Let denote the Hamilton quaternions. Set

Let be the point with , . There is a transitive action of given by with multiplication taking place in . This extends uniquely to an action of by requiring that the center act trivially. The stabilizer of is exactly , so we get an identification

Fix a triple with and . Set . Let with -action

Note that has central character . Set , so has central character .

Let be the right representation of on given by . There is a unique -bilinear pairing such that and for all .

**Definition.** *Let be an imaginary quadratic field. A cuspidal automorphic form of weight and level is a function such that*

*for all , , , , and , and such that satisfies the usual smoothness and cuspidality conditions, and has Casimir eigenvalues . Let denote the space of such functions.*

Note that if is a Hecke eigenform, say with for almost all places , there is a unique cuspidal automorphic representation such that for almost all , is one-dimensional and acts through the scalar . Furthermore, satisfies (and this determines uniquely in terms of ). In fact, we have

as -modules, where is the lowest -type of . Furthermore, fixing , there should be a Galois representation such that , and my normalization of things implies that should have Hodge-Tate weights at one embedding and at the other.

Let . My goal here is to *explicitly* define a canonical Hecke-equivariant linear injection

I hope the actual construction below will illuminate my use of the word “explicit”. Some version of this kind of thing lurks in the literature in many places, e.g. in a paper of Hida referenced below and in Urban’s thesis (Compositio. 99, 1995) (and in many other papers, but these were my primary references). A version of this over general number fields will appear in my book on *p*-adic *L*-functions, but the imaginary quadratic case is the key thing to understand, since once you understand how to do one complex place you just tensor madly over all complex places, and real places are easy.

For and , set . A straightforward calculation gives . It’s not hard to see that extends to a well-defined one-cocycle by setting for . If , then .

Set and . Following Hida’s 1993 Duke paper “p-ordinary cohomology groups for over number fields”, define by

Set . According to a calculation in Hida’s paper, we have for any , so trivially . The cocycle extends to by requiring the center to act trivially ( kills the ambiguity caused by ), and the formula then remains true for all .

Define functions for by setting

**Proposition**. *The functions satisfy the equivariance formula for any .*

Proof. Easy calculation; note that .

**Proposition**. *The functions satisfy the equivariance formula*

*for any .*

Proof. Set

An easy calculation gives

so replacing by and taking transposes everywhere we obtain

Comparing this to the definition of gives

and now the result is clear, since and .

**Proposition**. *The function*

*is well-defined, where is any element such that .*

Proof. The element is well-defined up to right multiplication by an arbitrary element . Set

We have

by the invariance of the pairing . An easier calculation likewise gives , so descends to a function on as claimed.

**Proposition**. *The functions satisfy the equivariance formula*

*for any .*

**Corollary**. *For any , the differential form*

*satisfies the equivariance relation for any and . In particular, defines a global section of the automorphic vector bundle on .*

**Definition**. *The Eichler-Shimura map is given by*