Interpolation of generalized Heegner classes along quaternionic Coleman families

1. Problem Statement and Context

The paper addresses a central problem in arithmetic geometry: the construction of Euler systems to study the Bloch–Kato conjecture for Galois representations attached to modular forms.

• Background: The Bloch–Kato conjecture relates the order of vanishing of $L$-functions to the size of Selmer groups. A primary tool for bounding these groups is the construction of Euler systems (norm-compatible families of cohomology classes).
• Existing Work:
  • For elliptic curves ($GL_2$), Heegner points provide such systems (Kolyvagin).
  • For higher weight modular forms, Generalized Heegner Cycles (Bertolini–Darmon–Prasanna, [BDP13]) were introduced.
  • Interpolation: Howard, Büyükboduk–Lei, and Jetchev–Loeffler–Zerbes ([JLZ21]) successfully interpolated these classes along Hida families (ordinary, slope 0) and Coleman families (finite slope) of elliptic modular forms, creating "Big Heegner classes."
• The Gap: While the theory of generalized Heegner cycles exists for quaternionic modular forms (associated with Shimura curves and indefinite quaternion algebras, as developed by Brooks [HB15]), the p-adic interpolation of these classes along Coleman families in the quaternionic setting had not been established.
• Objective: To construct Big Generalized Heegner Classes by interpolating p-adically the generalized Heegner classes associated with quaternionic modular forms along a Coleman (finite slope) family, adapting the methods of [JLZ21] to the indefinite quaternionic setting.

2. Methodology

The author employs a geometric and cohomological approach, bridging the gap between classical modular forms and p-adic families via the theory of motives and Shimura varieties.

A. The Geometric Setup

• Indefinite Quaternionic Setting: The work takes place over an indefinite quaternion algebra $B$ of discriminant $N^- > 1$. The modular forms are defined on Shimura curves $\tilde{X}_m$ (classifying QM abelian surfaces) rather than classical modular curves.
• Generalized Kuga–Sato Varieties: The author defines the generalized Kuga–Sato variety $W_{k,m}$ as a fiber product of the universal abelian surface $\tilde{A}_m$ over the Shimura curve.
• Generalized Heegner Cycles: Using an isogeny $\phi$ between a fixed CM abelian surface $A$ and a surface $A_{cp^n}$, the author constructs a cycle $\Delta^{[k]}_{\phi, m}$ in the Chow group of $W_{k,m}$.
• Lieberman's Trick: To simplify the cohomology, the author uses Lieberman's trick to map the cycle from the high-dimensional variety $W_{k,m}$ to the Shimura curve $\tilde{X}_m$, at the cost of working with a more complex coefficient sheaf involving symmetric tensors of the étale realization of the abelian surface.

B. The Interpolation Strategy

The core methodology follows the Jetchev–Loeffler–Zerbes approach:

1. Basis Vectors from Heegner Points: Instead of interpolating the cycles directly, the author interpolates the basis vectors in the representation space associated with the motive. These vectors are constructed from Heegner points on the Shimura curves.
2. Overconvergent Projectors: The author utilizes the Loeffler–Zerbes overconvergent projector ($\Pi_j$) to interpolate the basis vectors $e^{[k,j]}_{cp^n}$ (which depend on the weight $k$) into a distribution-valued vector $e^{[U,j]}_{cp^n}$ defined over the weight space $U$.
3. Big Gysin Map: A "Big Gysin map" is constructed to push forward these interpolated basis vectors into the cohomology of the Shimura curve, yielding the Big Generalized Heegner Classes.
4. Distribution Modules: The interpolation is performed within modules of distributions ($D_U$) that interpolate the symmetric tensor powers of the Galois representations.

C. The p-adic Jacquet–Langlands Correspondence

A crucial technical step is the use of the p-adic Jacquet–Langlands correspondence (Proposition 4.11). This establishes an isomorphism between the Coleman–Mazur eigencurve for elliptic modular forms and the Brasca–Buzzard eigencurve for quaternionic modular forms. This allows the author to transfer the existence of Coleman families and their properties from the elliptic case to the quaternionic case.

3. Key Contributions

1. Construction of Big Classes: The paper successfully defines the Big Generalized Heegner Class $z^{[F,j]}_{cp}$ associated to a Coleman family $F$ of quaternionic modular forms. This class lives in the Galois cohomology group $H^1(F_{cp}, V^*_F \otimes \dots)$, where $V^*_F$ is the "Big" Galois representation interpolating the dual representations of the specializations of $F$.
2. Explicit Reciprocity Law (Specialization): The paper proves that these big classes specialize correctly to classical generalized Heegner classes. Specifically, for a classical weight $k$ in the family, the specialization of the big class is proportional to the classical class $z^{[F_k, \chi]}_{cp^n, 1}$, scaled by a binomial factor $\binom{2r}{j}^{-1}$ and the eigenvalue of the $U_p$ operator.
3. Euler System Properties: The author establishes that the constructed classes satisfy norm-compatibility relations (Proposition 5.17). Specifically, the corestriction of the class at level $n+1$ to level $n$ is related to the class at level $n$ by the action of the dual Hecke operator $U'_p$ (which acts as the eigenvalue $a_p$). This confirms that the big classes form an Euler system.
4. Field of Definition: The paper refines the field of definition of these classes, showing they can be defined over the base field $K$ (or ring class fields) rather than just the larger fields required for the intermediate steps, by utilizing the universal character of the weight space.

4. Main Results

Theorem 5.21 (Specialization of Big Classes):
Let $F$ be a Coleman family passing through a p-stabilized quaternionic modular form $F_0$. Let $z^{[F,j]}_{cp}$ be the associated Big Generalized Heegner class. For any classical weight $k = 2r+2$ in the family and a Hecke character $\chi$ of infinity type $(2r-j, j)$, the specialization is:
$$\text{Spec}_k(z^{[F,j]}_{cp}) = a_p(F_k)^{-n} \binom{2r}{j}^{-1} z^{[F_k, \chi]}_{cp^n, 1}$$
where $a_p(F_k)$ is the $U_p$-eigenvalue of the specialization $F_k$.

Proposition 5.17 (Euler Relations):
The classes satisfy the norm compatibility:
$$\text{Norm}_{F_{cp^{n+1}}/F_{cp^n}}(z^{[F,j]}_{cp^{n+1}, 1}) = a_p \cdot z^{[F,j]}_{cp^n, 1}$$

5. Significance

• Extension of the Bloch–Kato Conjecture Toolkit: This work extends the powerful machinery of Euler systems from the elliptic setting to the quaternionic setting. This is vital for studying $L$-functions and Selmer groups of motives that do not arise from elliptic curves but from quaternionic modular forms (e.g., certain abelian surfaces with Quaternionic Multiplication).
• Foundation for Explicit Reciprocity Laws: The construction of these big classes is the necessary first step to proving an Explicit Reciprocity Law in the quaternionic setting. Such a law would relate the big Heegner classes to p-adic L-functions (specifically, BDP p-adic L-functions) interpolated along the Coleman family. The author notes that this reciprocity law is the subject of future work.
• Unification of Settings: By adapting the [JLZ21] approach to the indefinite quaternionic case, the paper demonstrates the robustness of the "Big Class" methodology, showing it applies even when the underlying geometry involves Shimura curves and QM abelian surfaces rather than modular curves.
• Technical Advancement: The paper provides a rigorous treatment of the "dimension halving" required in the quaternionic setting (via idempotents $e$ and $\bar{e}$) and the handling of the specific arithmetic of the ring class fields in the context of p-adic families.

In summary, this paper successfully bridges the gap between classical quaternionic Heegner cycles and p-adic families, providing the essential geometric objects (Big Generalized Heegner Classes) required to attack the Bloch–Kato conjecture for a broader class of automorphic representations.