Correspondence between Modular Forms of Even Integer Weight

Representations

For the background on automorphic representations of GL2(AQ) and how they

correspond to Hecke eigenforms, please refer to Section 2.5.

Let k be a positive odd integer with k ≥ 3. Let φ = P∞

n=1anqn ∈

Snew

k−1(N, χ) be a newform of weightk−1, level N and character χ.

Recall that we can associate toφ an automorphic representationρ. Let

ρp be the local component of ρ at a prime p. Recall that ρp is an irreducible

admissible representation of GL2(_Qp). Henceρp is either a principal series or a

supercuspidal representation or it is some twist of the Steinberg representation (sometimes also referred to as a special representation).

Recall that if φ =P∞

n=1anq

n _{is an eigenform, then we have defined its}

twist by a character µ to be the modular form φµ =

P∞

n=1anµ(n)q

n_{. Wald-}

spurger works with a different definition of twist:

Definition 4.2.1. Let φ be a newform of weight k−1 and character χ. Let

µ a Dirichlet character. We denote by φ⊗µ the (unique) newform of weight

k−1with characterχµ2 satisfyingλp(φ⊗µ) = µ(p)λp(φ)for almost all primes

p, where λp is the eigenvalue under Tp.

characters withp-power conductor. The following holds (see [45, Section III]): (i) ρp is supercuspidal if and only if for all µ ∈ ξp, the level of φ⊗ µ is

divisible by pand λp(φ⊗µ) = 0.

(ii) ρp is an irreducible principal series if and only if either

(a) there exists a characterµinξp such thatpdoes not divide the level

of φ⊗µ; or,

(b) there exist two distinct charactersµ1,µ2inξp such thatλp(φ⊗µ1)6= 0, λp(φ⊗µ2)6= 0.

(iii) ρp is a special representation if and only if the following conditions hold:

(a) for allµ∈ξp, the level of φ⊗µis divisible by p;

(b) there exists a unique µ inξp such thatλp(φ⊗µ)6= 0.

We shall need the following theorem which is extracted from the paper of Atkin and Li [1].

Theorem 4.2.2. (Atkin and Li) Let φ = P∞

n=1anq

n _{be a newform of weight}

k−1, character χ and level N. Let µ be a primitive character of conductor

m. Then

(a) If gcd(m, N) = 1 then φ⊗µ=φµ, and it is a newform of weight k−1, character χµ2 _{and level N m}2 _{([1, Introduction]).}

(b) Suppose µ is of q-power conductor where q | N and write N = qs_M where q _- M. Then φ⊗µ is a newform of weight k−1, character χµ2

and level qs0M for somes0 ≥0. Moreover,λp(φ⊗µ) =µ(p)λp(φ)for all primes p_- N ([1, Theorem 3.2]). In particular if s= 1 and χ is trivial, then for µ with conductor qr_{, r ≥ 1, it turns out that φ⊗µ = φ}

µ is a newform of level q2r_{M and character µ}2 _{([1, Corollary 4.1]).}

(c) Let q| N. Suppose φ is q-primitive and aq = 0. Then for all characters

µ of q-power conductor, φ⊗µ=φµ is a newform of level divisible by N (Recall that φ is q-primitive if φ is not a twist of any newform of level lower than N by a character of conductor equal to some power of q) ([1, Proposition 4.1]).

(d) Let N = qs_{M where q}

- M; let Q = qs. Let χQ be the Q-part 1 of the character χ. Ifs is odd and cond χQ≤

√

Q then φ is q-primitive. Now suppose q = 2. Then, if s = 2 then φ is always 2-primitive; if s is odd then φ is 2-primitive if and only if cond χQ <

√

Q; if s is even and

s ≥4 then φ is 2-primitive if and only if cond χQ = √

Q ([1, Theorem 4.4]).

We deduce the following corollaries which we will be using later.

Corollary 4.2.3. Let φ = P∞

n=1anqn ∈ Sk−new1(N) be a newform with trivial

character. Letρ2 be the local component at2of the corresponding automorphic

representation. Suppose either (i) N is odd; or

(ii) ν2(N) = 1 and a2 6= 0.

Then ρ2 is not supercuspidal.

Further if ν2(N)≥ 2 and φ is 2-primitive then ρ2 is supercuspidal. In

particular, if either ν2(N) = 2 or ν2(N) is odd then ρ2 is supercuspidal.

Proof. If N is odd, take µ to be the identity character. Thus µ∈ ξ2 and the level of φ ⊗µ is odd and hence ρ2 is not supercuspidal. If N = 2M such that M is odd and a2 6= 0, again taking µ as identity character we get that

λ2(φ⊗µ) =a2 6= 0 and thus ρ2 is not supercuspidal.

Let ν2(N) ≥ 2. Then a2 = 0 (see Theorem 2.2.12). If φ is 2-primitive then it follows using part (c) of the Theorem 4.2.2 that for any µ ∈ ξ2,

φ⊗µ = φµ is newform of level divisible by 2. Write T2(φµ) = P ∞ n=1bnq

n_.

By Proposition 2.2.5, bn = a2nµ(2n) +µ2(2)2k−2an/2µ(n/2) for all n. Thus

T2(φµ) = 0. Therefore, λ2(φ⊗µ) =λ2(φµ) = 0 and ρ2 is supercuspidal. Note that we have not yet used the condition that φ has trivial character, but we need it to prove the final statement which is indeed a direct application of part (d) of the Theorem 4.2.2.

Corollary 4.2.4. Let φ be as in the above corollary.

1_{Letχbe a Dirichlet character with modulusp}r1

1 · · ·p

rn

n where thepi are distinct primes. Thenχcan be written uniquely as a productQ_χ

pri_i whereχpri_i has modulusp ri

(i) If N = pM with M coprime to p and ap 6= 0, then ρp is a special representation.

(ii) If p_-N, then ρp is an irreducible principal series.

Proof. We first prove (i). By part (b) of the Theorem 4.2.2, for any µ ∈ ξp,

the level ofφ⊗µis divisible by p. Further if µis the identity character then

λp(φ⊗µ) = ap 6= 0; we claim that this is unique such character in ξp. Let

µ∈ξp be such that µis a character of conductor pr,r ≥1. Thenφ⊗µ=φµ

is a newform in Sk−1(p2rM, µ2) such that λp(φµ) = apµ(p) = 0 (see Theorem

2.2.12) and henceλp(φ⊗µ) = 0.

The proof of (ii) is obvious and does not require the condition that newformφ has trivial character.