[Paper Review] Algebraic points on Shimura curves of $\Gamma_0(p)$-type (III)

Keisuke Arai, Fumiyuki Momose|arXiv (Cornell University)|May 16, 2012

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper proves the non-existence of rational points on Shimura curves of Γ₀(p)-type over certain number fields for sufficiently large primes p, under mild conditions involving the absence of Hilbert class fields of imaginary quadratic fields and local non-splitting conditions on quaternion algebras. The key contribution is a generalization of classical non-existence results for modular curves to the Shimura curve setting, with explicit examples provided.

ABSTRACT

In previous articles, we classified the characters associated to algebraic points on Shimura curves of $\\Gamma_0(p)$-type, and over number fields in a certain large class we showed that there are at most elliptic points on such a Shimura curve for every sufficiently large prime number $p$. In this article, we prove the non-existence of elliptic points on Shimura curves of $\\Gamma_0(p)$-type under a mild assumption. We also give an explicit example.

Motivation & Objective

• To extend non-existence results for rational points on modular curves to Shimura curves of Γ₀(p)-type.
• To resolve the open question of whether elliptic points can exist on such Shimura curves over number fields.
• To establish conditions under which M₀ᴮ(p)(k) = ∅ for large primes p and number fields k.
• To provide an explicit example of a curve with no rational points under the given conditions.
• To clarify the role of quaternion algebras and Galois representations in obstructing rational points on these curves.

Proposed method

• Uses Galois representations associated to QM-abelian surfaces to analyze the action on p-torsion points.
• Applies the mod p cyclotomic character and character decomposition to study the structure of the Galois image.
• Employs the non-splitting condition of B⊗ℚ(√−q) ≇ M₂(ℚ(√−q)) to rule out rational points.
• Analyzes local behavior at primes of odd degree and unramified residual characteristics to derive global obstructions.
• Uses class field theory and Hilbert class field obstructions to rule out the existence of rational points.
• Applies explicit computations in specific cases (d=6, d=22) to verify the conditions and illustrate the theorem.

Experimental results

Research questions

• RQ1 Under what conditions do Shimura curves of Γ₀(p)-type have no rational points over a number field k for large p?
• RQ2 Can elliptic points exist on M₀ᴮ(p) over number fields when the base field avoids Hilbert class fields of imaginary quadratic fields?
• RQ3 How does the non-splitting of the quaternion algebra B⊗ℚ(√−q) obstruct rational points on M₀ᴮ(p)?
• RQ4 What is the role of the Galois representation ρ̄A,p in determining the existence of rational points on these curves?
• RQ5 Can explicit examples of such curves with no rational points be constructed under the given assumptions?

Key findings

• For any finite Galois extension k/ℚ not containing the Hilbert class field of any imaginary quadratic field, and satisfying mild local conditions, M₀ᴮ(p)(k) = ∅ for all sufficiently large primes p.
• The non-splitting condition B⊗ℚ(√−q) ≇ M₂(ℚ(√−q)) at a prime q of odd degree in k is sufficient to ensure the non-existence of rational points on M₀ᴮ(p)(k) for large p.
• Explicit examples are constructed for d=6 and d=22, where the conditions of the theorem are verified and M₀ᴮ(p)(k) = ∅ holds for p > 128.
• The curve Mᴮ has infinitely many rational points over k when d=6 or d=22, but M₀ᴮ(p) has no rational points over k for large p.
• The minimal norm of a prime of odd degree satisfying the non-splitting condition is 32, which bounds the range of p for which the theorem applies.