On the Iwasawa $λ$-invariant of the cyclotomic $\mathbb{Z}_2$-extension of a family of real quadratic fields in which $2$ splits

Abstract: We study Greenberg's conjecture for cyclotomic $\mathbb{Z}_2$-extensions of real quadratic fields. Let $K=\mathbb{Q}(\sqrt{pq})$, where $$p\equiv 1 \mod 8,\qquad q\equiv 9 \mod {16},\qquad \left(\frac{p}{q}\right)=-1.$$ Under the additional assumptions $$\left(\frac{2}{p}\right)_4 \left(\frac{2}{q}\right)_4 \left(\frac{pq}{2}\right)_4=-1$$ and $$\left(\frac{2}{p}\right)_4=-1 \quad\text{or}\quad \left(\frac{2}{q}\right)_4=-1,$$ we prove that $λ_2(K)=0$. The proof combines Greenberg's criterion for the split prime case with a capitulation argument modeled on Kumakawa. The main new input is a square-class computation of the Hasse unit index of the biquadratic extension $$K_2=\mathbb{Q}(\sqrt{pq}, \sqrt{2+\sqrt{2}})/\mathbf{Q}_1=\mathbb{Q}(\sqrt{2})$$, showing that $q(K_2)\le 2$.

• The paper demonstrates that the Iwasawa λ2-invariant vanishes in Q(√(pq)) under specific quartic residue conditions.
• It employs a synthesis of genus theory, capitulation arguments, and precise square-class and Hasse unit index computations.
• The study refines Greenberg’s conjecture by extending explicit arithmetic criteria to real quadratic fields with split prime 2.

The Iwasawa $λ$-Invariant in Cyclotomic $\mathbb{Z}_2$-Extensions Where $2$ Splits

Introduction

This paper addresses Greenberg's conjecture for cyclotomic $\mathbb{Z}_2$-extensions of real quadratic fields, focusing on cases where the prime $2$ splits and, more specifically, on the explicit family $K=\mathbb{Q}(\sqrt{pq})$ with $p\equiv 1\pmod{8}$, $q\equiv 9\pmod{16}$, and $(\tfrac{p}{q})=-1$. The main result establishes that the Iwasawa $\lambda$-invariant $\mathbb{Z}_2$0 vanishes under refined quartic-symbol conditions, thereby extending the landscape of known results on Greenberg's conjecture. The proof synthesizes genus theory, capitulation arguments, and a highly nontrivial square-class analysis of unit groups in biquadratic extensions.

Background and Framework

For a number field $\mathbb{Z}_2$1 and prime $\mathbb{Z}_2$2, the Iwasawa invariants $\mathbb{Z}_2$3, $\mathbb{Z}_2$4, and $\mathbb{Z}_2$5 pertain to the structure of the $\mathbb{Z}_2$6-primary part of the class group in the cyclotomic $\mathbb{Z}_2$7-extension $\mathbb{Z}_2$8. Greenberg conjectured $\mathbb{Z}_2$9 for totally real fields. While Ferrero and Washington established $2$0 for abelian $2$1 [FW79], the vanishing of $2$2 for real quadratic fields with splitting of $2$3 remains intricate due to the subtle behavior of strongly ambiguous ideal classes.

The study considers "nontrivial" cases—where $2$4 splits in $2$5 and the class number is even—excluding those covered by Iwasawa's theorem or genus theory. In the family considered, prior approaches proved insufficient due to complications from the structure of unit groups and ambiguous class groups.

Main Result and Reduction Strategy

The paper's principal theorem asserts: For $2$6 with

• $2$7,
• $2$8,
• $2$9,
• $\mathbb{Z}_2$0,
• and either $\mathbb{Z}_2$1 or $\mathbb{Z}_2$2, one has $\mathbb{Z}_2$3.

The proof applies Greenberg's criterion, which connects the vanishing of the Iwasawa $\mathbb{Z}_2$4-invariant to the capitulation of certain ambiguous classes in the cyclotomic tower. The challenge in the split $\mathbb{Z}_2$5 case is to guarantee that the only nontrivial ambiguous class in $\mathbb{Z}_2$6 (the $\mathbb{Z}_2$7-part of the class group of $\mathbb{Z}_2$8) capitulates in $\mathbb{Z}_2$9.

To reduce the question to a manageable invariant, the authors mimic Kumakawa's capitulation argument [Kum21], showing it suffices to bound the Hasse unit index $2$0 in the second layer $2$1 of the $2$2-extension over the quadratic field $2$3. The explicit computation $2$4 then yields the desired capitulation.

Tools and Computations

Genus Theory and Class Group Structure

The structure of the $2$5- and $2$6-ranks of the class group and narrow class group is navigated via the Rédei–Reichardt matrices [RR34] and sharply tuned use of genus theory. The explicit congruence conditions on $2$7, $2$8, and the quartic symbols serve to restrict the possible ambiguous ideal classes and ensure $2$9, generated by a prime above $K=\mathbb{Q}(\sqrt{pq})$0.

Strong and Weak Ambiguity, Norm Criteria

Leveraging Chevalley’s ambiguous class formula and Greenberg’s criterion, the paper identifies the persistence of ambiguous classes and tracks their behavior under norm maps across layers. The challenge is that for totally split primes, the triviality of ambiguous classes is not automatic.

The key computation involves the fundamental unit $K=\mathbb{Q}(\sqrt{pq})$1 of $K=\mathbb{Q}(\sqrt{pq})$2 and its non-norm behavior from $K=\mathbb{Q}(\sqrt{pq})$3 (the first layer), quantified via local Hilbert symbols and the $K=\mathbb{Q}(\sqrt{pq})$4-adic logarithm. The presence of specific quartic residue relationships between $K=\mathbb{Q}(\sqrt{pq})$5, $K=\mathbb{Q}(\sqrt{pq})$6, and $K=\mathbb{Q}(\sqrt{pq})$7 gives the necessary constraints.

Hasse Unit Index Computation

The central technical advance is a direct square-class computation of $K=\mathbb{Q}(\sqrt{pq})$8. The unit group structure of the biquadratic field $K=\mathbb{Q}(\sqrt{pq})$9 is analysed via explicit presentations, and the index $p\equiv 1\pmod{8}$0 is bounded by $p\equiv 1\pmod{8}$1 through a sequence of norm and square-class arguments.

The precise, explicit description of units in $p\equiv 1\pmod{8}$2, $p\equiv 1\pmod{8}$3, and $p\equiv 1\pmod{8}$4 (the three quadratic subfields of $p\equiv 1\pmod{8}$5 over $p\equiv 1\pmod{8}$6) and their interactions is essential. An analysis of potential norm obstructions ensures that the relevant unit index does not exceed $p\equiv 1\pmod{8}$7.

Implications and Further Directions

The results demonstrate that in "nontrivial" real quadratic fields with prescribed conductor data and splitting behavior of $p\equiv 1\pmod{8}$8, Greenberg's conjecture holds, conditional on explicit quadratic and quartic residue criteria. This represents a substantial advancement in the explicit arithmetic of Iwasawa invariants for fields outside the reach of prior systematic genus-theoretical or cyclotomic approaches.

The work's methodology clarifies that effective capitulation arguments in the split prime case rely upon the detailed governance of unit norm indices, not just class group rank constraints, highlighting a path for further generalizations. The explicit translation of class field-theoretic properties—particularly via the Rédei matrix and Hilbert symbol computation—shows promise for resolving remaining cases in the taxonomy of real quadratic fields where $p\equiv 1\pmod{8}$9 splits.

Prospective research includes extending the approach to related infinite towers, notably those arising from other $q\equiv 9\pmod{16}$0-split scenarios, and obtaining more refined or necessary/sufficient conditions involving higher power residue symbols or additional arithmetic invariants. The techniques might be adapted to refine Kumakawa-type results, making class group criteria fully explicit via further symbol computations.

Conclusion

The paper provides a detailed arithmetic criterion for the vanishing of the Iwasawa $q\equiv 9\pmod{16}$1-invariant in a family of real quadratic fields with split $q\equiv 9\pmod{16}$2, using a sequence of reductions culminating in the control of the Hasse unit index of a biquadratic extension. The argument showcases a deep interplay between genus theory, explicit norm computations, and Iwasawa theory, thereby enhancing our understanding of Greenberg's conjecture's explicit ramifications in the split-prime configuration (2605.09111).