Singular values of multiple eta-quotients for ramified primes

Abstract

We determine the conditions under which singular values of multiple η-quotients of square-free level, not necessarily prime to six, yield class invariants; that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index 2 k −1 when k 2 primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on X + 0 (p) for p prime and ramified.Let K = Q( √ ∆) be an imaginary-quadratic number field of discriminant ∆, and let O D be the order of discriminant D = c 2 ∆ and conductor c in K. For a modular function f and an argument τ ∈ K ⊆ C with τ > 0, we call the singular value f (τ ) a class invariant if it lies in the ring class field Ω c . This is the abelian extension of K with Galois group canonically isomorphic to Cl(O D ) through the Artin map σ : Cl(O D ) → Gal(Ω c /K), which sends a prime ideal representing an ideal class to its associated Frobenius automorphism. In this paper, we are interested in class invariants derived from multiple η-quotients, and we examine in particular cases where those generate a subfield of the ring class field.In § 1, we define the multiple η-quotients under consideration and collect their properties, in particular their transformation behaviour under unimodular matrices. We then proceed in § 2 to determine conditions under which their singular values lie in the ring class field and show how to compute their characteristic polynomials with respect to the number field extension Ω c /K using roots of an n-system, suitably normalised quadratic forms of discriminant D representing the class group. The results of § 3 are at the heart of this paper: we show that if some or all of the primes dividing the level of the multiple η-quotient are ramified in K, then the singular values lie in fact in a subfield L of Ω c of index a power of two; more precisely, the Galois group of Ω c /L is elementary abelian, so that Ω c is a compositum of linearly disjoint quadratic extensions of L. An alternative proof for the special case that all primes are ramified leads to an interesting generalisation in § 4, namely to functions of prime level invariant under the Fricke-Atkin-Lehner involution. We conclude with some class field-theoretic remarks in § 5, showing how this computationally less expensive construction of subfields of Ω c may be completed to obtain the full ring class field.

Schertz style class invariants for quartic CM fields

“…As for simple and double eta quotients in dimension 1, the process may be generalized to obtain multiple quotients of h k , cf. Enge and Schertz [11].…”

Short addition sequences for theta functions

“…In dimension 1, that is, in the context of elliptic curves, the Dedekind eta function is often more convenient [39,37,16,17,15]. It is a modular form of weight 1 2 and level 24 defined by…”

Generalized class polynomials

“…There has been continued interest in alternative class invariants ever since (e.g. [2,30,18,17,31,8,10,11,4,14,12,9]). None however matched, let alone surpassed, the factor 72 of Weber's functions.…”

“…based on Enge-Schertz[12]). Let C = (C, ψ) be a quotient over Q of X 0 (N ) and let D =F 2 D 0 < 0 be such that N | D, gcd(F, N ) = 1, and D ∈ {N, 4N }.…”