Indefinite Quaternion Algebra

• Indefinite quaternion algebra is a four-dimensional central simple algebra over Q, unramified at the infinite place and classified by its finite discriminant.
• Maximal and Eichler orders in these algebras structure ideals and norms, underpinning computational techniques in arithmetic geometry.
• Shimura curves and Hecke operators connect the algebra to modular forms, enabling explicit methods for computing CM points and optimal embeddings.

An indefinite quaternion algebra is a central simple algebra of dimension four over a number field, typically $\Q$, characterized by its ramification properties at archimedean and non-archimedean places. Specifically, a quaternion algebra $B$ over $\Q$ is called indefinite if it is unramified at the infinite (real) place, that is, $B \otimes_\Q \R \simeq M_2(\R)$. Such algebras are of fundamental importance in arithmetic, automorphic forms, the theory of Shimura curves, and computational number theory.

1. Algebraic Structure and Classification

A quaternion algebra over $\Q$ is an associative algebra of the form

$B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$

for $a, b \in \Q^\times$. The set of ramified places $\Ram(B)$ of a quaternion algebra $B$ consists of finite primes and possibly the real place, characterized by the Hilbert symbol: the local invariant $\operatorname{inv}_v(B) \in \{0, 1/2\}$ at each place $B$0. By definition, $B$1 is indefinite if and only if the infinite place $B$2, i.e., $B$3 (Cornut et al., 2010, Rickards, 2021).

Each indefinite quaternion algebra over $B$4 is classified up to isomorphism by its finite discriminant

$B$5

which is always the product of an even number of distinct finite primes. There is exactly one such algebra (up to isomorphism) for every square-free $B$6 with $B$7 (Rickards, 2021).

2. Orders, Ideals, and Norms

A maximal order $B$8 is a $B$9-lattice that is maximal with respect to inclusion among all $\Q$0-orders. For applications in arithmetic and geometry, one also considers Eichler orders of level $\Q$1 (with $\Q$2 prime to $\Q$3), which are intersections of two maximal orders optimally chosen at each finite place: $\Q$4 with $\Q$5 for $\Q$6, and special structures for $\Q$7 (Rickards, 2021).

Given a quaternion algebra $\Q$8 over a number field $\Q$9, every element $B \otimes_\Q \R \simeq M_2(\R)$0 can be written as $B \otimes_\Q \R \simeq M_2(\R)$1 with $B \otimes_\Q \R \simeq M_2(\R)$2. The reduced trace and norm are defined by

$B \otimes_\Q \R \simeq M_2(\R)$3

The reduced norm on each simple component of $B \otimes_\Q \R \simeq M_2(\R)$4 coincides with the usual determinant when $B \otimes_\Q \R \simeq M_2(\R)$5 split (Page, 2014).

The principal ideal problem in an indefinite quaternion algebra focuses on right ideals $B \otimes_\Q \R \simeq M_2(\R)$6, which are finitely generated $B \otimes_\Q \R \simeq M_2(\R)$7-lattices with prescribed right order. A core result (Eichler's theorem) provides a bijection between isomorphism classes of right ideals and generalized ideal class groups, reducing principality testing to the class group of the base field (Page, 2014).

3. Arithmetic and Geometry: Shimura Curves and CM Points

Shimura curves are moduli spaces attached to indefinite quaternion algebras, parametrizing abelian surfaces with quaternionic multiplication. For $B \otimes_\Q \R \simeq M_2(\R)$8 and an Eichler order $B \otimes_\Q \R \simeq M_2(\R)$9 (with associated open compact $\Q$0), the complex Shimura curve is

$\Q$1

where $\Q$2 is the upper half-plane.

A totally imaginary quadratic field $\Q$3 gives an algebraic torus $\Q$4 and leads to the concept of complex multiplication (CM) points: $\Q$5 For a point $\Q$6, the fine conductor $\Q$7 is determined by the intersection orders $\Q$8 and the corresponding structure from the second embedding, and the coarse conductor $\Q$9 specifies the field of definition via ring class field theory (Cornut et al., 2010).

4. Reduction Maps and Liftings

Reduction theory for Shimura curves at ramified or unramified primes provides a bridge from the geometry of indefinite quaternion algebras to that of their totally definite analogs. When $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$0 is a prime, and $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$1, the reduction map

$B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$2

is constructed, where $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$3 refer to the analogous data for the definite quaternion algebra $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$4 ramified appropriately at $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$5 and $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$6. This map is equivariant with respect to $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$7 action and relates CM points to supersingular points mod $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$8 (Cornut et al., 2010).

A lifting of the reduction map is constructed via an adelic morphism $B = (a, b \mid \Q) = \Q \oplus \Q i \oplus \Q j \oplus \Q ij,\quad i^2 = a,\, j^2 = b,\, ij = -ji,$9 with explicitly described local behavior at each place, inducing surjective fiber maps between CM sets: $a, b \in \Q^\times$0 The degree $a, b \in \Q^\times$1 of the fiber is determined by Galois orbits and local lattice invariants (Cornut et al., 2010).

5. Optimal Embeddings and Hecke Operators

For a real quadratic field $a, b \in \Q^\times$2 and Eichler order $a, b \in \Q^\times$3, an optimal embedding $a, b \in \Q^\times$4 is a ring homomorphism with $a, b \in \Q^\times$5. Two embeddings are equivalent if they are conjugated by a unit of norm one. The set of equivalence classes is denoted $a, b \in \Q^\times$6 (Rickards, 2021).

Hecke operators act naturally on formal sums of optimal embeddings. For $a, b \in \Q^\times$7 coprime to $a, b \in \Q^\times$8, define

$a, b \in \Q^\times$9

with weights computed via coset representatives of norm $\Ram(B)$0 elements in $\Ram(B)$1. The resulting structure mirrors the theory of modular symbols, and the Hecke algebra satisfies classical relations.

Geometrically, each optimal embedding corresponds to a primitive hyperbolic element and to a closed geodesic on the Shimura curve; the signed intersection number of pairs of geodesics gives a non-degenerate, skew-symmetric pairing, which under the identification with homology matches the intersection pairing there (Rickards, 2021).

6. Computational Aspects: Principal Ideal Problem

The principal ideal problem in indefinite quaternion algebras consists of, given a right ideal $\Ram(B)$2, deciding whether it is principal (i.e., $\Ram(B)$3 for some $\Ram(B)$4), and if so, finding such a generator. For indefinite algebras, the decision problem reduces to the analogous problem in the underlying number field via reduced norm maps and class field theory.

A substantial computational advancement is the introduction of a heuristically subexponential algorithm, inspired by Buchmann’s approach for class groups in number fields and leveraging:

• Smoothness with respect to a factor base of primes
• Local $\Ram(B)$5-reduction structures exploiting the Bruhat–Tits tree
• Global reductions built from collections of smooth units with prescribed norm relations

The overall expected running time is subexponential in the discriminant, $\Ram(B)$6, under standard heuristics concerning factor base completeness, smoothness probabilities, and unit group computations. Performance is confirmed by large-scale numerical experiments (Page, 2014).

7. Connections, Special Cases, and Applications

Indefinite quaternion algebras have deep connections with modular forms, Shimura varieties, and arithmetic geometry. The structure of Shimura curves over indefinite algebras generalizes the theory of modular curves, accommodating both real and imaginary quadratic fields. The special case of $\Ram(B)$7 yields classical modular curves and the study of Heegner points, with explicit correspondences between CM points and supersingular points at inert primes (Cornut et al., 2010).

The action of Hecke operators on optimal embeddings is linked to the classical theory of modular forms via the generating series

$\Ram(B)$8

which is shown to be a holomorphic cusp form of weight two and prescribed level. This modularity is made explicit using the Jacquet–Langlands correspondence and the theory of intersection pairings on Shimura curves (Rickards, 2021).

Applications include explicit computations of endomorphism rings of elliptic or QM abelian surfaces, the modularity of generating series arising from arithmetic geometry, and effective algorithms for ideal classes and embeddings in arithmetic applications.