Shimura Curves in Arithmetic Geometry

• Shimura curves are higher-genus algebraic curves defined as arithmetic quotients of quaternion algebras over totally real fields, with moduli interpretations for abelian surfaces with QM.
• They bridge concepts in modular forms, automorphic representations, and Galois representations through explicit uniformization and Hecke operator techniques.
• Their study uses analytic methods like Schwarzian ODEs and Atkin–Lehner involutions to derive finiteness results and applications in special cycles and Diophantine geometry.

Shimura curves are a distinguished class of higher-genus algebraic curves arising as arithmetic quotients associated to quaternion algebras over totally real fields, with deep connections to the theory of modular forms, automorphic representations, abelian varieties with quaternionic multiplication, and arithmetic geometry. They unify non-classical analogs of modular curves and play a central role in number theory, moduli problems, and arithmetic geometry, including explicit and uniformization theory, moduli of abelian varieties, and arithmetic applications such as the study of special cycles, Galois representations, and Diophantine conjectures.

1. Definition and Moduli Interpretation

Let $F$ be a totally real field and $B$ an “almost indefinite” quaternion $F$-algebra: $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$ at precisely one real embedding $\tau: F \to \mathbb{R}$ and a division algebra at all other real places. For any $\mathcal{O} \subset B$ (order) and compact open $K \subset B^\times(\mathbb{A}_f)$, the Shimura curve is defined as

$$X_K = \text{compactification of } \left( B^+(\mathbb{R}) \setminus \mathbb{H} \right) \times B^\times(\mathbb{A}_f)/K,$$

where $B^+(\mathbb{R})$ denotes elements of positive reduced norm at $\tau$ and $B$0 is the upper half-plane. The complex points satisfy

$B$1

and possess a canonical model over $B$2.

For definite $B$3, $B$4 parametrizes abelian surfaces $B$5 with a ring embedding $B$6 and level structure determined by $B$7.

Specializing to $B$8 and $B$9 indefinite over $F$0 (discriminant $F$1), the associated curves $F$2 parametrize abelian surfaces with QM by an Eichler order of level $F$3, and a cyclic subgroup $F$4 of order $F$5 stable under $F$6.

2. Automorphic Characterization and Finiteness

A major recent advance is the finiteness conjecture for modified diagonal cycles: for any rational point class $F$7 of degree $F$8 and Gross–Schoen modified diagonal $F$9, only finitely many congruence Shimura curves $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$0 of positive genus have $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$1 in $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$2. This is equivalent to the automorphic vanishing criterion: $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$3 is “good” if, for all triples $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$4 of irreducible, holomorphic, weight-2 automorphic representations in $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$5,

$$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$6

which implies vanishing of $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$7 (Qiu, 2023).

A refinement leverages Prasad’s trilinear form theorem, connecting such vanishing to the vanishing of local trilinear forms and root numbers via explicit representation-theoretic criteria at ramified and level-dividing places.

The finiteness results (for both quaternion algebra $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$8 and level $$B \otimes_{F, \tau} \mathbb{R} \cong M_2(\mathbb{R})$$9) use gonality bounds (Li–Yau–Zograf) and combinatorial analysis of Atkin–Lehner and genus 0 quotients, reducing to explicit finite cases (Qiu, 2023).

3. Uniformization, Moduli, and Explicit Models

Shimura curves admit canonical models defined over number fields using moduli problems of abelian varieties with quaternionic multiplication and level structure, as outlined above.

Explicit complex uniformization affords powerful analytic descriptions. In the classical case, modular curves $\tau: F \to \mathbb{R}$0 admit $\tau: F \to \mathbb{R}$1-expansions at cusps. In contrast, Shimura curves for $\tau: F \to \mathbb{R}$2 are compact, have no cusps, and uniformization uses the action of norm-one elements of an Eichler order on $\tau: F \to \mathbb{R}$3. The group of Atkin–Lehner involutions—indexed by exact divisors of $\tau: F \to \mathbb{R}$4—permutes these curves and leads to further moduli-theoretic identifications (Yang, 2012, Mercuri et al., 21 Jul 2025).

Explicit equations exist for particular Shimura curves; e.g., the discriminant $\tau: F \to \mathbb{R}$5 Shimura curve is presented as the affine intersection $\tau: F \to \mathbb{R}$6 over $\tau: F \to \mathbb{R}$7 (Lawson, 2013). Similarly, models in weighted projective space using icosahedral invariants relate certain Shimura curves to Hilbert modular and $\tau: F \to \mathbb{R}$8 moduli (Nagano, 2015).

4. Automorphic Forms and Hecke Theory

The space of holomorphic weight $\tau: F \to \mathbb{R}$9 automorphic forms on a Shimura curve $\mathcal{O} \subset B$0 is described via automorphic representations of the underlying quaternionic group. Hecke operators on $\mathcal{O} \subset B$1 are defined using double coset expressions, paralleling the classical case but requiring fundamentally different analytic techniques because of the lack of $\mathcal{O} \subset B$2-expansions.

Key advances include:

• Schwarzian ODE methods, supplying bases for automorphic forms and Hauptmodul (local parameter) calculations for genus zero curves.
• Evaluations of modular forms at CM (Heegner) points, integral over the chosen quaternionic order, enabling the computation of special values, and explicit modular parametrizations of elliptic curves.
• Jacquet–Langlands correspondence, transferring eigenvalue data between classical newforms and automorphic forms on Shimura curves, connecting representations on $\mathcal{O} \subset B$3 to those on quaternionic groups (Yang, 2012, Nelson, 2012).

Table: Comparison of Modular Curves and Shimura Curves

Feature	Modular Curves	Shimura Curves
Uniformization	$\mathcal{O} \subset B$4 with cusps	$\mathcal{O} \subset B$5 (compact)
Field of moduli	$\mathcal{O} \subset B$6, cyclotomic	Typically totally real $\mathcal{O} \subset B$7
Moduli of	Elliptic curves (plus level structure)	False elliptic curves (abelian surfaces with QM)
Automorphic forms	Classical modular forms	Automorphic forms on quaternion algebras
Atkin–Lehner involutions	Yes	Yes
$\mathcal{O} \subset B$8-expansions	Available	Not available

5. Arithmetic Applications and Special Cycles

Shimura curves serve as essential tools in several major arithmetic directions:

• Special cycles and diagonal cycles: Gross–Schoen cycles and their vanishing connect to the geometry of triple products, yielding finiteness statements that generalize Ogg's classical hyperelliptic curve classification to new “non-classical” curves, such as the genus $\mathcal{O} \subset B$9 $K \subset B^\times(\mathbb{A}_f)$0 and the genus $K \subset B^\times(\mathbb{A}_f)$1 Wiman curve $K \subset B^\times(\mathbb{A}_f)$2 (Qiu, 2023).
• Point-counts and automorphisms: Algorithms based on Ribet's isogeny and Eichler–Shimura theory enable effective point-counts on $K \subset B^\times(\mathbb{A}_f)$3 and their quotients, allowing identification of cases of maximum or optimal point-counts and complete characterization of automorphism groups as Atkin–Lehner except for explicit low-level exceptions (Mercuri et al., 21 Jul 2025).
• Gonality and Diophantine applications: Gonality bounds, such as Abramovich’s uniform lower bound, enable precise classification of possible low-degree maps (e.g., tetragonal curves) and exclude high-genus Shimura curves from violating conjectures like the bounded negativity conjecture for surfaces (Moeller et al., 2014, Mercuri et al., 21 Jul 2025).
• Shimura curves over positive characteristic: Via crystalline Hodge theory and explicit monodromy data, precise criteria for a curve in characteristic $K \subset B^\times(\mathbb{A}_f)$4 to be the reduction of a Shimura curve of Hodge (Mumford) type have been established, based on crystalline cycle-class conditions, maximal Higgs fields, and tensor decompositions of Dieudonné crystals (Xia, 2014, Xia, 2013).

6. Shimura Curves in Moduli and Prym Loci

Shimura curves are subject to strong constraints within the moduli of curves and abelian varieties:

• Torelli locus: It is conjectured (Oort’s conjecture) and proven for various classes that no Shimura curve (e.g., of Mumford type) lies generically in the Torelli locus $K \subset B^\times(\mathbb{A}_f)$5 for $K \subset B^\times(\mathbb{A}_f)$6, and specifically none exist in the hyperelliptic Torelli locus for $K \subset B^\times(\mathbb{A}_f)$7 except in explicit low-genus cases. When present, they correspond to families with strictly maximal Higgs fields; explicitly constructed examples exist for $K \subset B^\times(\mathbb{A}_f)$8 (Lu et al., 2013, Grushevsky et al., 2013).
• Prym loci: Systematic computer algebra methods yield all Shimura curves generically contained in unramified Prym loci (43 cases for $K \subset B^\times(\mathbb{A}_f)$9) and ramified Prym loci (9 families for Prym-dimension $$X_K = \text{compactification of } \left( B^+(\mathbb{R}) \setminus \mathbb{H} \right) \times B^\times(\mathbb{A}_f)/K,$$0). These are identified via fixed group actions and explicit invariants; most lie outside the Torelli locus (Colombo et al., 2017, Frediani et al., 2020).
• Jacobians of higher genus: Infinite families of Shimura curves generically contained in the locus of Jacobians of genus $$X_K = \text{compactification of } \left( B^+(\mathbb{R}) \setminus \mathbb{H} \right) \times B^\times(\mathbb{A}_f)/K,$$1 or $$X_K = \text{compactification of } \left( B^+(\mathbb{R}) \setminus \mathbb{H} \right) \times B^\times(\mathbb{A}_f)/K,$$2 curves have been explicitly constructed using cyclic covers and period matrix calculations, establishing new richness in totally geodesic curves with CM fixed parts (Grushevsky et al., 2015, Grushevsky et al., 2013).

7. Higher-Dimensional Shimura Varieties and Geodesic Curves

Shimura curves naturally embed as totally geodesic curves on Shimura surfaces (Hilbert modular or Picard modular surfaces). Recent work gives precise parameterizations of their commensurability classes:

• On Shimura surfaces $$X_K = \text{compactification of } \left( B^+(\mathbb{R}) \setminus \mathbb{H} \right) \times B^\times(\mathbb{A}_f)/K,$$3 ($$X_K = \text{compactification of } \left( B^+(\mathbb{R}) \setminus \mathbb{H} \right) \times B^\times(\mathbb{A}_f)/K,$$4 a Hermitian symmetric domain), totally geodesic curves correspond to $$X_K = \text{compactification of } \left( B^+(\mathbb{R}) \setminus \mathbb{H} \right) \times B^\times(\mathbb{A}_f)/K,$$5-Fuchsian subgroups arising from quadratic subfields and quaternion algebra data.
• There exist infinitely many pairwise incommensurable geodesic curves on such surfaces; their construction is parametrized by suitable quaternion algebras over quadratic subfields, via restriction of scalars and compatibility of base extensions (Chinburg et al., 2015).
• The supply of incommensurable totally geodesic Shimura curves has consequences in arithmetic geometry, including the generation of the fundamental group, the structure of algebraic cycles, and Diophantine “unlikely intersection” problems.

• (Qiu, 2023) C. Qiu, "Finiteness properties for Shimura curves and modified diagonal cycles" (2023)
• (Mercuri et al., 21 Jul 2025) Auricchio, Mercuri, Padurariu, Saia, Stirpe, "Point counts, automorphisms, and gonalities of Shimura curves" (2025)
• (Yang, 2012) Yang, "Computing modular equations for Shimura curves" (2012)
• (Nelson, 2012) Nelson, "Evaluating modular forms on Shimura curves" (2012)
• (Lu et al., 2013) Lu, Zuo, "On Shimura curves in the Torelli locus" (2013)
• (Chinburg et al., 2015) Chinburg, Stover, "Geodesic curves on Shimura surfaces" (2015)
• (Moeller et al., 2014) Möller, Toledo, "Bounded negativity of self-intersection numbers of Shimura curves on Shimura surfaces" (2014)
• (Grushevsky et al., 2013) Grushevsky, Möller, "Shimura curves within the locus of hyperelliptic Jacobians in genus three" (2013)
• (Grushevsky et al., 2015) Grushevsky, Möller, Salvati Manni, "Explicit formulas for infinitely many Shimura curves in genus 4" (2015)
• (Colombo et al., 2017) Colombo, Frediani, Hulek, Penegini, "Shimura curves in the Prym locus" (2017)
• (Frediani et al., 2020) Frediani, Grosselli, "Shimura curves in the Prym loci of ramified double covers" (2020)
• (Xia, 2013, Xia, 2014) Xia, "Crystalline Hodge cycles and Shimura curves," and "l-adic Monodromy and Shimura curves in positive characteristics" (2013, 2014)
• (Nagano, 2015) Nagano, "Icosahedral invariants and Shimura curves" (2015)
• (Lawson, 2013) Lawson, "The Shimura curve of discriminant 15 and topological automorphic forms" (2013)
• (Bayer et al., 2011) Bayer, Blanco-Chacón, "Quadratic modular symbols on Shimura curves" (2011)
• (Pasten, 2017) Baker et al., "Shimura curves and the abc conjecture" (2017)

These references collectively establish the foundational, structural, and computational properties of Shimura curves, their crucial role as bridges between geometry and arithmetic, and their singular position in the modern theory of automorphic forms, arithmetic geometry, and the explicit construction and analysis of higher-genus curves with controlled moduli and arithmetic properties.