Selberg conjecture

Let $\HH$ denote the upper half-plane, $SL(2,\ZZ)$ the group of integer matrices of determinant one and

$$\Gamma_0(N)= \left\{ \left(\begin{array}{rr} a & b \\ c & d \end{array}\right) \in SL(2,\ZZ) : c \equiv 0 \,(\operatorname{mod} N) \right\}.$$

Following H. Maass [a9], let $W_s(\Gamma_0(N))$ denote the space of bounded functions $f$ on $\Gamma_0(N) \backslash \HH$ that satisfy

$$\Delta f = \left( \frac{1-s^2}{4} \right) f,$$

where

$$\Delta = -y^2 \left( \frac{\partial^2}{\partial^2_x} + \frac{\partial^2}{\partial^2_y} \right)$$

is the Laplace–Beltrami operator (cf. also Laplace operator). Such eigenfunctions $f$ are called Maass wave forms. Since $\Delta$ in this context is essentially self-adjoint and non-negative (cf. also Self-adjoint operator), it follows that $(1-s^2)/4$ is real and not zero.

A. Selberg conjectured [a12] that there is a lower bound $\ell_1(N)$ for the smallest (non-zero) eigenvalue: For $N \not= 1$,

$$\ell_1(N) \geq 1/4.$$

This innocent looking conjecture is (cf. [a10]) one of the fundamental unsolved questions in the theory of modular forms (as of 2000; cf. also Modular form). For $SL(2,\ZZ)$ and for small values of $N$, it has been known for some time (Selberg, W. Roelcke). In general, it has many applications to classical number theory (see [a4] and [a12], for example). To back up his conjecture, Selberg also proved the following assertion:

$$\ell_1(N) \geq 3/16.$$

Selberg's approach was to relate this problem to a purely arithmetical question about certain sums of exponentials, called Kloosterman sums (cf. also Exponential sum estimates; Trigonometric sum). This allowed him to invoke results from arithmetic geometry. The key ingredient giving the estimate is a (sharp) bound on Kloosterman sums due to A. Weil [a13]. This bound, in turn, is a consequence of the Riemann hypothesis for the zeta-function of a curve over a finite field, which he had proven earlier (cf. also Riemann hypotheses). On the other hand, to go further than the theorem by this approach one needs to detect cancellations in sums of such Kloosterman sums, and arithmetic geometry offers nothing in this direction. This is the reason that the approach through Kloosterman sums has a natural barrier at $3/16$. It is interesting that H. Iwaniec [a5] has given a proof of Selberg's theorem which, while still being along the lines of Kloosterman sums, avoids appealing to Weil's bounds.

Presently (2000), Selberg's conjecture is part of the "Ramanujan–Petersson conjecture at infinity" . In other words, if interpreting the Ramanujan–Petersson conjecture as a statement about the irreducible representations of $p$-adic groups $GL(2,\QQ_p)$ inside a cuspidal representation of $GL(2,\mathcal{A})$ (see below), Selberg's conjecture will follow as a statement for $GL(2,\RR)$.

Indeed, first let $\QQ_p$ denote the completion of the rational field $\QQ$ with respect to the $p$-adic absolute value $|\cdot|_p$, $p < \infty$, and view $\RR$ as the completion with respect to $|\cdot|_\infty=|\cdot|$. By the adèles, denoted $\mathcal{A}$, one means the "restricted" direct product $\Pi^{\prime}_{p\leq \infty} \QQ_p$ (restricted so that almost every $x_p \in \QQ_p$ has $|x_p|_p \leq 1$, i.e., $x_p \in \ZZ_p$; cf. also Adèle). Secondly, let $L^0_2(Z_\mathcal{A}G_\QQ \backslash G_\mathcal{A})$ denote the space of measurable functions $\phi$ on $Z_\mathcal{A}G_\QQ \backslash G_\mathcal{A}$, where $G_\mathcal{A}=GL(2,\mathcal{A})=\Pi^{\prime}_{p\leq\infty}GL(2,\QQ_p)$ ( "restricted" means $g=\prod g_p$, with $g_p \in GL(2,\ZZ_p)$ for almost every $p$), $G_\QQ=GL(2,\QQ)$ imbedded diagonally in $G_\mathcal{A}$,

$$Z_\mathcal{A} =\left\{ \left(\begin{array}{rr} z & 0 \\ 0 & z \end{array}\right): z \in \mathcal{A} \right\}.$$

is the centre of $G_\mathcal{A}$,

$$\int_{Z_\mathcal{A}G_\QQ \backslash G_\mathcal{A}} |\phi(g)|^2 dg < \infty $$

and

$$ \int_{\QQ \backslash \mathcal{A}} \phi\left(\left(\begin{array}{rr} 1 & x \\ 0 & 1 \end{array}\right)g\right) dx = 0$$

for almost every $g$. Now, assuming $f \in W_s(\Gamma_0(N))$ is an eigenfunction of all Hecke operators $T(p)$, one can define, in a one-to-one way, a function $\phi_f \in L^0_2(Z_\mathcal{A}G_\QQ \backslash G_\mathcal{A})$ such that the $G_\mathcal{A}$-module $\pi_f=\otimes_p \pi_p$ generated by the right $G_\mathcal{A}$-translates of $\phi_f$ is an irreducible subrepresentation of $L^0_2(Z_\mathcal{A}G_\QQ \backslash G_\mathcal{A})$. Then Selberg's conjecture states that the representation $(\pi)_\infty$ of $GL(2,\RR)$ is a principal series $\pi(i t_1, i t_2)$ with trivial central character and

$$\frac{1-s^2}{4} = \frac{1+4(t_1-t_2)^2}{4} \geq \frac{1}{4}.$$

In other words, complementary series $\pi(s_1,s_2)$ with $s_1-s_2=2s$ between $0$ and $1$ (and $1-s^2/4$ between $0$ and $1/4$) should not occur (cf. [a2]; see also Irreducible representation; Principal series). In this context the Ramanujan–Petersson conjecture says that for (almost every) $p$ the same conclusion holds, i.e., for $p$ a class-one representation, that is, $p \not| N$, $\pi_p(s_{1,p},-s_{1,p})$ satisfies (cf. [a11])

$$\operatorname{Re}(s_{1,p}) = 0 .$$

P. Deligne proved the original Ramanujan conjecture [a1] when $\pi_\infty$ is a holomorphic discrete series of weight $k$. For example, when $f$ equals Ramanujan's $\Delta(z)=\sum_{n=1}^{\infty} \tau(n) e^{2 i \pi n z}$, the condition (a1) implies

$$|\tau(p)| \leq p^{11/2} (p^{s_{1,p}}+p^{s_{2,p}}) \leq 2 p^{11/2}$$

the famous Ramanujan inequality. In general, Deligne was able to exploit algebraic-geometric interpretations of the classical Ramanujan–Petersson identities. Note that for Selberg's conjecture one again assumes that $\pi_\infty$ is of class-one and $\pi_\infty=\pi(s_{1,\infty},s_{2,\infty})$ with

$$\operatorname{Re}(s_{i,\infty})=0.$$

It was with this modern representation-theoretic point of view that progress was made on Selberg's theorem.

First, consider the mapping

$$ \operatorname{Sym}^m : GL(2,\CC) \longrightarrow GL(m+1,\CC)$$

defined by action of $GL(2,\CC)$ on symmetric tensors of rank $m$. It was conjectured by R.P. Langlands [a8] that there should be a corresponding mapping $\operatorname{Sym}^m_\star$ that (roughly) maps cuspidal automorphic representations of $GL(2)$ to those of $GL(m+1)$ ; moreover, whenever $\pi$ corresponds to class-one representations indexed by

$$ \left(\begin{array}{rr} \alpha_p & 0 \\ 0 & \beta_p \end{array}\right) $$

(including possibly $p=\infty$), $\operatorname{Sym}^m_\star(\pi)$ should correspond to $\operatorname{Sym}^m\left(\begin{array}{rr} \alpha_p & 0 \\ 0 & \beta_p \end{array}\right)$. This conjecture, for all $m$, implies both the Selberg and the Ramanujan–Petersson conjecture. In 1978, S. Gelbart and H. Jacquet [a3] proved Langlands' conjecture for $m=2$; for Selberg's conjecture, this simply replaced the equality in his theorem by an inequality. Then, in 1994 [a7], W. Luo, Z. Rudnick and P. Sarnak used $\operatorname{Sym}^2_\star$ and analytic properties of L-functions to go well beyond Selberg's conjecture:

$$\ell_1(N) \geq \frac{171}{784} \simeq 0.2181...$$

And in 2000, H. Kim and F. Shahidi [a6] proved Langlands' conjecture for $m=3$ and established $\ell_1(N) \geq 0.22837...$, i.e.,

$$|\operatorname{Re}(s_{p,i})| \leq \frac{5}{34}$$

Either Selberg's conjecture will continue to be proved along the lines of Langlands' conjecture, or by entirely new ideas.

There is a far-reaching generalization of Selberg's conjecture to $GL(n)$: If $\pi=\otimes_{p\leq\infty} \pi_p$ is an irreducible cuspidal automorphic representation of $GL(n,\mathcal{A})$, then every class-one local representation of $GL(n,\QQ_p)$ is "tempered" .

References

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