[Paper Review] On the Rasmussen-Tamagawa conjecture for QM-abelian surfaces

Keisuke Arai|arXiv (Cornell University)|Nov 3, 2012

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper proves the Rasmussen-Tamagawa conjecture for QM-abelian surfaces over number fields of higher degree, generalizing previous results over imaginary quadratic fields. It establishes finiteness of isomorphism classes of QM-abelian surfaces with bounded ramification under specific Galois-theoretic and quaternion algebra conditions, using local-global principles and class field theory.

ABSTRACT

In the previous article, we showed the Rasmussen-Tamagawa conjecture for QM-abelian surfaces over imaginary quadratic fields. In this article, we generalize the previous work to QM-abelian surfaces over number fields of higher degree. We also give several explicit examples.

Motivation & Objective

• To extend the Rasmussen-Tamagawa conjecture on finiteness of abelian varieties with bounded ramification to number fields of degree greater than 2.
• To establish finiteness of isomorphism classes of QM-abelian surfaces over Galois extensions of Q not containing Hilbert class fields of imaginary quadratic fields.
• To provide explicit criteria involving splitting primes and non-isomorphism of quaternion algebras over quadratic extensions.
• To verify the conjecture under conditions ensuring the existence of suitable primes and local solubility of associated quadratic forms.

Proposed method

• Use of the maximal pro-p extension unramified outside p to control the Galois representation of p-power torsion points.
• Application of the Hasse principle to reduce global solubility of quadratic forms to local solubility at all places.
• Employment of class field theory and Galois descent to analyze the structure of the extension K(A[p^∞]) over K(μ_p).
• Construction of explicit examples via computation of ramification indices, residual degrees, and number of primes above a given prime in number fields.
• Use of Hensel’s lemma and p-adic analysis to determine local solubility of equations defining the moduli space of QM-abelian surfaces.
• Verification of non-isomorphism between B⊗Q(√−q) and M2(Q(√−q)) using splitting behavior of primes in quadratic extensions.

Experimental results

Research questions

• RQ1 Under what conditions on a number field K is the set of K-isomorphism classes of QM-abelian surfaces with bounded ramification finite?
• RQ2 Does the Rasmussen-Tamagawa conjecture hold for QM-abelian surfaces over Galois extensions of Q of degree >2?
• RQ3 Can the finiteness result be established when K does not contain the Hilbert class field of any imaginary quadratic field and a suitable prime q splits completely in K?
• RQ4 What role does the non-splitting of the quaternion algebra B over Q(√−q) play in ensuring finiteness of the moduli space?
• RQ5 How can local solubility of the defining equations for QM-abelian surfaces be used to deduce global finiteness?

Key findings

• The set 𝒜(K,2)_B of K-isomorphism classes of QM-abelian surfaces over a Galois extension K of Q is finite if K does not contain the Hilbert class field of any imaginary quadratic field and there exists a prime q splitting completely in K such that B⊗Q(√−q) is not isomorphic to M2(Q(√−q)).
• For K = Q(√3,√−5), Q(ζ5), or Q(ζ17), and d ∈ {6,10,22}, the set M^B(K) is infinite unless (d,K)=(22, Q(ζ5)), implying infinitely many K-isomorphism classes of QM-abelian surfaces.
• When (d,K) ≠ (22, Q(ζ5)), the quaternion algebra B⊗Q K is isomorphic to M2(K), ensuring that the moduli space has rational points over K.
• The field K = Q(√3,√−5) is not the Hilbert class field of any imaginary quadratic field, as its subfields Q(√−5) and Q(√−15) have class number >1 and the extension is ramified over primes above 3 and 2.
• For K = Q(ζ5) and K = Q(ζ17), the only quadratic subfields are Q(√5) and Q(√17), respectively, neither of which has class number one, so K does not contain any Hilbert class field.
• The existence of a prime q splitting completely in K with B⊗Q(√−q) ≇ M2(Q(√−q)) ensures that the associated Galois representations satisfy the required filtration condition, enabling the application of the main finiteness theorem.