Ray Class Field Research Articles

Explicit upper bounds on the average of Euler–Kronecker constants of narrow ray class fields

Explicit upper bounds on the average of Euler–Kronecker constants of narrow ray class fields

On special values of generators of modular function fields

Let N be a positive integer and ζN be a primitive Nth root of unity. Let X(N) be the modular curve of level N and FX(N),Q(ζN) be its field of meromorphic functions with Fourier coefficients in Q(ζN). Let u and v be nonconstant functions in FX(N),Q(ζN) which generate a sufficiently large subfield of FX(N),Q(ζN) over Q(ζN). In this paper, we give sufficient conditions for generating the ray class field of an imaginary quadratic field K modulo N over K(ζN) by the special values of u and v.

Analogue of Ramanujan’s function k(τ) for the cubic continued fraction

We study the modularity of the function [Formula: see text], where [Formula: see text] is Ramanujan’s cubic continued fraction. It is an analogue of Ramanujan’s function [Formula: see text], where [Formula: see text] is the Rogers–Ramanujan continued fraction. We first prove the modularity of [Formula: see text] and express [Formula: see text] and [Formula: see text] in terms of [Formula: see text]. Subsequently, we find modular equations of [Formula: see text] of level [Formula: see text] for every positive integer [Formula: see text] by using affine models of modular curves. Finally, we demonstrate that the value of [Formula: see text] generates the ray class field over an imaginary quadratic field modulo 2 for some [Formula: see text] in an imaginary quadratic field.

Frobenius lifts and elliptic curves with complex multiplication

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.

Recovering affine curves over finite fields from L-functions

Let $K$ be the function field of a curve over a finite field of odd characteristic. We investigate using $L$-functions of Galois extensions of $K$ to effectively recover $K$. When $K$ is the function field of the projective line with four rational points removed, we show how to use $L$-functions of a ray class field to effectively recover the removed points up to automorphisms of the projective line. When $K$ is the function field of a plane curve, we show how to effectively recover the equation of that curve using $L$-functions of Artin-Schreier extensions of $K$.

Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

Cohomology groups of Fermat curves via ray class fields of cyclotomic fields

The absolute Galois group of the cyclotomic field K=Q(ζp) acts on the étale homology of the Fermat curve X of exponent p. We study a Galois cohomology group which is valuable for measuring an obstruction for K-rational points on X. We analyze a 2-nilpotent extension of K which contains the information needed for measuring this obstruction. We determine a large subquotient of this Galois cohomology group which arises from Heisenberg extensions of K. For p=3, we perform a Magma computation with ray class fields, group cohomology, and Galois cohomology which determines it completely.

On Siegel invariants of certain CM-fields

We first construct Siegel invariants of some CM-fields in terms of special values of theta constants, which would be a generalization of Siegel–Ramachandra invariants of imaginary quadratic fields. We further describe Galois actions on these invariants and provide a numerical example to show that this invariant really generates the ray class field of a CM-field.

Modularity of Galois traces of ray class invariants

After Zagier’s significant work (in: Bogomolov and Katzarkov (eds) Motives, polylogarithms and hodge theory, part I, International Press, Somerville, 2002) on traces of singular moduli, Bruinier and Funke (J Reine Angew Math 594:1–33, 2006) generalized his result to the traces of singular values of modular functions on modular curves of arbitrary genus. In class field theory, the extended ring class field is a generalization of the ray class field over an imaginary quadratic field. By using Shimura’s reciprocity law, we construct primitive generators of the extended ring class fields by using Siegel functions of arbitrary level $$N\ge 2$$ and identify their Galois traces with Fourier coefficients of weight 3/2 harmonic weak Maass forms. This would extend the results of Jeon et al. (Math Ann 353:37–63, 2012) and Jung et al. (Modularity of Galois traces of Weber’s resolvents, under revision).

Generating ray class fields of real quadratic fields via complex equiangular lines

For certain real quadratic fields $K$ with sufficiently small discriminant we produce explicit unit generators for specific ray class fields of $K$ using a numerical method that arose in the study of complete sets of equiangular lines in $\mathbb{C}^d$ (known in quantum information as symmetric informationally complete measurements or SICs). The construction in low dimensions suggests a general recipe for producing unit generators in infinite towers of ray class fields above arbitrary real quadratic $K$, and we summarise this in a conjecture. There are indications [19,20] that the logarithms of these canonical units are related to the values of $L$-functions associated to the extensions, following the programme laid out in the Stark Conjectures.

The Lambda Invariants at CM Points

Abstract In this paper, we show that $\lambda (z_1) -\lambda (z_2)$, $\lambda (z_1)$, and $1-\lambda (z_1)$ are all Borcherds products on $X(2) \times X(2)$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $\lambda (\frac{d+\sqrt d}2)$, $1-\lambda (\frac{d+\sqrt d}2)$, and $\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$, with the latter under the condition $(d_1, d_2)=1$. Finally, we use these results to show that $\lambda (\frac{d+\sqrt d}2)$ is always an algebraic integer and can be easily used to construct units in the ray class field of ${\mathbb{Q}}(\sqrt{d})$ of modulus $2$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest.

SIC-POVMs and the Stark Conjectures

Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. We prove that, if there exists a set of real units in a certain ray class field (depending on $d$) satisfying certain algebraic properties, a SIC-POVM exists, when $d$ is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at $s=0$ and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.

Composition law and complex multiplication

Let K be an imaginary quadratic field of discriminant dK, and let n be a nontrivial integral ideal of K in which N is the smallest positive integer. Let QN(dK) be the set of primitive positive definite binary quadratic forms of discriminant dK whose leading coefficients are relatively prime to N. We adopt an equivalence relation ∼n on QN(dK) so that the set of equivalence classes QN(dK)/∼n can be regarded as a group isomorphic to the ray class group of K modulo n. We further establish an explicit isomorphism of QN(dK)/∼n onto Gal(Kn/K) in terms of Fricke invariants, where Kn denotes the ray class field of K modulo n. This would be a certain extension of the classical composition theory of binary quadratic forms, originated and developed by Gauss and Dirichlet.

Simplified exact SICS

In the standard basis, exact expressions for the components of SIC vectors (belonging to a symmetric informationally complete positive operator valued measure or POVM) are typically very complicated. We show that a simple transformation to a basis adapted to the symmetries of a fiducial SIC vector can result in a massive reduction in complexity. We rely on a conjectural number theoretic connection between SICs in dimension dj and SICs in dimension dj+1 = dj(dj − 2). We focus on the sequence 5, 15, 195, … We rewrite Zauner’s exact solution for the SIC in dimension 5 to make its simplicity manifest and use our adapted basis to convert numerical solutions in dimensions 15 and 195 to exact solutions. Comparing to the known exact solutions in dimension 15, we find that the simplification achieved is dramatic. The proof that the exact vectors are indeed SIC fiducial vectors, also in dimension 195, is guided by the standard ray class hypothesis about the algebraic number fields generated by the SICs. In the course of the calculation, we introduce SIC adapted generators for the ray class field. We conjecture that our result generalizes to every dimension in the particular sequence we consider.

Modular equations of a continued fraction of order six

Abstract We study a continued fraction X(τ) of order six by using the modular function theory. We first prove the modularity of X(τ), and then we obtain the modular equation of X(τ) of level n for any positive integer n; this includes the result of Vasuki et al. for n = 2, 3, 5, 7 and 11. As examples, we present the explicit modular equation of level p for all primes p less than 19. We also prove that the ray class field modulo 6 over an imaginary quadratic field K can be obtained by the value X2 (τ). Furthermore, we show that the value 1/X(τ) is an algebraic integer, and we present an explicit procedure for evaluating the values of X(τ) for infinitely many τ’s in K.

On a problem of Hasse and Ramachandra

AbstractLetKbe an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal ofK. Hasse and Ramachandra asked whether the ray class field ofKmodulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = (N) for any positive integerNexcluding 2, 3, 4 and 6.

Binary quadratic forms and ray class groups

AbstractLetKbe an imaginary quadratic field different from$\open{Q}(\sqrt {-1})$and$\open{Q}(\sqrt {-3})$. For a positive integerN, letKNbe the ray class field ofKmodulo$N {\cal O}_K$. By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for whichKNabis the maximal abelian extension ofKunramified outside prime ideals dividing$N{\cal O}_K$.

Construction of class fields over imaginary biquadratic fields

Let K be an imaginary biquadratic field, K1, K2 be its imaginary quadratic subfields and K3 be its real quadratic subfield. For integers N > 0, µ ≥ 0 and an odd prime p with gcd(N,p) = 1, let K(Npµ) and (Ki)(Npµ) for i = 1,2,3 be the ray class fields of K and Ki, respectively, modulo Np µ . We first present certain class fields ^ K 1,2 N,p,µ of K, in the sense of Hilbert, which are generated by ray class invariants

Constructing ray class fields of a real quadratic field using elliptic curves

Constructing ray class fields of a real quadratic field using elliptic curves

Primitive and totally primitive Fricke families with applications (II)

We find necessary and sufficient conditions for a Fricke family of level N (≥2) to be primitive or totally primitive. Let K be an imaginary quadratic field of discriminant d K other than Q ( − 1 ) and Q ( − 3 ) . As applications of Fricke families, we show that if | d K | is sufficiently large, then the special values of a primitive Fricke family generate the ray class field K ( N ) modulo N over K . Moreover, we construct a primitive generator of K ( N ) over K in terms of the special values of classical Fricke functions for every K which would be a partial answer to a question of Hasse and Ramachandra.