[Paper Review] On the Frey-Mazur conjecture over low genus curves

Benjamin Bakker, Jacob Tsimerman|arXiv (Cornell University)|Sep 25, 2013

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes a geometric analog of the Frey-Mazur conjecture for fake elliptic curves—abelian surfaces with quaternionic multiplication—over function fields of low-genus curves. By analyzing the geometry of modular surfaces $ Z^D(p) $ and using Riemann-Hurwitz bounds on curve genus, it proves that for sufficiently large $ p $, any two such abelian surfaces with isomorphic $ p $-torsion Galois representations are $ \mathcal{O}_D $-isogenous, provided the base curve has genus less than a fixed $ k $.

ABSTRACT

The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their $p$-torsion Galois representations is one-to-one over function fields of small genus complex curves for sufficiently large $p$ relative to the genus.

Motivation & Objective

• To establish a geometric analog of the Frey-Mazur conjecture in the context of abelian surfaces with quaternionic multiplication over function fields.
• To show that for sufficiently large $ p $, isomorphism of $ p $-torsion Galois representations implies $ \mathcal{O}_D $-isogeny over low-genus curves.
• To analyze the geometry of the modular surface $ Z^D(p) $ and its Hecke divisors to rule out non-Hecke curves of low genus.
• To bound the genus of curves in $ X^D(p) \times X^D(p) $ via Riemann-Hurwitz and CM point incidence, leveraging results from Hwang-To and Andre-Deligne.
• To demonstrate that the constant $ N $ in the main theorem depends only on the gonality of the base curve, extending prior results.

Proposed method

• Lift a curve $ B \to Z^D(p) $ to a curve $ C \to X^D(p) \times X^D(p) $, where $ X^D(p) $ parametrizes abelian surfaces with $ \mathcal{O}_D $-action and level structure.
• Apply Riemann-Hurwitz to the projection $ \pi: C \to X^D(p)_0 $ to derive a lower bound on $ g(C) $ in terms of the canonical divisor $ K_{X^D(p)_0 \times X^D(p)_0} $.
• Use the Zariski density of monodromy for non-Hecke curves (via Andre-Deligne) to show that such curves must have large bidegree, implying large genus.
• Bound the ramification divisor of $ C \to B $ by analyzing incidence at singular points of $ X^D(p) \times X^D(p) \to Z^D(p) $, particularly Heegner and anti-Heegner CM points.
• Apply hyperbolic metric estimates to show that Heegner CM points are well-separated except near low-degree Hecke curves, and use volume bounds from Hwang-To to constrain curve incidence.
• Combine genus lower bounds from projections and upper bounds from CM multiplicity to derive a contradiction for large $ p $, proving all such curves must lie on Hecke divisors.

Experimental results

Research questions

• RQ1 For a smooth quasiprojective complex curve $ B $ of genus $ g < k $, when are two $ \mathcal{O}_D $-module isomorphic $ p $-torsion local systems on abelian surfaces over $ B $ induced by $ \mathcal{O}_D $-isogenies?
• RQ2 What is the asymptotic behavior of the genus of curves in $ X^D(p) \times X^D(p) $ that do not factor through Hecke curves, as $ p \to \infty $?
• RQ3 How does the incidence of curves with high multiplicity at CM points (Heegner and anti-Heegner) constrain the geometry of $ Z^D(p) $?
• RQ4 Can the constant $ N $ in the main theorem be taken to depend only on the gonality of $ B $, rather than its genus?
• RQ5 To what extent do the techniques used for compact Shimura curves extend to non-compact or higher-genus cases?

Key findings

• For any $ k > 0 $, there exists $ N > 0 $ such that any smooth curve $ B \to Z^D(p) $ of genus $ g(B) < k $ must factor through a Hecke divisor if $ p > N $, proving the main theorem.
• The genus of a curve $ C \subset X^D(p)_0 \times X^D(p)_0 $ lying over a non-Hecke curve $ B \to Z^D(p) $ is bounded below by a linear function of $ C \cdot K_{X^D(p)_0 \times X^D(p)_0} $, while the upper bound grows asymptotically slower due to $ \mathrm{mult}_{\mathrm{CM}}(C) = o(C \cdot K) $.
• Heegner CM points are well-separated under the hyperbolic metric, except near low-degree Hecke curves, which allows volume-based constraints on curve incidence.
• The absence of Hecke curves through anti-Heegner CM points is overcome by proving analogous volume bounds for conjugate Hecke curves.
• The constant $ N $ in the main theorem can be taken to depend only on the gonality of $ B $, as shown via extension of techniques from [BT14].
• The result holds over $ \overline{\mathbb{F}}_\ell $ for sufficiently large $ \ell $, by standard specialization arguments.