On the simultaneous $3$-divisibility of class numbers of quadruples of real quadratic fields

Abstract: In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain tuples of real quadratic fields. At the end, we give an application of this result to produce some elliptic curves having a $3$-torsion subgroup.

• The paper establishes an infinite family of quadruples of real quadratic fields with all class numbers divisible by 3 using new parametric constructions.
• It employs explicit cubic polynomial parametrizations and Galois-theoretic methods to secure unramified cyclic cubic extensions crucial for 3-divisibility.
• The results link class number divisibility to the arithmetic of elliptic curves, opening avenues for extensions to larger tuples or other primes.

Simultaneous $3$-Divisibility of Class Numbers in Quadruples of Real Quadratic Fields

Introduction

This paper establishes the existence of infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. Existing literature offers numerous constructions for divisibility results on class numbers in both real and imaginary quadratic fields, primarily for individual fields or pairs, with some work on larger tuples in the imaginary quadratic case. However, prior to this work, simultaneous divisibility for quadruples of real quadratic fields remained unresolved. The analytic and constructive framework introduced here leverages both explicit parametrization techniques and Galois-theoretic methodologies, with further applications to the arithmetic of torsion subgroups in elliptic curves.

Background and Motivation

Class numbers of quadratic fields serve as central invariants in algebraic number theory, encoding the structure of ideal class groups. Investigating their divisibility elucidates deep connections to the arithmetic of field extensions and provides insight into conjectures like that of Iizuka, which predicts the existence of arbitrarily long chains of quadratic fields (real or imaginary) with class numbers divisible by a prescribed prime. For imaginary quadratic fields, the landscape is more thoroughly explored, and for small divisibility (notably $3$-divisibility), various infinite families are known. For real quadratic fields, the question is more delicate: the interplay between unramified extensions and class field theory, particularly via the Spiegelungssatz and cyclic cubic (or $S_3$) extensions, underpins most constructions.

Main Results

The main achievement is the construction of infinitely many quadruples of real quadratic fields of the explicit forms

$$\left( \mathbb{Q}(\sqrt{D}),\ \mathbb{Q}\left(\sqrt{216000 D^3 + 457200 D^2 + 322580 D + 75866}\right),\ \mathbb{Q}\left(\sqrt{432 D^3 + 1080 D^2 + 900 D + 223}\right),\ \mathbb{Q}\left(\sqrt{40500 D^3 + 89100 D^2 + 65340 D + 16215}\right) \right)$$

with $D\in \mathbb{N}$, and with the property that for each quadruple, $3$ divides all four class numbers.

The proofs employ two major techniques:

• Parametrized families of irreducible cubic polynomials whose splitting fields realize cyclic cubic extensions that are unramified over the associated quadratic fields (via Kishi–Miyake and Kishi constructions).
• Explicit analysis of ramification behavior at $3$, verifying unramifiedness at the critical prime.

For each field in the quadruple, the divisibility by $3$ follows from the existence of an unramified cyclic cubic extension—by Artin reciprocity, this forces $3$ to divide the class number.

The infinitude of such quadruples is established by ruling out diophantine obstructions with a Siegel-type argument, ensuring the defining cubic polynomials yield infinitely many non-isomorphic quadratic fields of the desired forms.

Construction of Unramified Cyclic Cubic Extensions

Kishi–Miyake's explicit criterion is leveraged to construct unramified cyclic cubic extensions of quadratic fields: for relatively prime pairs $(u, v)$, the discriminant of $Z^3-uvZ-u^2$ produces the required sequence of radicands for quadratic fields. The irreducibility and non-square conditions on the discriminant, together with careful $3$-adic analysis (ramification criteria via the Llorente–Nart lemma), underlie the proof of unramifiedness.

Moreover, the second method, building on Kishi's parameterization with elements $\alpha$ in the ring of integers whose norms are perfect cubes but which are not themselves cubes, provides an alternative description, particularly useful for one of the fields in each quadruple. The Hilbert class field and Artin reciprocity are pivotal: if there is a nontrivial unramified cyclic cubic extension, this guarantees class number divisibility by $3$.

Application to Elliptic Curve Torsion

The concluding section connects these number-theoretic results to the arithmetic of elliptic curves. For the elliptic curve defined by

$y^2 = 40500x^3 + 89100x^2 + 65340x + 16215$

(and analogous models for the other defining polynomials), it is shown that the associated $\mathbb{Q}$-rational points admit a $3$-torsion subgroup. The argument proceeds via the Nagell–Lutz theorem, the distribution of $3$-torsion in the specializations over quadratic fields with class numbers divisible by $3$, and the structure of the Picard group across the relative family of curves. The infinitude of such quadratic fields, together with the isotriviality of the family, enforces the existence of $3$-torsion on the generic fiber.

Implications and Future Directions

This result provides the first infinite family of quadruples of real quadratic fields with simultaneously $3$-divisible class numbers, pushing forward the program laid out by Iizuka for both real and imaginary cases. The construction techniques, grounded in explicit Galois theory and class field theory, are robust and potentially extensible to larger tuples or to other primes $p > 3$, as has been done in the imaginary case. Practically, such results have consequences for the construction of abelian unramified extensions, understanding the behavior of torsion in the context of arithmetic geometry (notably in the context of the Langlands program and the arithmetic of elliptic curves), and for exploring the effective parametrization of arithmetic invariants over infinite field families.

Speculative extensions include generalizations to $p$-divisibility for higher $p$, simultaneous divisibility by multiple primes, and the explicit realization of associated Galois representations with controlled ramification.

Conclusion

The paper rigorously constructs infinite families of quadruples of real quadratic fields with class numbers divisible by $3$, thereby resolving a significant open instance of simultaneous divisibility for tuples in the real case. The interplay between explicit polynomial parameterizations, Galois-theoretic extension criteria, and applications to elliptic curves underscores the deep interrelations between classical number theory, arithmetic geometry, and explicit class field theory. This breadth provides fertile ground for continued research into simultaneous divisibility phenomena in number fields and their arithmetic avatars.