Indefinite Quaternion Algebras

• Indefinite quaternion algebras are four-dimensional central simple algebras over number fields that split at least one infinite place.
• They underpin the structure of Shimura curves, enabling the study of optimal embeddings, Eichler orders, and Hecke operator actions in arithmetic geometry.
• They are vital in modern research, linking algebraic geometry, number theory, and automorphic forms through concrete arithmetic reduction and correspondence methods.

An indefinite quaternion algebra over a number field, most classically over $\mathbb{Q}$, is a central simple algebra of dimension four that is unramified at at least one infinite place. Indefinite quaternion algebras form a fundamental class of arithmetic objects with deep connections to algebraic geometry, number theory, and automorphic forms. Their arithmetic and geometric properties, especially as they relate to Shimura curves and the action of Hecke operators on optimal embeddings, are of central interest in both theoretical and computational arithmetic geometry.

1. Definition and Algebraic Structure

Let $F$ be a totally real field (often $F = \mathbb{Q}$), and let $B$ be a quaternion algebra over $F$. Concretely, for $a, b \in \mathbb{Q}^{\times}$,

$B = \left( \frac{a, b}{\mathbb{Q}} \right)$

is the $\mathbb{Q}$-algebra generated by $i, j$ with relations $i^2 = a$, $F$0, $F$1. For each place $F$2 of $F$3 ($F$4 archimedean or finite), the algebra $F$5 is either isomorphic to $F$6 (split) or to the unique division quaternion algebra over $F$7 (ramified). The set of finite places where $F$8 is ramified, together with the infinite places, fully determines the algebra up to isomorphism. The discriminant $F$9 is the product of the finite places where $F = \mathbb{Q}$0 ramifies.

An algebra $F = \mathbb{Q}$1 is called indefinite if it splits at at least one infinite (archimedean) place, i.e., $F = \mathbb{Q}$2 for some real place $F = \mathbb{Q}$3. This is in contrast to totally definite quaternion algebras, which are division algebras at every archimedean place (Cornut et al., 2010, Rickards, 2021).

2. Eichler Orders and Optimal Embeddings

Within an indefinite quaternion algebra $F = \mathbb{Q}$4 over $F = \mathbb{Q}$5 of discriminant $F = \mathbb{Q}$6, an Eichler order $F = \mathbb{Q}$7 of level $F = \mathbb{Q}$8 is the intersection of two suitably chosen maximal orders, with reduced discriminant $F = \mathbb{Q}$9. For a real quadratic field $B$0, let $B$1 be the order of discriminant $B$2.

An embedding $B$3 is called optimal (of level $B$4) if

$B$5

The set $B$6 of optimal embeddings (up to $B$7-conjugacy, with $B$8 being the norm-one units of $B$9) parametrizes foundational arithmetic and geometric objects associated with $F$0 and its Shimura curve (Rickards, 2021).

3. Shimura Curves, CM Points, and Geodesic Correspondence

Indefinite quaternion algebras are linked to the theory of Shimura curves, which generalize modular curves. Fix $F$1 an indefinite quaternion algebra over $F$2, and identify $F$3. The group $F$4 acts discretely on the upper half-plane $F$5, with quotient

$F$6

the (possibly compact) Shimura curve of discriminant $F$7 and level $F$8. Each optimal embedding class $F$9 corresponds to a primitive hyperbolic element in $a, b \in \mathbb{Q}^{\times}$0, whose axis in $a, b \in \mathbb{Q}^{\times}$1 projects to a closed geodesic $a, b \in \mathbb{Q}^{\times}$2. This establishes a bijective correspondence: $a, b \in \mathbb{Q}^{\times}$3 (Rickards, 2021).

Complex multiplication (CM) points on $a, b \in \mathbb{Q}^{\times}$4 correspond, via double coset parametrizations, to data involving embeddings of quadratic fields into $a, b \in \mathbb{Q}^{\times}$5. For a CM field $a, b \in \mathbb{Q}^{\times}$6 and an $a, b \in \mathbb{Q}^{\times}$7-embedding $a, b \in \mathbb{Q}^{\times}$8, the set of CM points at level $a, b \in \mathbb{Q}^{\times}$9 is

$B = \left( \frac{a, b}{\mathbb{Q}} \right)$0

with $B = \left( \frac{a, b}{\mathbb{Q}} \right)$1 and $B = \left( \frac{a, b}{\mathbb{Q}} \right)$2 (Cornut et al., 2010).

4. Hecke Operators and Generating Series

A natural action of Hecke operators $B = \left( \frac{a, b}{\mathbb{Q}} \right)$3 exists on the set of formal sums of optimal embeddings. For $B = \left( \frac{a, b}{\mathbb{Q}} \right)$4,

$B = \left( \frac{a, b}{\mathbb{Q}} \right)$5

where $B = \left( \frac{a, b}{\mathbb{Q}} \right)$6 counts certain conjugacy classes related to elements of norm $B = \left( \frac{a, b}{\mathbb{Q}} \right)$7 in $B = \left( \frac{a, b}{\mathbb{Q}} \right)$8, with $B = \left( \frac{a, b}{\mathbb{Q}} \right)$9 equivalence classes of optimal embeddings.

Associating to each embedding its closed geodesic, this formalism allows for the construction of generating series via the signed intersection numbers: $\mathbb{Q}$0 where $\mathbb{Q}$1 is the signed intersection form on $\mathbb{Q}$2. The main modularity result asserts that such generating series are classical cuspidal modular forms of weight 2 and level $\mathbb{Q}$3, new at all primes dividing $\mathbb{Q}$4 (Rickards, 2021).

5. Arithmetic Reduction, Liftings, and Correspondence

For an indefinite quaternion algebra $\mathbb{Q}$5 and a finite place $\mathbb{Q}$6 not in the set of ramified places, reduction maps can be defined from the set of CM points on $\mathbb{Q}$7 to supersingular points of definite quaternion algebras $\mathbb{Q}$8. The lifting map $\mathbb{Q}$9, introduced in (Cornut et al., 2010), provides a correspondence between CM points on indefinite quaternion algebras and CM points on associated totally definite algebras $i, j$0, for a suitable set $i, j$1 of places satisfying parity and ramification constraints.

Given Eichler orders $i, j$2, $i, j$3, and fixed conductors, there is a precise arithmetic correspondence:

• The fine and coarse conductors associated to CM points transform by taking the prime-to-$i, j$4 components under $i, j$5.
• The lifting map $i, j$6 is uniquely characterized by certain commutative diagrams involving reduction maps and is both Galois and Hecke equivariant.
• The size of Galois orbits of CM points with fixed conductor is governed by products of local factors and ring class field degrees, enabling equidistribution results (Cornut et al., 2010).

A table summarizing key objects:

Object	Construction/Notation	Role in Arithmetic/Geometry
Indefinite algebra	$i, j$7, $i, j$8	Central simple algebra, $i, j$9
Eichler order	$i^2 = a$0	Intersection of two maximal orders
Optimal embedding	$i^2 = a$1, $i^2 = a$2	Parametrizes closed geodesics/CM points
Shimura curve	$i^2 = a$3	Quotient of upper half-plane
Hecke operator	$i^2 = a$4	Acts on optimal embedding classes
Lifting map	$i^2 = a$5	Correspondence between CM points

6. Classical Examples and Broader Connections

Specializing to $i^2 = a$6 and $i^2 = a$7 recovers the setting of classical modular curves. For example, Heegner points on $i^2 = a$8 correspond via optimal embeddings. Reduction maps at primes inert in $i^2 = a$9 yield supersingular points on the corresponding definite algebra, with bijections established between Heegner points of given conductor reducing to a supersingular locus, and $F$00-conjugacy classes of optimal embeddings $F$01. This encapsulates the classical Deuring-Gross-Zagier correspondence for reductions of CM points (Cornut et al., 2010).

The lifting maps $F$02 unify diverse reduction constructions (including Deligne–Rapoport and Carayol degenerations) within a single framework, systematically accounting for conductor-matching and local orbit counts. These results are vital in analytic and arithmetic applications, such as subconvexity, non-vanishing of toric periods, and the equidistribution of Heegner points. The methodology extends to higher-dimensional Shimura varieties with toric subvarieties whenever a corresponding definite form is available (Cornut et al., 2010).

7. Research Directions and Open Problems

Current research actively investigates the interplay between the arithmetic of indefinite quaternion algebras, Shimura varieties, and automorphic forms:

• The modularity properties of generating series arising from intersection theory on Shimura curves highlight connections to the Langlands program and the arithmetic of L-functions.
• The explicit orbit correspondences obtained via lifting maps fuel advances in $F$03-adic uniformization, equidistribution, and distributional properties of CM points.
• Generalizations to higher dimension and other types of algebras and their associated Shimura varieties remain active avenues for development, especially in the context of special cycles and higher automorphic forms (Cornut et al., 2010, Rickards, 2021).