Counterexamples for (weak) multiplicity one

Hi.

For a Dirichlet character $\chi$ mod $N$, the (weak) multiplicity one theorem states the following:

Let $f, g \in S_k^{\text{new}}(\Gamma_0(N), \chi)$ be such that for every $m \in \mathbb{N}$ s.t. $(m,N)=1$, we have $$T(m)f = \lambda_m f,~~ T(m)g = \lambda_m g$$ i.e. $f,g$ are simultaneous eigenforms with the same eigenvalues for all the $m$-th Hecke operators, then actually, provided that $f \neq 0, g \neq 0$, $f$ and $g$ coincide up to a constant, i.e. $\mathbb{C}f = \mathbb{C}g$.

What i am wondering about is: How can one construct counterexamples when one drops the keyword 'new'? I am in particular interested in $S_k(\Gamma_0(p))$ (trivial character) for odd primes $p$. Is there a general 'method' for doing that?

Fabian Werner

• How do you define Hecke operators on old forms? Commented Mar 6, 2013 at 16:16
• I do not understand the question: Hecke operators are defined on the space $M_k(\Gamma_0(N))$ and oldforms are elements of that space... Commented Mar 6, 2013 at 16:23
• Fabian, read my response. Commented Mar 6, 2013 at 16:25
• But become commutative algebra only when restricted to newforms (Atkin-Lehner). So the question is rather, how do diagonalize the family of Hecke operators (necessary for finding joint eigenfunctions) if you do not consider newforms? Commented Mar 6, 2013 at 16:30
• Wait, what? Do we talk about two different Hecke algebras maybe? I mean the one generated by all the $T(m)$ where $m$ runs through the natural numbers (lets say we consider the abstract sums of double cosets, not even the operators on modular forms). This Hecke algebra is commutative and hence their induced operator-algebra is commutative on the whole space $M_k(\Gamma_0(N))$. Do i still misunderstand something? Commented Mar 6, 2013 at 16:44

Very concretely: if $f\in S_k^{\text{new}}(\Gamma_0(N), \chi)$, and $d>1$ is an integer, then $f(z)$ and $f(dz)$ are elements of $S_k(\Gamma_0(dN), \chi)$ with the same Hecke eigenvalues at $m$ coprime with $dN$, yet they are not multiples of each other. Of course this is the same what Marc was saying, but in more classical terms.

In particular, if $dN=p$ (i.e. $d=p$, $N=1$, $\chi$ is trivial), then you get a counterexample for $S_k(\Gamma_0(p))$.

I suggest that you read this famous paper by Atkin and Lehner.

• Or Casselman's representationtheoretic interpretation of Atkin-Lehner theorie. Math.Ann.206, which I prefer:) Commented Mar 6, 2013 at 16:26
• Ah ok, this is something i can understand :) Thank you very much! Commented Mar 6, 2013 at 16:38

Modular Hecke cusp forms are vectors of irreducible cuspidal automorphic representations.

If you do not consider new forms, there might two distinc functions being vectors in the same irreducible cuspidal automorphic representation. The restriction to newforms makes the associated vector/function unique.

This is also the only choice you have.

• Thanks for your response. I am completely unaware of the repn-theoretic part of modular forms, so could you tell me: is it possible to explicitly compute these counterexamples? Commented Mar 6, 2013 at 16:21
• GH explains how to explicate this, but you were asking about a general method to begin with. Commented Mar 6, 2013 at 17:15