# Multiplicity one conjecture

I recently became interested in Maass cusp forms and heared people mentioning a "multiplicity one conjecture". As far as I understood it, it says that the dimension of the space of Maass cusp form for fixed eigenvalue should be at most one.

Since Maass cusp forms always are defined for a Fuchsian lattice, I wonder 1) for which lattices this conjecture had been conjectured? 2) what is the motivation for this conjecture? 3) to whom this conjecture is due? 4) is it published somewhere? 5) is it proven in some cases?

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In Sarnak's article http://web.math.princeton.edu/sarnak/baltimore.pdf, he recalls one famous conjecture (Conjecture I, due to himself) that $\Gamma \backslash \mathbb{H}$ should have very few Maass forms for "most" Fuchsian lattices $\Gamma$.

Btw, he attributes the simplicity conjecture (Conjecture 3) for $SL_2(\mathbb{Z})$ to Cartier.

Wolpert (Theorem I) has shown that Sarnak's conjecture would follow if the simplicity conjecture holds for the congruence subgroup $\Gamma(2)$.

Also GH last conjecture that the multiplicity is uniformly bounded in $N$ would suffice for Sarnak's conjecture, but current knowledge is that the multiplicity of an eigenvalue of magnitude $T$ is at most $\ll_N \sqrt{T}/ \log T$ and not $\ll_N 1$.

In fact, Sarnak conjectures that there exist $\Gamma$ with only finitely many Maass wave forms, but this does not follow from the simplicity conjecture for $\Gamma(2)$.

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This conjecture is usually stated for $\mathrm{SL}_2(\mathbb{Z})$, and it is widely open. I think it is folklore, and is stated in several papers, e.g. in Luo: Nonvanishing of $L$-values and the Weyl law (before (3)).

The motivation, I think, is similar as with the conjecture for the multiplicity of the Riemann zeta zeros. The belief is that there is no "accidental" algebraic independence among the eigenvalues of the Laplacian or the zeros of an automorphic $L$-function. For example, the Laplacian eigenvalue $1/4$ is expected to "come from" an even Galois representation, while the zero $1/2$ is expected to "come from" rational points of infinite order on an abelian variety. For $\mathrm{SL}_2(\mathbb{Z})$ or $\zeta(s)$ we don't know of any object that would "impose" any algebraic independence on the data, hence we believe that in those cases the data is entirely transcendental.

For congruence subgroups $\Gamma_0(N)$ the "multiplicity one conjecture" is false, because the eigenvalue $1/4$ is known to occur with multiplicity for some $N$'s. The known examples come from even Galois representations. I think it is safe to believe that the multiplicities are bounded for any $N$.

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Is it also know for newforms of $\Gamma_0(N)$? –  Marc Palm Jul 19 '12 at 14:29
I think the eigenvalue 1/4 occurs with multiplicity even among newforms of some level. All we need is two even Galois representations with the same Artin conductor. –  GH from MO Jul 19 '12 at 19:16
Sorry for bumping this old question, but is the multiplicity bounded uniformly with respect to $N$? –  user31814 May 15 at 18:07
@user31814: I think the multiplicity is bounded (conjecturally) for a fixed $N$, but it can get arbitrary large (provably) for varying $N$. –  GH from MO May 15 at 18:11
@GHfromMO, thanks! Is there a reference for what you said? –  user31814 May 16 at 1:36

Multiplicity one refers to something else, related but much weaker.

For the analogue question for lattices, there are trivial counter examples: Induction by steps for example suggests on the level of Lie groups $$Ind_{\Gamma(N)} ^{PSL_2(\mathbb{R})} 1 \cong Ind_{PSL_2(\mathbb{Z})} ^{PSL_2(\mathbb{R})} Ind_{\Gamma(N)}^{PSL_2(\mathbb{Z})} 1$$ and e.g. by the Peter-Weyl theorem, we know that $$Ind_{\Gamma(N)}^{PSL_2(\mathbb{Z})} 1$$ the multiplicity of an irreducible representation equals its dimension.

Note that GH's example is less trivial, since $$Ind_{\Gamma_0(N)}^{PSL_2(\mathbb{Z})} 1$$ decomposes with multiplicity one.

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Could you elaborate to what it does refer, please? –  Ruedi Meier Aug 15 '12 at 17:49
For every prime, you get also an eigenvalue. Multiplicity one theorem states that their exists only one eigenfunction with all the same eigenvalues. More rigorously put, you find this here: en.wikipedia.org/wiki/Multiplicity-one_theorem –  Marc Palm Sep 4 '12 at 9:33