GROUP GL(2, Fq[T ])
ERNST-ULRICH GEKELER
Abstract. We introduce para-Eisenstein series as a second ana-logue of classical elliptic Eisenstein series in the framework of Drin-feld modular forms and show that they share many properties with ordinary Eisenstein series.
Introduction. In the well-known analogy between the respective arithmetics of the rational number field Q and the rational function field K = Fq(T ) over a finite field Fq, the part of classical modular
forms is played by Drinfeld modular forms, certain rigid-analytic func-tions on Drinfeld’s upper half-plane. See e.g. [4], [6], [7] for some results as well as for a discussion of similarities and differences of both theories.
On both sides Eisenstein series are crucial in that they generate the rings of modular forms for the modular groups Γclass := SL(2, Z) or
Γ := GL(2, Fq[T ]), respectively. The occurrence of Eisenstein series
in the classical theory is (at least) twofold: As coordinates of elliptic curves (e.g., the coefficients g2, g3 in a Weierstraß equation) and as
co-efficients in the Weierstraß ℘-function ([16] p. 157), where these data depend on a lattice Λ = Zω + Z in C.
All the named objects have their function field counterparts: ellip-tic curves E correspond to rank-two Drinfeld modules φ, which, like E = C/Λ, are uniformized by a lattice Λ in the function field version C∞ of C; the quantities g2, g3 correspond to coefficients of φ, and the
complex ℘-function ℘Λ to the rigid analytic function eΛ of the lattice
Λ in C∞.
However, the two roles of classical Eisenstein series break up on the function field side into two different families of modular forms. While the coefficients of φ are still described by Eisenstein series of the clas-sical shape (introduced in the Drinfeld module context by David Goss [10], [12]), the coefficients of eΛ are of different nature. We baptize
them para-Eisenstein series, since they still share many features with ordinary (or “ortho-”) Eisenstein series as studied by Goss.
The aim of the present note is to develop some properties of these: ele-mentary identities, congruence properties modulo primes p of Fq[T ] and
Fq[T ]), and to parallel them with similar properties of ortho-Eisenstein
series. The principal results are Theorems 3.5 and 4.3.
Acknowledgement. The author would like to thank the referee for his or her careful reading and suggestions, which made the paper eas-ier to read. It was finished during a visit at the National Center for Theoretical Sciences in Hsinchu, Taiwan, whose hospitality is gratefully acknowledged.
Notation.
Fq = finite field with q elements
A = Fq[T ], the polynomial ring over Fq in an indeterminate T ,
with quotient field K = Fq(T )
K∞ = Fq((T−1)), the completion of K with respect to the ∞-adic
valuation
| | = the absolute value on K∞, normalized by |T | = q
C∞ = completed algebraic closure of K∞ with respect to
the unique extension of | | to a fixed algebraic closure K∞
Ω = P1(C
∞) − P1(K∞) = C∞− K∞ the Drinfeld upper
half-plane
| |i : Ω −→ R the “imaginary part” function; |z|i = infx∈K∞|z − x|
Γ = GL(2, A), the Drinfeld modular group, which acts on Ω through fractional linear transformations
C∞{τ } (resp. C∞{{τ }}) the non-commutative polynomial ring
(resp. power series ring) over C∞ with commutation rule
τ x = xqτ for constants x ∈ C ∞.
We identify C∞{τ } (resp. C∞{{τ }}) with the ring
(multi-plication defined by insertion) of Fq-linear polynomials
(resp. power series ) in a variable X through X i aiτi = X i aiXq i . [k] = Tqk
− T ∈ A, the product of the monic irreducibles in A of degree d dividing k, if k ∈ N, and [0] = 0
Lk = Y 1≤i≤k [i], Dk = Y 1≤i≤k
[i]qk−i for k ≥ 1, L0 = D0 = 1
1. Background on Drinfeld modular forms (see [3] for more de-tails).
A lattice Λ in C∞is a finitely generated (hence free) discrete A-submodule
of C∞. With such a Λ, we associate its exponential function eΛ :
C∞−→ C∞, which is defined as the everywhere and locally uniformly
convergent product
eΛ(z) := z
Y
06=λ∈Λ
It has an additive (also everywhere convergent) expansion (1.1) eΛ(z) = X k≥0 αkzq k =Xαkτk
with coefficients αk ∈ C∞, and satisfies a functional equation
(1.2) eΛ(T z) = φT(eΛ(z))
with some φT = φΛT ∈ C∞{τ } of shape
(1.3) φT(X) = T X + g1Xq+ · · · + grXq
r
= T τ0 + · · · + grτr,
where gr 6= 0 and r is the rank of Λ as an A-module.
The rule Λ 7−→ φΛ
T establishes a bijective correspondence between
A-lattices Λ of rank r and Drinfeld A-modules of rank r over C∞. We
will only need the two special cases:
(a) Λ = L := πA has rank one, and is scaled (through the choice of the constant π) such that the associated Drinfeld module φΛ is the Carlitz
module ρ, defined by
(1.4) ρT = T X + Xq = T + τ.
Here the exponential function is
(1.5) eL(z) =
X
k≥0
Dk−1zqk,
as is immediate from (1.2). Note that π is defined up to a (q − 1)-th root of unity; hence only πq−1 is well-defined through (1.4). Many
explicit formulas for πq−1 = −Tq+ T − T−(q2−2q) + · · · are available, see [4] (4.9), (4.10), (4.11).
(b) Λ = π(Aω + A) with some ω ∈ Ω and the constant π above. Here φ = φΛ = φω has rank two, and is given by
(1.6) φT = T X + gXq+ ∆Xq
2
= T + gτ + ∆τ2 with 0 6= ∆ ∈ C∞.
A Drinfeld modular form for Γ of weight k ∈ N ∪ {0} (and type zero: there will be no other “types” in this paper) is a holomorphic function f : Ω −→ C∞ subject to
(1.7)
(i) f (γ(z)) = (cz + d)kf (z) for γ = a b
c d ∈ Γ, z ∈ Ω;
(ii) f (z) =P
i≥0aisi(z) with some power seriesP aisi with positive
convergence radius in s(z) = e1−qL (πz), the identity being valid for sufficiently large imaginary parts |z|i.
By abuse of notation, we often identify f = P aisi ∈ C∞[[s]], since
f is uniquely determined by its s-expansion. Letting ω vary over Ω, the quantities g = g(ω) and ∆ = ∆(ω) in (1.6) become functions from
Ω to C∞, and in fact, modular forms of respective weights q − 1 and
q2− 1. The forms g and ∆ are algebraically independent and generate
the C∞-algebra
(1.8) M := M
k≥0
Mk = C∞[g, ∆]
of all modular forms (D. Goss [11]), where Mk is the C∞-vector space
of modular forms of weight k, which vanishes for k 6≡ 0 (mod q − 1). Our normalization is such that
(1.9) g(z) = X i≥0 aisi, ∆(z) = X i≥1 bisi
with a0 = 1, b1 = −1, and all the coefficients ai, bi lie in A. Moreover,
g and ∆ generate the A-algebra MAof modular forms which have their
s-expansion coefficients in A.
Other instances of modular forms are:
(1.10) Consider the Fq-algebra homomorphism
φω : A −→ C∞{τ } a 7−→ φω a uniquely determined by φω T = T + g(ω)τ + ∆(ω)τ2, and write φωa = X 0≤i≤2 deg a `i(a, ω)τi.
Then `i(a, ·) defines a modular form of weight k = qi− 1, a so-called
coefficient form.
(1.11) The Eisenstein series Ek(ω) := π−k
X
(0,0)6=(a,b)∈A×A
1 (aω + b)k
defines a non-zero element of Mk, provided that 0 < k ≡ 0( mod q − 1).
(1.12) For Λ = π(Aω + A) as before, write eΛ(z) = X i≥0 αi(ω)zq i .
Once again, αi is a modular form of weight k = qi−1, a para-Eisenstein
series. We will study some of its arithmetical properties. 1.13 Remark: Recall that classical Eisenstein series
Ek(ω) = (2πi)−k
X
(0,0)6=(a,b)∈Z×Z
1 (aω + b)k
occur both as coefficients forms attached to elliptic curves (e.g. the coefficients g2, g3 in a Weierstraß equation) and as the coefficients of the
Weierstraß ℘-function ℘Λof Λ = Zω+Z. Since the exponential function
eΛ through its functional equation uniformizes the Drinfeld module φΛ
in the same way as ℘Λ uniformizes the elliptic curve E = C/Λ, the αi
provide a function field analogue of classical Eisenstein series different from the one described in (1.11). This explains the terminology used. From (1.2) and (1.6) we get the formula
(1.14) [k]αk= gαqk−1+ ∆α q2
k−2,
valid for k ≥ 1, where αk = 0 for k < 0 and α0 = 1. (Recall that
[k] = Tqk − T .)
The modular invariant is the function
(1.15) j := g
q+1
∆ : Ω −→ C∞.
It is Γ-invariant and identifies the quotient space Γ \ Ω biholomorphi-cally with the affine line over C∞. Accordingly, if f ∈ Mk is a modular
form of weight k, where k = a(q2 − 1) + b(q − 1) with a ∈ N 0 and
0 ≤ b ≤ q, there exists a unique polynomial ϕ(X) = ϕf(X) ∈ C∞[X]
such that
(1.16) f = ϕ(j)∆agb.
We call it the companion polynomial of f and its zeroes the j-zeroes of f .
2. The para-Eisenstein series mk. The Eisenstein series of weight
qk− 1 are particularly important. We normalize them as follows:
(2.1) gk(z) := (−1)k+1LkEqk−1(z) (Lk = [k][k − 1] · · · [1]).
Then gk has its s-coefficients in A, with absolute term 1, and satisfies
the recursion (2.2) gk = −[k − 1]gk−2∆q k−2 + gk−1gq k−1 (k ≥ 2)
with g0 = 1, g1 = g (see [4] 6.9). It is easily checked that its companion
polynomial γk= ϕgk satisfies
(2.3) γk(X) = Xλ(k)γk−1(X) − [k − 1]γk−2(X) (k ≥ 2),
γ0 = γ1 = 1, γk is monic of degree ν(k), gk = γk(j)∆ν(k)gχ(k), where
λ(k) = q
k−1+ (−1)k
q + 1 , ν(k) =
qk− qχ(k)
q2− 1 , χ(k) = 0(1)
if k is even (odd), respectively.
Since the lattice π(Aω + A) degenerates to πA = L for |ω|i −→ ∞, i.e.,
s(ω) −→ 0, the constant term of αk(z) equals the k-coefficient Dk−1 of
eL(z). We therefore normalize the para-Eisenstein series
which has constant term 1. In analogy with (2.2) and (2.3), we have the recursions derived from (1.14):
(2.5) mk= gmqk−1+ [k − 1] q∆mq2
k−2
and for the companion polynomials µk = ϕmk:
(2.6) µk(X) = Xχ(k−1)µ q
k−1(X) + [k − 1]
qX(q−1)χ(k)µq2
k−2(X),
both valid for k ≥ 2, with m0 = 1, m1 = g, µ0 = µ1 = 1. Since g and
∆ have their s-coefficients in A, the same holds for the mk. Again, µk
is monic of degree ν(k) = deg γk, and mk = µk(j)∆ν(k)gχ(k).
The first few of the polynomials γk, µk are as follows.
2.7 Table. k γk(X) µk(X) 2 X − [1] X + [1]q 3 Xq− [1]Xq−1− [2] Xq+ [2]qXq−1+ [1]q2 4 Xq2+1− [1]Xq2− [2]Xq2−q+1 Xq2+1+ [3]qXq2+ [2]q2Xq2−q+1 −[3]X + [1][3] +[1]q3X + [3]q[1]q3
Defining the support supp(f ) of a polynomial (or power series) f as the set of exponents with nonvanishing coefficients, we observe that supp(γk) and supp(µk) agree, although the recursions (2.3) and (2.6)
are rather different. This is not accidental, and will result from Theo-rem 3.5. A first step towards the proof of this fact is:
2.8 Proposition. Let S(k) = supp(γk) and T (k) = supp(µk) be the
supports. We have for k ≥ 2
(i) S(k) = S(k − 1) + λ(k)∪ S(k − 2);·
(ii) T (k) = qT (k − 1) + χ(k − 1)∪ q· 2T (k − 2) + (q − 1)χ(k);
(iii) |S(k)| = |T (k)| = Fk := the k-th Fibonacci number defined by
F1 = 1, F2 = 2, Fk = Fk−1+ Fk−2 (k ≥ 3).
Proof. Consider the recursions (2.3) and (2.6).
(i) Since λ(k) > deg γk−2 = ν(k − 2), there is no cancellation of terms
in (2.3), hence S(k − 1) + λ(k) and S(k − 2) are disjoint.
(ii) Similarly, the two sets in the right hand side of eq. (ii) are disjoint since they belong to different residue classes modulo q.
(iii) is immediate from (i) and (ii).
2.9 Proposition. All the zeroes of gk and mkare simple. Equivalently,
the polynomials γk and µk are separable.
Proof. The equivalence of the two statements results from the fact that the canonical mapping from Ω to Γ \ Ω is unramified off elliptic points (i.e., those where j(z) 6= 0) and g has simple zeroes at elliptic points ([4] 5.15).
The separability of γk is shown in [6] 7.7, 7.8. That of µk will result
in a similar way from Theorem 3.5. However, there is a simple direct proof as follows. The derivative of µk(X) is µqk−1(X) if k is even and
−[k − 1]qXq−2µq2
k−2(X) if k is odd. Assume that µk(x) = 0 = µ 0 k(x),
where x 6= 0 and k ≥ 3. In both cases (k even / k odd) we conclude from (2.6) that 0 = µk(x) = µk−1(x) = µk−2(x). Again by (2.6), this
implies 0 = µk−3(x) = · · · = µ1(x), which however is not the case.
Therefore there are no multiple roots of µk.
2.10 Remark. There are two important qualitative differences be-tween the ortho-Eisenstein series gk and the para-Eisenstein series mk.
The non-elliptic j-zeroes x of gk (i.e., zeroes of γk) all satisfy |x| = qq,
or, what amounts to the same, gk(z) = 0 with non-elliptic z in the
standard fundamental domain F ⊂ Ω of Γ implies |z| = |z|i = 1 (see
[6] 6.7). This is similar to the corresponding property of classical Eisen-stein series for SL(2, Z) (see [15]). However, the j-zeroes x of mk(which
have been determined in [7]) are in general larger than qq in absolute value, and max{|x| | x zero of µk} −→ ∞ as k −→ ∞.
A second difference is the behavior under Hecke operators. While gk is
always an eigenform with simple eigenvalues ([4] 7.2), mk is in general
not an eigenform, as can be seen e.g. from m2 = [2]∆ + g2,
where ∆ and g2 are eigenforms with different eigenvalues.
3. p-adic congruences. Let p be an irreducible monic prime poly-nomial in A of degree d. We use the same symbol for the prime ideal generated by p and write Fp for the finite A-field A/p, Fp for its
alge-braic closure, and F(2)p for the quadratic extension of Fp in Fp.
For the reader’s convenience, we recall some of the relevant facts about supersingularity of Drinfeld modules, which are strikingly similar to the corresponding facts about supersingularity of elliptic curves. Miss-ing definitions and more details can be found in [5]. Let L be a field subextension of Fp/Fp. A Drinfeld module φ over L is supersingular if
and only if one of the following equivalent conditions is satisfied: (i) φ has no p-torsion over Fp;
(ii) the “multiplication-by-p” map φp is purely inseparable;
(iii) the ring End
Fp(φ) of endomorphisms of φ over Fpis non-commutative.
(In this case it is an order, in fact a maximal A-order, in a cer-tain quaternion algebra over K.)
Supersingularity of φ depends only on the Fp-isomorphism class of φ,
and therefore on its j-invariant j(φ) ∈ Fp. The set Σ(p) ⊂ Fp of
the Galois conjugation of F(2)p over Fp. Moreover, we have
0 ∈ Σ(p) ⇔ d odd ⇔ χ(d) = 1 and |Σ(p) − {0}| = ν(d) = deg γd= deg µd.
Let ssp(X) be the polynomial Q06=j∈Σ(p)(X − j) ∈ Fp[X]. Then in fact
(3.1) γd(X) ≡ ssp(X) (mod p).
This is implicit in [4] Cor. 12.3 and explicit in [1] and [6]. A similar congruence in the framework of classical modular forms and elliptic curves is due to Swinnerton-Dyer [17], see also [14]. A crucial step in the proof of (3.1) is the congruence, valid for k ≥ 0 (see [4] 6.11): (3.2) gk+d(s) ≡ gk(s)q
d
≡ gk(sq
d
) (mod p),
where we abuse notation to write gk(s) for the power series expansion
of gk in s. In particular,
(3.3) gd≡ 1 (mod p).
In view of [d] =Q
(deg p)|dp, the two preceding congruences hold in fact
modulo [d]. Finally, we have the periodicity relation (3.4) γk+d(X) ≡ Xχ(k)λ(d+1)γq
d
k (X)γd(X) (mod p),
valid for k ≥ 0 ([6] 7.6). We show similar p-adic properties of the para-Eisenstein series mk and their companions µk(X).
3.5 Theorem. Let p be a prime of A of degree d. (i) The polynomial µd(X) satisfies
µd(X) ≡ γd(X) ≡ ssp(X) (mod p).
We further have for k ≥ 0 the following congruences (mod p): (ii) mk+d(s) ≡ mk(s);
(iii) µk+d(X) ≡ Xχ(d)λ(k+1)µq
k
d (X)µk(X).
Proof. It is known ([4] (2.6)–(2.9)) that the series P
k≥0αk(ω)τ k and
−P
k≥0Eqk−1(ω)τkare inverses of each other in C∞{{τ }}. Since up to
the normalizations (2.1) and (2.4) Eqk−1 and αk agree with gk and mk,
respectively, we get X i,j≥0 i+j=k (−1)j+1 Dk DiLq i j mig qi j = 0 for k ≥ 1. The coefficients Dk/(DiL qi
k−i) always belong to A and are divisible by
each prime q of A of degree k if 0 < i < k, as is easily seen from their definitions, see [13] Theorem 3.1.5. In particular,
md= X 0≤i<d (−1)d−i+1 Dd DiLq i d−i migq i d−i.
As mi, gj belong to A[[s]] and Dd/Ld ≡ (−1)d+1 (mod p) ([4] 11.4),
we find
(3.6) md≡ gd≡ 1 (mod p)
as a power series. Assertion (i) is now a formal consequence of (3.6) and Theorem 12.1 of [4]. Viz, let Gd(X, Y ) and Md(X, Y ) be the
polynomials in A[X, Y ] defined by Gd(g, ∆) = gd, Md(g, ∆) = md, and
˜
Gd, ˜Mdtheir respective reductions in Fp[X, Y ]. Then ˜Md− 1 lies in the
kernel of the homomorphism : Fp[X, Y ] −→ Fp[[s]] which to X resp.
Y associates the s-expansion ( mod p) of g resp. ∆. However, ker() is generated by ˜Gd−1 ([4] (12.1)+(12.2)). Comparing leading coefficients
in X, we find ˜Md = ˜Gd. In view of gd = Gd(g, ∆) = γd(j)∆ν(d)gχ(d)
and similarly, md= Md(g, ∆) = µd(j)∆ν(d)gχ(d), (i) follows.
From (2.5), (3.6), [d] ≡ 0, [d + k] ≡ [k] (all the congruences are (mod p)), we find md+1≡ g ≡ m1, which finally implies md+2≡ m2, md+3≡
m3 ..., hence (ii). The last assertion (iii) now follows from (2.6) through
a standard induction on k (distinguishing the 4 cases k/d even/odd,
respectively), which we omit.
3.7 Corollary. md ≡ 1 (mod[d])
Proof. By the theorem, the congruence holds modulo all primes p of
degree dividing d.
3.8 Corollary. The supports S(k) of γk and T (k) of µk agree.
Proof. The two sets have the same size by (2.8), hence it suffices to show an inclusion between them. A view on (2.3) reveals that the non-vanishing coefficients of γk are products of terms [i] with 0 < i < k,
which are incongruent to zero modulo each prime q of degree k. The assertion now results from γk ≡ µk modulo q and the fact that for each
k there exists a prime q of degree k.
4. Balancedness. Let Aff(Fq) be the group of matrices of shape u, v0, 1
over Fq (u 6= v). It acts naturally on A by T 7−→ uT + v.
4.1 Definition [8]. A polynomial or power series f (s) = P cksk in
A[[s]] is balanced if the (visibly equivalent) conditions hold:
(a) each ck as a polynomial in T satisfies ck(uT + v) = ukck(T ) for u, v
0, 1 ∈ Aff(Fq);
(b) (i) ck ∈ Fq[Tq− T ], the ring of invariants under shifts T 7−→
T + v; (ii) if ck =
P
jck,jTj with ck,j ∈ Fq then ck,j 6= 0 implies j ≡
k (mod q − 1);
(c) f is invariant under the action of Aff(Fq) on A[[s]] that extends
the natural action on “constants” in A and satisfies u, v0, 1(s) = u−1s.
In particular, the set of balanced power series is a subring of A[[s]]. It will turn out that most of our modular forms are balanced as power series in s.
Fix an element u, v0, 1 of Aff(Fq), and let T0 = uT + v, another
uni-formizer for our ring A. We may calculate all the relevant quantities, labelled by a prime ( )0 with respect to the coordinate T0 and relate them to the corresponding quantity w.r.t. T . Let α ∈ Fq satisfy
αq−1 = u. Then
(4.2)
(i) [k]0 = u · [k]
(ii) ρ0 = α ◦ ρ ◦ α−1 (i.e., ρ0T = α ◦ ρT ◦ α−1)
(iii) (πq−1)0 = u · πq−1.
Here (i) is obvious, (ii) is ρ0T0 = (uT + v) + τ = α(uT + v + uτ )α−1 =
α ◦ ρT0 ◦ α−1, and (iii) follows from either one of the formulas (4.9),
(4.10), (4.11) for πq−1 of [4].
4.3 Theorem. The following modular forms are balanced as power series in s: (i) g, (ii) [k]∆ (k ∈ N), (iii) gk, (iv) mk.
Proof. We have g = g1 and
g(ω)k = (−1)k+1Lkπ1−q
k X
(0,0)6=(a,b)∈A2
1 (aω + b)qk−1.
Here the lattice sum is intrinsic (independent of the choice of the co-ordinate T of A). In view of (4.2), we have L0k = ukL
k and (πq
k−1
)0 = uk(πqk−1), which together gives (iii) and thus (i). Now from (2.2) we
have
∆ = 1 [1](g
g+1− g 2),
which implies ∆0 = u−1∆. Together with (4.2)(i), the balancedness of [k]∆ results. Finally, (iv) comes out from (2.5) and the fact that the balanced elements of A[[s]] from a subring. Remark. The present argument is essentially from [8]; however, due to different normalizations, some formulas differ from the correspond-ing ones given there.
There are similar invariance properties for the polynomials γk and µk.
4.4 Proposition. If we endow the ring A[X] = Fq[T, X] with the
degree graduation modulo q − 1, the polynomials γk and µk are
homoge-neous of degree [k/2] = k/2 resp. (k − 1)/2 if k is even resp. odd. That is, regarded as polynomials in T and X, γk(uT, uX) = u[k/2]γk(T, X)
Proof. The assertion could be formally derived from (4.3); however it is much easier to refer to the recursions (2.2) and (2.5), from which it
comes out in a straightforward fashion.
4.5 Remark. The functions f = ∆ and f = gk satisfy k ∈ supp(f ) ⇒
k ≡ 0, 1(mod q), see [4] 6.10. That property also holds for the forms mk, as is evident from (2.5). Hence the results (4.3) and (4.4) imply
strong restrictions on the nature of the coefficients of gk, mk, γk, µk.
These, together with bounds on the degrees of the coefficients, suffice to determine some of the coefficients e.g. of the gk [2]. A similar study
of coefficients of the mk would be desirable.
Resum´e. We tabulate properties of classical Eisenstein series Ek for
SL(2, Z) and the corresponding properties of the ortho-Eisenstein series gk and para-Eisenstein series mk for Γ = GL(2, A), along with
proper-ties of the respective companion polynomials. Here p ≥ 5 is a natural prime and p a prime of A of degree d. We don’t give precise statements in the classical case (which would require more notation), but content ourselves with giving references.
Ek gk mk
recursion of Eisenstein series [17] p. 19 (2.2) (2.5) of companion polynomials to be worked out (2.3) (2.6)
simplicity of non-elliptic zeroes yes yes yes
location of zeroes z in standard |z| = 1, [15] |z| = 1, [6] different, [7] fundamental domain F
Hecke eigenform property yes yes no
congruences mod p resp. mod p Ep−1≡ 1, [17] gd≡ 1 md≡ 1
of companions [9] Thm. 2.2 γd≡ ss℘ µd ≡ ss℘
periodicity of companions [9] Thm. 2.4 (3.4) (3.5)(iii) reduced mod p, mod p
References
[1] G. Cornelissen: Geometric properties of modular forms over rational function fields. Thesis Universiteit Gent 1997.
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[17] H.P.F. Swinnerton-Dyer: On `-adic representations and congruences for co-efficients of modular forms. Lect. Notes Math. vol. 350, 1-55, Springer 1973
Ernst-Ulrich Gekeler
FR 6.1 Mathematik, Universit¨at des Saarlandes
