Eichler cohomology

In [a2], M. Eichler conceived the "Eichler cohomology theory" (but not the designation) while studying "generalized Abelian integrals" (now called "Eichler integrals" ; see below).

The setting for this theory is that of automorphic forms, with multiplier system, on a discrete group of fractional-linear transformations (equivalently, of -matrices; cf. also Automorphic form; Fractional-linear mapping). One may assume that consists of real fractional-linear transformations, that is, that fixes , the upper half-plane. A fundamental region, , of is required to have finite hyperbolic area; this is equivalent to the two conditions (taken jointly):

i) is finitely generated;

ii) each real point of is a parabolic point (a cusp) of , that is, it is fixed by a cyclic subgroup of with parabolic generator.

Let and let be a multiplier system in weight with respect to . Since is integral, this means simply that for all , and is multiplicative:

(a1)

For and a function on , define the slash operator

(a2)

In this notation, the characteristic transformation law satisfied by an automorphic form on of weight and multiplier system can be written

(a3)

Let denote the vector space of automorphic forms on of weight and multiplier system , the collection of satisfying (a3) and that are holomorphic on and meromorphic at each parabolic cusp of (in the usual local variable, cf. also Analytic function; Meromorphic function). One says that is an entire automorphic form if is holomorphic at each parabolic cusp. An entire automorphic form is called a cusp form if vanishes at each parabolic cusp. As usual, denotes the space of entire automorphic forms and denotes the subspace of cusp forms. For groups of the kind considered here, a suitable version of the Riemann–Roch theorem shows that has finite dimension over .

To describe the genesis of Eichler cohomology it is helpful to introduce the Bol identity [a1]:

(a4)

where , , is any fractional-linear transformation of determinant , and is a function with derivatives ((a4) is easily derived from the Cauchy integral formula, cf. Cauchy integral theorem, or proved by induction on ). As a consequence of (a4), if , then .

There is a second consequence of (a4), more directly relevant to the case under consideration: if and (for example, ), then satisfies

(a5)

where is a polynomial in of degree at most . is called an Eichler integral of weight and multiplier system , with respect to , and with period polynomials , . Eichler integrals generalize the classical Abelian integrals (cf. Abelian integral), which occur as the case , . As an immediate consequence of (a5), satisfies the cocycle condition

(a6)

Consider the cocycle condition for in the space of polynomials of degree at most . A collection of polynomials satisfying (a6) is called a cocycle in . A coboundary in is a collection such that

(a7)

with a fixed polynomial . Note that defined by (a7) satisfies (a6). The Eichler cohomology group is now defined to be the quotient space: cocycles in modulo coboundaries in .

To state Eichler's cohomology theorem of [a2] one must introduce the notion of a "parabolic cocycle" . Let be the (necessarily finite) set of inequivalent parabolic cusps in . For , let be the stabilizer of in with parabolic generator (cf. also Stabilizer). One says that the cocycle is parabolic if the following holds: For each , , there exists a such that .

Coboundaries are of course parabolic cocycles, so one may form the quotient group: parabolic cocycles in modulo coboundaries in . This is a subgroup of , called the parabolic Eichler cohomology group and denoted by .

Eichler's theorem [a2], p. 283, states: The vector spaces and are isomorphic under a canonical mapping.

The discussion above, leading to (a6), shows how to associate a unique element of to , by forming a -fold anti-derivative of . The key to the proof of Eichler's theorem lies in the construction of a suitable mapping from to . Eichler accomplishes this by attaching to an element of with poles in , and then passing to the cocycle of period polynomials of a -fold anti-derivative of . The mapping from to is then defined by means of . The proof that is one-to-one follows from Eichler's generalization of the Riemann period relation for Abelian integrals to the setting of Eichler integrals.

The proof can be completed by showing that . The essence of Eichler's theorem is that every parabolic cocycle can be realized as the system of period polynomials of some unique Eichler integral of weight and multiplier system , with respect to .

R.C. Gunning [a3] has proved a related result, from which Eichler's theorem follows as a corollary: The vector spaces and are isomorphic under the mapping of Eichler's theorem.

Proving Gunning 's theorem first and then deriving Eichler's theorem from it has the advantage that the calculation of is substantially easier than that of ; this, because in there is no restriction on the elements of associated to the parabolic generators , .

There are various proofs of Gunning 's theorem and its corollary, in addition to those in [a2], [a3]. See, for example, [a4], [a11], [a14]. (G. Shimura [a14] has refined Eichler's theorem by working over the real rather than the complex field.) In [a6], Chap. 5, [a7], [a8], and [a13], analogous results are proved for the more general situation in which is a finitely generated Kleinian group. I. Kra has made further contributions to this case ([a9], [a10]).

The literature contains several results describing the cohomology groups and that arise when the space of polynomials is replaced by a larger space of analytic functions [a3], Thm. 3, [a5], Thms. 1; 2, [a7], Thm. 5. Gunning [a3], Thms. 4; 5, discusses and , for , as well as . For an overview see [a5].

References

[a1]	G. Bol, "Invarianten linearer Differentialgleichungen" Abh. Math. Sem. Univ. Hamburg , 16 : 3–4 (1949) pp. 1–28
[a2]	M. Eichler, "Eine Verallgemeinerung der Abelschen Integrale" Math. Z. , 67 (1957) pp. 267–298
[a3]	R.C. Gunning, "The Eichler cohomology groups and automorphic forms" Trans. Amer. Math. Soc. , 100 (1961) pp. 44–62
[a4]	S.Y. Hussemi, M.I. Knopp, "Eichler cohomology and automorphic forms" Illinois J. Math. , 15 (1971) pp. 565–577
[a5]	M.I. Knopp, "Some new results on the Eichler cohomology of automorphic forms" Bull. Amer. Math. Soc. , 80 (1974) pp. 607–632
[a6]	I. Kra, "Automorphic forms and Kleinian groups" , Benjamin (1972)
[a7]	I. Kra, "On cohomology of Kleinian groups" Ann. of Math. , 89 : 2 (1969) pp. 533–556
[a8]	I. Kra, "On cohomology of Kleinian groups - II" Ann. of Math. , 90 : 2 (1969) pp. 576–590
[a9]	I. Kra, "On cohomology of Kleinian groups - III" Acta Math. , 127 (1971) pp. 23–40
[a10]	I. Kra, "On cohomology of Kleinian groups - IV" J. d'Anal. Math. , 43 (1983-84) pp. 51–87
[a11]	J. Lehner, "Automorphic Integrals with preassigned period polynomials and the Eichler cohomology" A.O.L. Atkin (ed.) B.J. Birch (ed.) , Computers in Number Theory, Proc. Sci. Research Council Atlas Symp. no. 2 , Acad. Press (1971) pp. 49– 56
[a12]	J. Lehner, "Cohomology of vector-valued automorphic forms" Math. Ann. , 204 (1973) pp. 155–176
[a13]	J. Lehner, "The Eichler cohomology of a Kleinian group" Math. Ann. , 192 (1971) pp. 125–143
[a14]	G. Shimura, "Sur les intégrales attachées aux formes automorphes" J. Math. Soc. Japan , 11 (1959) pp. 291–311