[Paper Review] Darmon's points and quaternionic Shimura varieties

Jérôme Gärtner|arXiv (Cornell University)|Apr 17, 2011

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper generalizes Darmon's conjectural construction of Stark-Heegner points to non-ATR quadratic extensions using quaternionic Shimura varieties, establishing an Abel-Jacobi map via holomorphic differential forms and conjecturing a Gross-Kohnen-Zagier-type formula linking special values of L-functions to the heights of these points. The key contribution is a refined Gross-Zagier conjecture for Darmon's points in an adelic, non-CM setting.

ABSTRACT

In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.

Motivation & Objective

• To extend Darmon's conjectural construction of Stark-Heegner points beyond the ATR (anti-torsion) condition, allowing for general quadratic extensions K/F with K not CM.
• To define a generalized Abel-Jacobi map using holomorphic differential forms on quaternionic Shimura varieties associated to a Hilbert modular form via the Jacquet-Langlands correspondence.
• To conjecture a Gross-Zagier-type formula relating the height of Darmon's points to special values of Rankin-Selberg L-functions.
• To establish a connection with Kudla's theory of cycles on orthogonal Shimura varieties, providing a geometric interpretation of the construction.
• To formulate a conjecture linking the arithmetic of Darmon's points to Fourier coefficients of Hilbert modular forms of half-integral weight.

Proposed method

• Construct a quaternionic Shimura variety Sh_H(G,X) associated to a totally real field F, a quadratic extension K/F, and a quaternion algebra B/F that splits at r real places and ramifies at d−r places.
• Define a real cycle T_b of dimension r−1 on Sh_H(G,X)(C) via the action of (K⊗R)^×_+/(F⊗R)^× on (C\R)^r, and lift it to a complex r-cycle Δ_b with boundary proportional to T_b.
• Use Matsushima and Shimura's theorem to ensure the existence of such a cycle Δ_b with ∂Δ_b ∈ Z·T_b.
• Attach a holomorphic differential form ω_φ of degree r to a Hilbert modular eigenform φ on B via the Jacquet-Langlands correspondence.
• Introduce a character β of the connected components of (K⊗R)^×_+/(F⊗R)^× to define a modified differential form ω_φ^β whose periods are conjecturally a lattice homothetic to the Néron lattice of E.
• Define a point P_b^β ∈ E(C) as the image under the Weierstrass uniformization of the complex number ∫_{Δ_b} ω_φ^β, independent of the choice of Δ_b.

Experimental results

Research questions

• RQ1 Can Darmon’s construction of Stark-Heegner points be generalized beyond the ATR condition, removing the narrow class number one and ATR assumptions?
• RQ2 What is the precise geometric and arithmetic role of quaternionic Shimura varieties in the construction of special points on elliptic curves over totally real fields?
• RQ3 How do the periods of the differential form ω_φ^β relate to the Néron lattice of the elliptic curve E/F?
• RQ4 Is there a Gross-Kohnen-Zagier-type formula that relates the height of Darmon’s points to special values of L-functions?
• RQ5 Can the arithmetic of Darmon’s points be linked to Fourier coefficients of Hilbert modular forms of half-integral weight?

Key findings

• The construction defines a point P_b^β ∈ E(C) as the image of ∫_{Δ_b} ω_φ^β under the Weierstrass uniformization, independent of the choice of cycle Δ_b.
• The paper conjectures that P_b^β lies in E(K^{ab}) and transforms under the Galois action via β(a_∞), generalizing Darmon’s original conjecture.
• A conjectural Gross-Zagier formula is proposed, relating the height of P_b^β to the central value L'(π×χ, 1/2), where χ is a Hecke character.
• The paper establishes a relation to Kudla’s theory of cycles on orthogonal Shimura varieties, suggesting a geometric interpretation of the construction.
• For rank 1 elliptic curves, the integers [P_t] measuring the size of Darmon’s points are conjectured to be proportional to Fourier coefficients of a Hilbert modular form of weight 3/2.
• The paper conjectures that [P_t] are proportional to square roots of L(E_{-D_0t}, 1), analogously to the Gross-Kohnen-Zagier theorem.