin’s global reciprocity law asserts that φF K is surjective; and moreover that there ~ exists an integral ideal f of K divisible precisely by the finite primes in S, such that the kernel of φF K always contains the group Pf of principal ideals generated by elements α ~ of K satisfying:

● ordv α − 1 ordv f for all v dividing f; and ( ) ≥ ( ) ● α is totally positive at the real primes of K which ramify in F .

The smallest such f is called the conductor of F K. The quotient group IS Pf is called ~ ~ the ray class group of K modulo f, and thus Artin’s law asserts that the ray class group modulo f is mapped onto Gal F K by the Artin map. Moreover, for every such ideal f, ( ~ ) the existence theorem of global class field theory shows that there exists a finite abelian extension Ff K, unramified outside S, such that the kernel of the Artin map for this ~ extension is precisely Pf. This field, unique up to isomorphism, is called the ray class field of K modulo f.

4.2. Properties of ray class fields in the context of SICs. Fix a square-free in- teger D 2 and an associated dimension d 4 as explained in the previous section, ≥ ≥ defining as always our base field to be K Q √D . Recall that d′ d if d is odd or 2d = ( ) = if d is even, and that ζd′ is an arbitrary primitive d′-th root of unity. Write d′ for the modulus d′∞1∞2 and let mod× denote multiplicative congruence. Let σD be the unique non-trivial element of Gal K Q . Write R for the ray class field of K modulo d′ with ( ~ ) ramification allowed at both infinite primes; similarly we write R0, R1 and R2 when ramification is allowed respectively at no infinite primes, just at ∞1, and just at ∞2. We GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 9 have illustrated this notation and many of the conclusions of the next proposition in the field tower diagram in Figure 4.1.

1 ′ ′ − Proposition 9. (i) R contains the d′-th roots of unity ζd , and ζd ζd′ R0. ⟨ ⟩ + ∈ (ii) R0 is a non-trivial extension of K. Hence so are R1, R2, R.

(iii) R0 and R are Galois over Q.

(iv) R1 and R2 have isomorphic Galois groups over K.

(v) R is the compositum R1R2, and R0 R1 R2. = ⋂ Proof. R is the fixed field of the Artin symbols α , R K attached to the integral (( ) ~ ) principal ideals α where α 1mod×d′. The extension K ζd′ K is unramified outside ( ) ≡ ( )~ primes dividing d′, so to show that ζd′ R it suffices to show that these Artin symbols ∈ fix it. This action translates [22, X §1] via the restriction map from K ζd′ K down ( )~ to Q ζd′ Q, to raising ζd′ to the power NK~Q α . But d′ is fixed by σD and so (i) follows ( )~ ( ) from:

σD σD α 1 mod×d′ Ô⇒ α 1 mod×d′ Ô⇒ αα 1 mod×d′ Ô⇒ NK Q α 1 d′ . ≡ ≡ ≡ ~ ( ) ≡ ( ) 1 ′ − As a direct consequence, ζd ζd′ R0. It follows that to prove (ii) we need only show 1 + ∈ ′ − that ζd ζd′ Q √D , which follows from the definitions of D, d′ and basic cyclotomic + ~∈ ( ) theory.

Now lift σD to some σˆD Gal Q Q . In terms of moduli,σ ˆD fixes d′ and d′ and ∈ ( ~ ) interchanges d′∞1 and d′∞2. The field σˆD R sits inside a normal closure of R Q: it ( ) ~ −1 is an abelian extension of K with conductor d′ and with Galois group σˆDGal R K σˆD ( ~ ) of order R K . So σˆD R R and R Q is Galois. Similarly for R0. Finally (iv), (v) [ ∶ ] ( ) = ~ follow from a consideration of the moduli under the mapσ ˆD. 

Interestingly, R0 is abelian over Q in dimensions d 4, 5, 7, 8. However this is never = true of R. Moreover, we always have Gal R R0 Z 2Z × Z 2Z, as we now show. ( ~ )≃ ~ ~

Proposition 10. The groups Gal R R1 , Gal R1 R0 , Gal R R2 , Gal R2 R0 all ( ~ ) ( ~ ) ( ~ ) ( ~ ) have order 2.

We shall prove this after pointing out some consequences for the structure of the SIC

fields. Write γj for the unique non-trivial element of Gal Rj R0 for j 1, 2: that ( ~ ) = is, γj Gal Rj K is complex conjugation under every complex embedding of Rj . Let Γ12 ∈ ( ~ ) be the subgroup of Gal R K of order 2 generated by the product γ1γ2, and recall ( ~ ) the notation M G for the elements of a G-module M fixed by the action of G. By

Proposition 9, any liftingσ ˆD of σD swaps γ1 and γ2.

Corollary 11. (i) R is non-abelian over Q. 1 (ii) R1, R2 are non-Galois over Q, having R1 K real places and 2 R1 K pairs of [ ∶ ] [ ∶ ] complex places.

(iii) R R1 ζk R2 ζk for any k 2 which is a divisor of d′. = ( ) = ( ) > GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 10

R R1R2 2 = 1 ∞ ∞

R1 R0 ζd′ R2 ( ) 1 2 ∞ ∞ R0

d′

HK

1

K Q √D = ( )

Q

Figure 1. Tower of ray class field extensions above K, for a fixed finite mod- ulus d′ and the four possible combinations of infinite places. The moduli are ′ ‘cumulative’ going up the tower from K, so for example R1 has modulus d 1. The Hilbert class field HK of K is always the ray class field of modulus 1. The∞ CM field R0 ζd′ is not strictly a ray class field itself, in general. ( )

Γ12 (iv) With k as in (iii), R0 ζk R is a CM-field. R1, R2 and R0 ζk are the three ( ) = ( ) quadratic fields between R0 and R.

Proof. (of Corollary). If R Q were abelian then the inner automorphismσ ˆD would be ~ σˆD trivial. But R is the compositum of R1 and R2 R1 , so R R1 R2, contradicting = = = Proposition 10. This proves (i). For (ii): R1 is Galois over K and R1 R0 is non-trivial; 1 ~ so it has 2 R1 K pairs of complex places over ∞1. The corresponding R1 K places [ ∶ ] [ ∶ ] over ∞2 are real. This argument is symmetric in R1, R2 by Proposition 9. For (iii) just combine the facts that R1 is not totally complex, R R1 has degree 2, and K ζd′ R. 1 ~ ( ) ⊆ ′ −  Since ζd ζd′ R0 this also implies (iv) . + ∈

Proof. (of Proposition 10). Let hK denote the class number of ZK. If m is any modulus 1 of K we denote by Um the subgroup of UK ZK× consisting of units 1mod×m. This has 1 = ≡ finite index UK ∶ Um in UK. Express m m0m as a product of its finite and infinite [ ] = ∞ parts respectively. We denote by Φ the generalized Euler totient function, so Φ m0 is ( ) the order of the multiplicative group of the ring ZK m0. By analogy we write Φ m ∞ rm ~ ( ) for the size of the signature group ±1 where rm is the number of real places included { } in m . The formula [22, VI §1] for the order hm of the ray class group of K of modulus m ∞ is:

hK Φ m0 Φ m ∞ (15) hm ( ) (1 ) . = UK ∶ Um [ ] GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 11

Let uf and uD be defined as in section 3. Since K is a real quadratic field, 1 1 1 Z ′ ′ ′ × ± Ud Ud 1 Ud ZK× uf 1 ⊆ ∞ ⊆ ⊆ = { } 1 1 ′ ′ with each inclusion being of finite index; and similarly with Ud 2 replacing Ud 1 . So 1 1 1 1∞ ∞ ′ ′ ′ ′ for our purposes it suffices to show that Ud Ud 1 Ud 2 Ud , or equivalently: = ∞ = ∞ = 1 1 (16) Ud′ Ud′ , = for then the signature factor Φ m in (15) will tell the whole story of the growth in ( ∞) the size of the ray class groups as we successively add real places into the modulus. 1 1 Now (16) says that every unit in Ud′ is totally positive. Since d′ Z, Ud′ is closed 1 1 − 1 ∈ 1 under the action of σD. Hence NK Q Ud′ Ud′ . But 1 Ud′ because d′ 2. So Ud′ ~ ‰ Ž ⊆ ∉ > is a rank 1 torsion-free abelian group, all of whose elements have absolute norm +1. In particular their signature must either be +, + or −, − , so we are reduced to showing 1 ( ) ( ) that no unit in Ud′ can be totally negative. By construction uD is the first totally positive power of uf , so any totally negative 1 − r unit in Ud′ must be of the form uD for some r Z. Suppose that such a unit exists and ∈ that r 1 is minimal. Let j be the positive integer such that d dj, in the sense of (11). ≥ r r − = Since both uD and uD− 1 mod dj′ it follows that ≡ r r (17) dr uD + uD− + 1 −1 mod dj′ . = ≡ There are two cases. If 3 j, then the congruences C1, C2 imply d1 dj′ and conse- r − ∤ − S quently uD 1 mod d1. Moreover (17) shows dr 1 mod d1, so a second application ≡ ≡ 2 3 √5 of C0, C1, C2 forces d1 4 and 3 r. We therefore have D 5 and uD uf +2 , = S r = = = which has order 3 inside ZK 4 . Since 3 r this means uD 1 mod d1, a contradiction. ~( ) S ≡ On the other hand, if 3 j then write j 3tq for t,q 1 and 3 q. It follows from C1, S r − = ≥ ∤ C2 that d3t dj′ , implying uD 1 mod d3t . So if such a totally negative unit exists S ≡ for j 3tq then it must also exist for j 3t. Redefine r to be minimal for this relation: = = 2 then the order of uD modulo d3t is 2r, which is even. But d3c T3∗ dc dc dc − 3 + 3 = ( ) = ( ) for any integer c and so d3t is odd, which by Lemma 12 below implies that the order of uD modulo d3t is also odd, a contradiction. 

Finally we prove the technical lemma used in the last proof.

′ Lemma 12. The order of u modulo d is 3r dr (i.e. it is 3r if d is odd and 6r if d is D r′ dr r r even).

r Proof. Consider the absolute minimal polynomial for uD: 2 r r (18) X − dr − 1 X + 1 X − uD X − uD− . ( ) = ( )( ) − 3r Multiplying by X 1 shows that uD 1 mod dr for all r. When dr is even, we know ( ) ≡ from C2 that d2r is divisible by dr′ 2dr and thus: = 2r 2r 2 2 X − uD X − uD− X − d2r − 1 X + 1 X + X + 1mod2dr. ( )( ) = ( ) ≡ − 6r Once again, multiplying by X 1 yields uD 1 mod dr′ , for any r. ( ) ≡ GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 12

′ It remains to show that the power 3r dr is minimal in each case. We first show that 3r is dr q minimal modulo dr for all r. Let q N be minimal such that uD 1 mod dr. It is easy to ∈ q 2− − ≡+ see that q 3r. Now the minimal polynomial of uD is X dq 1 X 1, so this must vanish S ( ) modulo dr at X 1. In other words dr divides into dq −3, proving that r q since dj j 1 = < ( ) ≥ is a strictly increasing sequence. So 3 q, since otherwise q 3r Ô⇒ q r Ô⇒ q r. S S S ≤ Writing q 3q0 we see that q0 r and so q0 r 3q0 q. This forces q0 r or q0 r 2 = S ≤ < = = = ~ and we now show in fact q0 r. If r is odd we are done. If r is even it follows r − 2 2=r r from equation (18) that uD uD uD 1 0 mod dr, which translates via equation (11) 3r ( + + ) ≡ u 2 d r d d r d q r to D 1 2 mod r. But 2 0 mod r, so again 0 . ≡ − ≡~ = q′ Finally assume that dr is even and let q′ N be the minimal integer such that uD ∈ ≡ 1 mod dr′ 2dr. Again we must have q′ 6r; moreover q′ 3r would (by reduction = S < modulo dr) contradict the proof above for dr. So either q′ 3r or q′ 6r. Using (11), (18) 3r − = = as above uD 1 dr mod d2r, dr 0mod2dr and the proof is complete again observing ≡ ≡~ that dr even Ô⇒ 2dr d2r.  S

5. Linking unitary geometry and arithmetic via subgroups of GL2 Z d′Z ( ~ ) Let v be a fiducial vector taken from one of the orbits in the list (7) that is centred as 2 in §2. This section and the next are devoted to a study of the function fv∶ Z d′Z → R ( ~ ) defined by

v, ∆jv (19) fv j ( ) Tr Πv∆j, ( ) = v, v = ( ) where the ∆j are defined as in (6). Unlike the components of the fiducial vector the values of the function fv are independent of both the scaling and the basis. It follows 0 from Proposition 1 and (6) that the values fv j for j 0 mod d are all elements of R ( ) ≢ ‰ Ž with modulus 1 . In fact we have found empirically that it is possible to choose v so √d 1 + that the field Q fv obtained by adjoining to Q all of the values of fv is equal to R1. ( ) We say that a centred fiducial vector for which that is true is strongly centred. Such fiducials may be characterized as follows: ● If 3 d every centred fiducial is strongly centred. ∤ ● If 3 d let n d 3 and B ∆n,2n. Then for each centred fiducial v exactly one of S = ~ = the three vectors v,Bv,B2v is strongly centred. From now on it will be assumed without comment that fiducials are strongly centred.

Lemma 13. If v is a strongly-centred fiducial and g is any element of Gal R K then 2 ( ~ ) g permutes the set fv j ∶ j Z d′Z . { ( ) ∈ ( ~ ) } Proof. Immediate consequence of Theorem 8 in ref. [4]. 

2 Let L be the smallest subfield of R1 fixing fv Z d′Z setwise. In all of our examples, (( ~ ) ) L K with index L ∶ K equal to the number of Sp2 Z d′Z -orbits of fiducials [4, §7]; ⊃ [ ] ( ~ ) see the remark at the end of this section. In addition to the Galois action, GL2 Z d′Z ( ~ ) GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 13

2 acts naturally on functions f∶ Z d′Z → R1 by translation via gf j f gj . What ( ~ ) ( )( ) = ( ) follows is a tantalizing observation linking these two distinct actions, strengthening a statement in §7 of [4]. Define

S det F F ∶ F S0 , = {( ) ∈ } S S where 0 ESp2 Z d′Z is the stabilizer defined in Sec. 2. As shown in [4], is the ⊂ ( S~ ) stabilizer of fv, i.e. g ESp2 Z d′Z ∶ gfv fv . = { ∈ ( ~ ) = } Proposition 14. For all orbits listed in (7) L is the Hilbert class field HK of K, and

(20) M S S Gal R1 L , ( ) ~ ≃ ( ~ ) where M S is a maximal abelian subgroup of GL2 Z d′Z containing S. For the or- ( ) ( ~ ) bits of Fz-type in (7) M S is in fact C S , the centralizer of S inside GL2 Z d′Z ; ( ) ( ) ( ~ ) for the orbits of Fa-type M S is conjugate to one of three possible subgroups that are ( ) characterized in Ref. [5].

We are grateful to John Coates for the following observation, which we hope to address in a forthcoming paper:

Remark 15. The appearance of HK in (20) cannot be a coincidence. The LHS is ostensibly a geometrically defined abelian subgroup of GL2 Z d′Z . On the other hand ( ~ ) the RHS contains information about a subgroup of a ray class group, and therefore also potentially about the ideal class group K of K. But this cannot be true in general, since C the structure of K is very erratic as D varies. We can therefore be confident that L will C typically contain HK.

Our empirical observations suggest that L may be identical with HK in all dimensions, not just those in (2). It follows that the Gal L K -set of distinct Sp2 Z d′Z -orbits may ( ~ ) ( ~ ) actually be a K-set. Preliminary numerical results communicated to us by Andrew Scott C for certain higher dimensions, wherein K is much larger than for the dimensions in (2), C provide additional evidence for this speculation.

6. Canonical units associated to the ray class fields

We now proceed to link invariants of the SICs with canonical units associated to the 0 ray class fields. The condition of equiangularity means that when j 0 mod d, the 1 ≢ ‰ Ž inner products fv j defined in Eq. (19) all have absolute value . It follows that the ( ) √d 1 numbers + 1 if j 0 mod d iθv j ⎧ 0 (21) e ( ) ⎪ ≡ ‰ Ž = ⎨√d + 1fv j otherwise ⎪ ( ) are all in U 1 . We refer to them⎩ as normalized inner products, and they are the subject ( ) of this section. The fact that we are now restricting ourselves to strongly centred fiducials means that the numbers fv j are all in R1. However, this may not be true of the iθv j ( ) numbers e ( ). We therefore need to introduce the extension field 1 R1 √d 1 and F = ( + ) GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 14 its ring of integers Z 1 . It is also convenient to define R √d 1 (see below for the F F = ( + ) relation between 1, and R1, R). We then have F F Proposition 16. Fix any d in Eq. (2) and consider a strongly-centred SIC fiducial on an EAut0 H -orbit. ( ) (i) Every one of the normalized inner products lies in the unit group Z× 1 . F Let ud be a normalized inner product of maximal degree nd over K √d 1 . + 3 ( ) (ii) ud generates 1 over Q √d 1 and therefore over K √d 1 . F ( + ) ( + ) (iii) Given any complex embedding of 1, the Gal 1 K -conjugates of ud all lie F (F ~ ) on U 1 .

( ) nd 1 (iv) The minimal polynomial fd x of ud is a reciprocal polynomial: that is, x fd x ( ) ( ) = fd x . ( ) Remark 17. For more on the Galois theory of fields generated by reciprocal polynomials see [21].

Proof. As with the main proposition, this is proven through lengthy computations using Magma [7]. The main steps are again indicated in Section 7. 

Corollary 18. Let ud be a normalized inner product of maximal degree nd over K √d 1 . ( + ) Then all of its images under the action of Gal 1 K √d 1 appear as normalized inner (F ~ ( + ) products for the same SIC. In particular, 1 ∶ K √d 1 nd. [F ( + )] = Proof. Immediate consequence of Proposition 16 and Lemma 13. 

We now examine the relation between 1, and R1, R. Observe at the outset F F that √d 1 K if and only if either d + 1 or d − 3 is a perfect square. + ∈ Proposition 19. Suppose d 4 and neither d + 1 nor d − 3 is a perfect square. Then ≥ (i) If d is even then √d 1 R0. + ∈ (ii) If d 1 4 then √d 1 R0. ≡ ( ) + ~∈ (iii) If d 3 4 then √d 1 R0 iff the extension K √d 1 K is everywhere unrami- ≡ ( ) + ∈ ( + )~ fied. In particular √d 1 R0 if the class number of K is 1 (that is, if HK K). + ~∈ = Proof. By assumption K √d 1 is a totally real biquadratic extension of Q, with exactly ( + ) three distinct proper quadratic subextensions K Q √D , Q √d 1 and Q √d 3 . If p = ( ) ( + ) ( − ) is an odd prime divisor of the square-free part of d + 1 then p will ramify in K Q but not ~ in K √d 1 K. So K √d 1 K is unramified at every place not lying over the prime 2. ( + )~ ( + )~ When d is even, d′ 2d is divisible by at least 22 whereas d 1 is odd, so the 2- = + primary part of the conductor of K √d 1 K divides that of d′ [10, §17F]. So √d 1 R0, ( + )~ + ∈ proving (i).

When d is odd, so d′ d, the only cases where √d 1 can possibly lie in the ray class = + field R0 are those where 2 does not ramify in the extension K √d 1 K. This proves ( + )~ 3Except in orbit 12b where we also need to adjoin √3. GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 15 assertion (iii) for any odd d, with the consequence that the extension HK K must be ~ non-trivial if √d 1 R0. + ∈ To prove (ii), if d 1 mod 4 then we claim that the unique prime of K above 2 ≡ always ramifies again in K √d 1 K: meaning that √d 1 can never lie in R0. First of ( + )~ + all, Q √d 1 Q and Q √d 3 Q are each ramified over 2, since both d 1 and d 3 2 mod4. ( + )~ ( − )~ + − ≡ Also d 1 d 3 12 mod16 and since 22 d 1 d 3 it follows that D 3 mod4. ( + )( − ) ≡ S ( + )( − ) ≡ So 2 is ramified in all three quadratic subextensions Q √D , Q √d 1 , Q √d 3 . There- ( ) ( + ) ( − ) fore [10, §17F] the inertia group at 2 must be the whole Galois group Gal K √d 1 Q , ( ( + )~ ) proving the claim. 

Corollary 20. For the dimensions in list (2) it follows that √d 1 R0 except for d 5, + ∈ = 9, 11, 13, 17, 21. (The case d 11 follows from the fact that the class number of Q √6 = ( ) is 1 ).

Since R0 is the maximal totally real subfield of R, it follows therefore that √d 1 R + ∉ for d 5, 9, 11, 13, 17, 21. We conclude = ● If d 4, 6–8, 10, 12, 14–16, 18–20, 24 , 28, 30, 35, 39, 48 then 1 R1, R. = F = F = ● If d 5, 9, 11, 13, 17, 21 then 1, are degree 2 extensions of R1, R such that = F F 1 ∩ R R1. F = The next result is striking but the size of the necessary calculations has prevented us from testing dimensions 16, 17, 18, 20, 21, 30 and 39 in list (2). Let U denote the group of units of the ring of integers of the field 1 under any one of its complex embeddings, and F let V be the subgroup generated by the normalized inner products. By construction V is contained in the unit circle subgroup U U 1 of U. We would like to compare the ∩ ( ) relative ranks of V and U U 1 . Let Nd 1 K , so Nd nd or 2nd according to ∩ ( ) = [F ∶ ] = whether √d 1 R0 or not. Part (ii) of the Corollary to Proposition 10 implies that 1

+ ∈ Nd F has Nd real places and 2 pairs of complex places; so by Dirichlet’s unit theorem U is an 3Nd abelian group of rank 2 1. By the same Proposition R0 √d 1 is the maximal real − ( + ) subfield of 1 and has index 2: so it is a consequence of the lemma in §5 of [23] that the

F Nd unit circle subgroup U U 1 of U has Z-rank 2 . ∩ ( ) The following proposition is split into two parts. The first part holds for the set of dimensions 4–15, 19, 24, 28, 35 and 48 for which we were able to calculate the group V. The second part holds for the smaller set of dimensions 4–8, 12, 19 for which we were able to calculate both the group V and the group U.

Proposition 21. (A) In dimensions 4–15, 19, 24, 28, 35 and 48, V is generated by the

Gal 1 K -orbit of ud. One has (F ~ ) rank U ∩ U 1 , d 7, 15, 19, 35; (22) rank V ⎪⎧ ⎪ 1 ( ∩ ( )) = ( ) = ⎨ 2 rank U U 1 , d 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 24, 28, 48. ⎪ ( ( )) = (B) In dimensions⎩ 4–8, 12 and 19, V is also a direct summand of U ∩ U 1 . In ( ) particular, U ∩ U 1 V in dimensions 7 and 19. ( ) = GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 16

Proof. Again, this is proven via lengthy computations using Magma [7], and the main steps are indicated in Section 7. Note that the calculation of U, in those dimensions where it was possible, used the function IndependentUnits in Magma, which assumes the validity of the generalized Riemann hypothesis. 

Finally, preliminary calculations by one of us (GM) together with Steve Donnelly and reported in the first draft of this paper indicated that the logarithms of these canonical units may be related to the values of L-functions associated to the extensions, following the programme laid out in the Stark Conjectures. Subsequent work by Gene Kopp has numerically confirmed this conjecture for dimension d 5 [19] and some higher = dimensions [20], via the values at s 0 of the first derivatives of the partial L-functions = attached to the ray classes.

7. Verification of Propositions 1, 3, 16, and 21

A full account of the verification of these propositions would be very tedious. In this section we confine ourselves to an indication of the main steps.

7.1. Propositions 1, 3. The first obstacle we face is the definition of the field Q E , ( ) in terms of the ratios of components of the vectors v defining the lines. Conceptually speaking this is the most natural way of associating a field to a set of equiangular lines. However, it is cumbersome to work with. To obtain a computationally more convenient definition we work with the projectors vv† (23) Πv = v, v ( ) ∗ vj vk instead of the vectors v. Let Q Πv be the field generated by v,v of Πv, and let L be ( ) ( ) the smallest subfield of C containing Q Πv for every Cv E. It is straightforward to ( ) ∈ verify that L is Galois if and only if Q E is Galois, in which case L Q E . We can ( ) = ( ) further simplify the computations by observing the following. The fact that the lines form an orbit under the action of the group generated by the matrices Xd, Zd specified by Eq. (21) means that L Q Πv,ζd , the field generated by the matrix elements of = ( ) th 2πi d the single projector Πv together with the d root of unity ζd e ~ . The problem of = verifying Proposition 1 thus reduces to showing that for each dimension d in the list (2) there is a fiducial projector Πv such that Q Πv,ζd is the ray class field R. ( ) At this point we ought to comment on a seeming discrepancy between what is said in the previous paragraph, and what is said in, for example, ref. [5], where the SIC field is defined to be Q Πv,ζ2d . In fact, the discrepancy is only apparent, since for every known ( ) SIC the two fields are identical. Indeed, it is immediate that Q Πv,ζ2d Q Πv,ζd when d+1 ( ) = ( ) − 2 d is odd, since in that case one has ζ2d ζd . If, on the other hand, d is even we appeal = to the fact that every known SIC fiducial Πv is stabilized by a canonical order 3 Clifford GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 17 unitary [5, 8] to deduce that

k (24) Q ζd Tr ∆j Πv∶ k, j1, j2 Z dZ and j1j2 0 mod2 ‰{ ( ) ∈ ~( ) = }Ž k Q ζd Tr ∆j Πv∶ k, j1, j2 Z dZ , = ‰{ ( ) ∈ ~( )}Ž and to deduce in turn that

k (25) ζ2d Tr XdZdΠv Q ζd Tr ∆j Πv∶ k, j1, j2 Z dZ and j1j2 0 mod2 ( ) ∈ ‰{ ( ) ∈ ~( ) = }Ž Q Πv,ζd . ⊆ ( ) Since Tr XdZdΠv is also in Q Πv,ζd it follows that ζ2d Q Πv,ζd and, consequently, ( ) ( ) ∈ ( ) that Q Πv,ζ2d Q Πv,ζd . This point incidentally makes clear what might otherwise ( ) = ( ) be doubted: that the introduction of 2d th roots of unity is unavoidable when d is even. ( ) It is shown in ref. [5] that Q Πv,ζ2d is invariant under the action of the extended ( ) Clifford group. The argument in the last paragraph means that the same is true of

Q Πv,ζd . This proves Proposition 3. ( ) The remainder of the verification of Proposition 1 is most conveniently explained by means of an example. The expressions for fiducial vectors are usually extremely complicated, as can be seen from a casual inspection of refs. [12, 26, 33]. However, in dimension 7 one has a SIC fiducial vector with the remarkably simple formula4

v0 ⎛v1⎞ (26) v ⎜ ⋮ ⎟ , = ⎜ ⎟ ⎜v6⎟ ⎝ ⎠ where −2√2 if n 0, ⎪⎧ (27) vn ⎪ n = = ⎨1 + i »4√2 − 5 otherwise, ⎪ 7 ⎪ ‹  n ⎩ being the Legendre symbol. The formula being so simple, this is the natural example ‹7 to choose.

Using Magma [7] we compute the ray class fields R1, R. We find that R is a degree 2 extension of K ζ7 . In particular R∶ K 12, R1∶ K 6. We then calculate the matrix ( ) [ ] = [ ] = elements of Πv and show that they all belong to R. It follows that Q Πv,ζ7 R. ( ) ⊆ Finally, we show that Q Πv,ζ7 is a degree 2 extension of K ζ7 . We conclude that ( ) ( ) Q Πv,ζ7 R, thereby verifying Proposition 1 for the case d 7. ( ) = = 7.2. Propositions 16, 21. We begin by observing that (i) & (ii) Ô⇒ (iii) Ô⇒ (iv).

Indeed, the abelian structure imposes a unique complex conjugation operator σc ∈ Gal 1 K upon every complex embedding. So for any g Gal 1 K , (F ~ ) ∈ (F ~ ) 1 1 σc g ud g σc ud g ud− g ud − . ( ( )) = ( ( )) = ( ) = ( ( )) 4There exists a fiducial vector in dimension 19 with a similarly simple formula [1, 26]. The role of the Legendre symbol in these expressions is intriguiing, as is the fact that every component apart from the first has the same absolute value. For the question, whether there exist other examples of such fiducials, see ref. [18]. GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 18

Since Gal 1 K is transitive on the roots, fd is a product of reciprocal polynomials of (F2 ~ ) 1 the form x − u + u− x + 1, hence is itself reciprocal. ( ) We next show that if (i) and (ii) hold for one strongly-centred fiducial on a given 0 EAut H -orbit then they hold for all. Indeed, let v, v′ be two strongly centred fiducials ( ) on the same orbit. It follows from the analysis in ref. [5] that

′ k1j2 k2j1 iθv j ∶ 2 iθv p j ∶ 2 (28) e ( ) j Z d′Z ζd − e ( ( )) j Z d′Z ™ ∈ ( ~ ) ž = ™ ∈ ( ~ ) }

2 0 for some k Z d′Z and permutation p fixing the point 0 . The fact that both fiducials ∈ ( ~ ) k1 k2 ‰ Ž are strongly centred means ζd , ζd R1. Let g be any element of Gal R Q mapping kj λkj ∈ ( ~ ) R1 to R2. Then g ζd ζd for some integer λ coprime to d. In view of Corollary 11 λk( 1 ) λk= 2 this means that ζd , ζd are both real. It follows that if k1, k2 are both zero if d is odd, and multiples of d 2 if d is even. We conclude that ~

iθ ′ j 2 iθv p j 2 (29) e v ( )∶ j Z d′Z sje ( ( ))∶ j Z d′Z ™ ∈ ( ~ ) ž = ™ ∈ ( ~ ) } for some set of signs sj. The claim is now immediate. The problem thus reduces to establishing parts (i) and (ii) for one particular strongly centred fiducial. Since the fiducial specified by Eq. (26) is strongly-centred we illustrate the calculations with that. It follows from Corollary 20 that 1 R1, R. Also we F = F = know from the previous sub-section that R1∶ K 6, R∶ K 12. Using the function [ ] = [ ] = IndependentUnits in Magma we find that U ∩ U 1 is generated by ( ) 1 π (30) v1 7 + i»35 + 28√2 + −14 − i»−308 + 224√2 cos = 14 ‹‹  ‹  ‹ 7  2π + 14√2 + i»−56 + 112√2 cos ‹  ‹ 7  1 π (31) v2 i»−7 + 14√2 + 7 − 7√2 − i»−7 + 14√2 cos = 7 ‹ ‹  ‹ 7  2π + −7 − i»−77 + 56√2 cos ‹  ‹ 7  1 π (32) v3 7 − 7√2 + i»−7 + 14√2 + 14√2 + i»−56 + 112√2 cos = 14 ‹‹  ‹  ‹ 7  2π 14 − 14√2 − i»−28 + 56√2 cos ‹  ‹ 7  together with −1. On the other hand the normalized inner products take the 9 distinct values

(33) 1, v1, v2, v3, −v1v2v3, v1∗, v2∗, v3∗, −v1∗v2∗v3∗

where vj∗ is the complex conjugate of vj. This establishes Proposition 21 and part (i) of Proposition 16 for the case d 7. Using Magma one finds that the minimal polynomials = GENERATING RAY CLASS FIELDS OF REAL QUADRATIC FIELDS ... 19 over K of these numbers are

(34) f1 x x − 1 , ( ) = ( ) (35) f2 x x + v1v2v3 x + v1∗v2∗v3∗ ( ) = ( ) ( ) x2 + √2 − 1 x + 1, = Š  (36) f3 x x − v1 x − v2 x − v3 x − v1∗ x − v2∗ x − v3∗ ( ) = ( ) ( ) ( ) ( ) ( ) ( ) x6 + √2 − 2 x5 + √2 − 2 x4 + −5√2 + 9 x3 = Š  Š  Š  + √2 − 2 x2 + √2 − 2 x + 1. Š  Š  The fact that f3 x is irreducible, has the same degree as R1 K and splits over R1 ( ) ~ means that its roots generate R1. Since K K √d + 1 Q √d + 1 this proves part = ( ) = ( ) (ii) of Proposition 16 for the case d 7. = Finally, let us remark that in those dimensions where it is impracticable to compute U it is still possible to compute V by applying an integer relation algorithm to the logarithms of the normalized inner products.

Acknowledgments

It is a pleasure to thank John Coates and Andrew Scott for their continuing support and guidance. Also thanks to Steve Donnelly for his help with the Magma code and with interpreting the number-theoretic results; to James McKee and Chris Smyth for explaining various aspects of the theory of reciprocal units; and to Brian Conrad and to an anonymous referee for comments on an earlier draft. This research was supported by the Australian Research Council via EQuS project number CE11001013, and SF acknowledges support from an Australian Research Council Future Fellowship FT130101744 and JY from National Science Foundation Grant No. 116143. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

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Centre for Engineered Quantum Systems, School of Physics, University of Sydney E-mail address: [email protected]

Centre for Engineered Quantum Systems, School of Physics, University of Sydney E-mail address: [email protected]

Controlled Quantum Dynamics Theory Group, Imperial College, London E-mail address: [email protected]

Institute for Quantum Computing, Department of Combinatorics & Optimization, Uni- versity of Waterloo and Perimeter Institute for Theoretical Physics, Waterloo, ON E-mail address: [email protected]