Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 457

Shimura curves

We now need to introduce what are known as Shimura curves. Recall that
in Section 10.2 we defined quaternion algebras as (noncommutative) rings of
the form

Q = Q + Q α + Q β + Q αβ,

where

α 2 ,β 2 ∈ Q , βα = −αβ.

We omit the requirement from Section 10.2 that α 2 < 0and β 2 < 0since
we want to consider indefinite quaternion algebras as well. Let r be a prime
(possibly ∞ )andlet Q r be the ring obtained by allowing r -adic coecients
in the definition of Q . As we mentioned in Section 10.2, there is a finite set
of primes r , called the ramified primes, for which Q r has no zero divisors. On
the other hand, when r is unramified, Q r is isomorphic to M 2 ( Q r ), the ring
of 2 × 2 matrices with r -adic entries.
Given two distinct primes p and q , there is a quaternion algebra B that is

ramified exactly at p and q .Inparticular, B is unramified at ∞ ,so

B ∞ = M 2 ( R ) .

Corresponding to the integer M = N/pq , there is an order O⊂B , called

an Eichler order of level M (an order in that has rank 4

as an additive abelian group; see Section 10.2). Regarding O as a subset of
B ∞ = M 2 ( R ), define

B is a subring of B

Γ ∞ = O∩SL 2 ( R ) .

Then Γ ∞ acts on by linear fractional transformations. The Shimura curve

C is defined to be H modulo Γ ∞ .
There is another description of C , analogous to the one given above for
X 0 ( N ). Let

H

. Consider pairs ( A, B ), where A
is a two-dimensional abelian variety (these are algebraic varieties that, over
C , can be described as C 2 mod a rank 4 lattice) and B is a subgroup of
A isomorphic to Z M ⊕ Z M . We restrict our attention to those pairs such
that

O max be a maximal order in B

O max
maps B to B . When we are working over C , such pairs are in one-to-one
correspondence with the points on C . In general, over arbitrary fields, such
pairs correspond in a natural way to points on an algebraic curve, which we
again denote C .
Let J be the Jacobian of C . The description of C in terms of pairs ( A, B )
means that we can define an action of the Hecke operators on J , similarly to
what we did for the modular curves.
Let J [ ]bethe -torsion of the Jacobian J of C .Itcanbeshownthat
the representation ρ occurs in J [ M ], so there is a space V isomorphic to the

O max is contained in the endomorphism ring of A and such that