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Let $\HH$ denote the upper half-plane, $SL(2,\ZZ)$ the group of integer matrices of determinant one and
 
Let $\HH$ denote the upper half-plane, $SL(2,\ZZ)$ the group of integer matrices of determinant one and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s1302103.png" /></td> </tr></table>
+
$$\Gamma_0(N)= \left\{ \left(\begin{array}{rr}
 +
a & b \\
 +
c & d
 +
\end{array}\right) \in SL(2,\ZZ) : c \equiv 0 \,(\operatorname{mod} N) \right\}.$$
  
Following H. Maass [[#References|[a9]]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s1302104.png" /> denote the space of bounded functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s1302105.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s1302106.png" /> that satisfy
+
Following H. Maass [[#References|[a9]]], let $W_s(\Gamma_0(N))$ denote the space of bounded functions $f$ on $\Gamma_0(N) \backslash \HH$ that satisfy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s1302107.png" /></td> </tr></table>
+
$$\Delta f = \left( \frac{1-s^2}{4} \right) f,$$
  
for
+
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s1302108.png" /></td> </tr></table>
+
$$\Delta = -y^2 \left( \frac{\partial^2}{\partial^2_x} + \frac{\partial^2}{\partial^2_y} \right)$$
  
the Laplace–Beltrami operator (cf. also [[Laplace operator|Laplace operator]]). Such eigenfunctions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s1302109.png" /> are called Maass wave forms. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021010.png" /> in this context is essentially self-adjoint and non-negative (cf. also [[Self-adjoint operator|Self-adjoint operator]]), it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021011.png" /> is real and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021012.png" />.
+
is the Laplace–Beltrami operator (cf. also [[Laplace operator|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|Self-adjoint operator]]), it follows that $(1-s^2)/4$ is real and not zero.
  
A. Selberg conjectured [[#References|[a12]]] that there is a lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021013.png" /> for the smallest (non-zero) eigenvalue: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021014.png" />,
+
A. Selberg conjectured [[#References|[a12]]] that there is a lower bound $\ell_1(N)$ for the smallest (non-zero) eigenvalue: For $N \not= 1$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021015.png" /></td> </tr></table>
+
$$\ell_1(N) \geq 1/4.$$
  
This innocent looking conjecture is (cf. [[#References|[a10]]]) one of the fundamental unsolved questions in the theory of modular forms (as of 2000; cf. also [[Modular form|Modular form]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021016.png" /> and for small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021017.png" />, it has been known for some time (Selberg, W. Roelcke). In general, it has many applications to classical number theory (see [[#References|[a4]]] and [[#References|[a12]]], for example). To back up his conjecture, Selberg also proved the following assertion:
+
This innocent looking conjecture is (cf. [[#References|[a10]]]) one of the fundamental unsolved questions in the theory of modular forms (as of 2000; cf. also [[Modular form|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 [[#References|[a4]]] and [[#References|[a12]]], for example). To back up his conjecture, Selberg also proved the following assertion:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021018.png" /></td> </tr></table>
+
$$\ell_1(N) \geq 3/16.$$
  
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|Exponential sum estimates]]; [[Trigonometric sum|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 [[#References|[a13]]]. This bound, in turn, is a consequence of the Riemann hypothesis for the [[Zeta-function|zeta-function]] of a curve over a finite field, which he had proven earlier (cf. also [[Riemann hypotheses|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021019.png" />. It is interesting that H. Iwaniec [[#References|[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.
+
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|Exponential sum estimates]]; [[Trigonometric sum|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 [[#References|[a13]]]. This bound, in turn, is a consequence of the Riemann hypothesis for the [[Zeta-function|zeta-function]] of a curve over a finite field, which he had proven earlier (cf. also [[Riemann hypotheses|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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021020.png" />-adic groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021021.png" /> inside a cuspidal representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021022.png" /> (see below), Selberg's conjecture will follow as a statement for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021023.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021024.png" /> denote the completion of the rational field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021025.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021026.png" />-adic absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021028.png" />, and view <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021029.png" /> as the completion with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021030.png" />. By the adèles, denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021031.png" />, one means the "restricted" direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021032.png" /> (restricted so that almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021033.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021034.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021035.png" />; cf. also [[Adèle|Adèle]]). Secondly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021036.png" /> denote the space of measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021039.png" /> ( "restricted" means <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021040.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021041.png" /> for almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021042.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021043.png" /> imbedded diagonally in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021044.png" />,
+
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|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 $G_\mathcal{A}$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021045.png" /></td> </tr></table>
+
$$Z_\mathcal{A} =\left\{ \left(\begin{array}{rr}
 +
z & 0 \\
 +
0 & z
 +
\end{array}\right): z \in \mathcal{A} \right\}.$$
  
is the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021046.png" />,
+
is the centre of $G_\mathcal{A}$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021047.png" /></td> </tr></table>
+
$$\int_{Z_\mathcal{A}G_\QQ \backslash G_\mathcal{A}} |\phi(g)|^2 dg < \infty $$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021048.png" /></td> </tr></table>
+
$$ \int_{\QQ \backslash \mathcal{A}} \phi\left(\left(\begin{array}{rr}
 +
1 & x \\
 +
0 & 1
 +
\end{array}\right)g\right) dx = 0$$
 +
 
 +
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
  
for almost every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021049.png" />. Now, assuming <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021050.png" /> is an eigenfunction of all Hecke operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021051.png" />, one can define, in a one-to-one way, a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021052.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021053.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021054.png" /> generated by the right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021055.png" />-translates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021056.png" /> is an irreducible subrepresentation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021057.png" />. Then Selberg's conjecture states that the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021059.png" /> is a principal series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021060.png" /> with trivial central character and
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021061.png" /></td> </tr></table>
+
$$\frac{1-s^2}{4} = \frac{1+4(t_1-t_2)^2}{4} \geq \frac{1}{4}.$$
  
In other words, complementary series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021062.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021063.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021065.png" /> (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021066.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021068.png" />) should not occur (cf. [[#References|[a2]]]; see also [[Irreducible representation|Irreducible representation]]; [[Principal series|Principal series]]). In this context the Ramanujan–Petersson conjecture says that for (almost every) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021069.png" /> the same conclusion holds, i.e., for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021070.png" /> a class-one representation, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021072.png" /> satisfies (cf. [[#References|[a11]]])
+
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. [[#References|[a2]]]; see also [[Irreducible representation|Irreducible representation]]; [[Principal series|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. [[#References|[a11]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021073.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$\operatorname{Re}(s_{1,p}) = 0 .$$
  
P. Deligne proved the original Ramanujan conjecture [[#References|[a1]]] when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021074.png" /> is a holomorphic discrete series of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021075.png" />. For example, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021076.png" /> equals Ramanujan's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021077.png" />, the condition (a1) implies
+
P. Deligne proved the original Ramanujan conjecture [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021078.png" /></td> </tr></table>
+
$$|\tau(p)| \leq p^{11/2} (p^{s_{1,p}}+p^{s_{2,p}}) \leq 2 p^{11/2}$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021079.png" /> is of class-one and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021080.png" /> with
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021081.png" /></td> </tr></table>
+
$$\operatorname{Re}(s_{i,\infty})=0.$$
  
 
It was with this modern representation-theoretic point of view that progress was made on Selberg's theorem.
 
It was with this modern representation-theoretic point of view that progress was made on Selberg's theorem.
Line 57: Line 69:
 
First, consider the mapping
 
First, consider the mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021082.png" /></td> </tr></table>
+
$$ \operatorname{Sym}^m : GL(2,\CC) \longrightarrow GL(m+1,\CC)$$
  
defined by action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021083.png" /> on symmetric tensors of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021084.png" />. It was conjectured by R.P. Langlands [[#References|[a8]]] that there should be a corresponding mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021085.png" /> that (roughly) maps cuspidal automorphic representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021086.png" /> to those "of GLm+ 1" ; moreover, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021087.png" /> corresponds to class-one representations indexed by
+
defined by action of $GL(2,\CC)$ on symmetric tensors of rank $m$. It was conjectured by R.P. Langlands [[#References|[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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021088.png" /></td> </tr></table>
+
$$
 +
\left(\begin{array}{rr}
 +
\alpha_p & 0 \\
 +
0 & \beta_p
 +
\end{array}\right)
 +
$$
  
(including possibly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021089.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021090.png" /> should correspond to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021091.png" />. This conjecture, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021092.png" />, implies both the Selberg and the Ramanujan–Petersson conjecture. In 1978, S. Gelbart and H. Jacquet [[#References|[a3]]] proved Langlands' conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021093.png" />; for Selberg's conjecture, this simply replaced the equality in his theorem by an inequality. Then, in 1994 [[#References|[a7]]], W. Luo, Z. Rudnick and P. Sarnak used <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021094.png" /> and analytic properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021095.png" />-functions to go well beyond Selberg's conjecture:
+
(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 [[#References|[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 [[#References|[a7]]], W. Luo, Z. Rudnick and P. Sarnak used $\operatorname{Sym}^2_\star$ and analytic properties of [[L-function]]s to go well beyond Selberg's conjecture:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021096.png" /></td> </tr></table>
+
$$\ell_1(N) \geq \frac{171}{784} \simeq 0.2181...$$
  
And in 2000, H. Kim and F. Shahidi [[#References|[a6]]] proved Langlands' conjecture for $m=3$ and established <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021098.png" />, i.e.,
+
And in 2000, H. Kim and F. Shahidi [[#References|[a6]]] proved Langlands' conjecture for $m=3$ and established $\ell_1(N) \geq 0.22837...$, ''i.e.'',
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s13021099.png" /></td> </tr></table>
+
$$|\operatorname{Re}(s_{p,i})| \leq \frac{5}{34}$$
  
 
Either Selberg's conjecture will continue to be proved along the lines of Langlands' conjecture, or by entirely new ideas.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s130210101.png" /> is an irreducible cuspidal automorphic representation of $GL(n,\mathcal{A})$, then every class-one local representation of $GL(n,\QQ_p)$ is "tempered" .
+
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====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–307 {{MR|0340258}} {{ZBL|0314.14007}} {{ZBL|0287.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Gel'fand, M. Graev, I. Piatetski-Shapiro, "Representation theory and automorphic functions" , W.B. Saunders (1969) {{MR|233772}} {{ZBL|0718.11022}} {{ZBL|0138.07201}} {{ZBL|0136.07301}} {{ZBL|0121.30601}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Gelbart, H. Jacquet, "A relation between automorphic representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s130210104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s130210105.png" />" ''Ann. Sci. École Norm. Sup.'' , '''11''' (1978) pp. 471–552 {{MR|0533066}} {{ZBL|0406.10022}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Hejhal, "The Selberg trace formula II" , ''Lecture Notes in Mathematics'' , '''1001''' , Springer (1983) {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H. Iwaniec, "Selberg's lower bound for the first eigenvalue of congruence groups" , ''Number Theory, Trace Formula, Discrete Groups'' , Acad. Press (1989) pp. 371–375 {{MR|993327}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> H. Kim, F. Shahidi, "Functorial products for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s130210106.png" /> and functorial symmetric cube for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130210/s130210107.png" />" ''C.R. Acad. Sci. Paris'' , '''331''' : 8 (2000) pp. 599–604 {{MR|1799096}} {{ZBL|1002.11044}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> W. Luo, Z. Rudnick, P. Sarnak, "On Selberg's eigenvalue conjecture" ''Geom. Funct. Anal.'' , '''5''' (1995) pp. 387–401 {{MR|1334872}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R.P. Langlands, "Problems in the theory of automorphic forms" , ''Lectures in Modern Analysis and Applications'' , ''Lecture Notes in Mathematics'' , '''170''' , Springer (1970) pp. 18–86 {{MR|0302614}} {{ZBL|0225.14022}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> H. Maass, "Nichtanalytishe Automorphe Funktionen" ''Math. Ann.'' , '''121''' (1949) pp. 141–183 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P. Sarnak, "Selberg's eigenvalue conjecture" ''Notices Amer. Math. Soc.'' , '''42''' : 4 (1995) pp. 1272–1277 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> I. Satake, "Spherical functions and Ramanujan's conjecture" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''IX''' , Amer. Math. Soc. (1966) pp. 258–264 {{MR|211955}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> A. Selberg, "On the estimation of Fourier coefficients of modular forms" , ''Proc. Symp. Pure Math.'' , '''VIII''' , Amer. Math. Soc. (1965) pp. 1–15 {{MR|0182610}} {{ZBL|0142.33903}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> A. Weil, "On some exponential sums" ''Proc. Nat. Acad. Sci.'' , '''34''' (1948) pp. 204–207 {{MR|0027006}} {{ZBL|0032.26102}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Deligne, "La conjecture de Weil I" ''Publ. Math. IHES'' , '''43''' (1974) pp. 273–307 {{MR|0340258}} {{ZBL|0314.14007}} {{ZBL|0287.14001}} </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> I. Gel'fand, M. Graev, I. Piatetski-Shapiro, "Representation theory and automorphic functions" , W.B. Saunders (1969) {{MR|233772}} {{ZBL|0718.11022}} {{ZBL|0138.07201}} {{ZBL|0136.07301}} {{ZBL|0121.30601}} </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Gelbart, H. Jacquet, "A relation between automorphic representations of $GL(2)$ and $GL(3)$" ''Ann. Sci. École Norm. Sup.'' , '''11''' (1978) pp. 471–552 {{MR|0533066}} {{ZBL|0406.10022}} </TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Hejhal, "The Selberg trace formula II" , ''Lecture Notes in Mathematics'' , '''1001''' , Springer (1983) {{MR|}} {{ZBL|0543.10020}} </TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> H. Iwaniec, "Selberg's lower bound for the first eigenvalue of congruence groups" , ''Number Theory, Trace Formula, Discrete Groups'' , Acad. Press (1989) pp. 371–375 {{MR|993327}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top"> H. Kim, F. Shahidi, "Functorial products for $GL(2) \times GL(3)$ and functorial symmetric cube for $GL(2)$" ''C.R. Acad. Sci. Paris'' , '''331''' : 8 (2000) pp. 599–604 {{MR|1799096}} {{ZBL|1002.11044}} </TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top"> W. Luo, Z. Rudnick, P. Sarnak, "On Selberg's eigenvalue conjecture" ''Geom. Funct. Anal.'' , '''5''' (1995) pp. 387–401 {{MR|1334872}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a8]</TD> <TD valign="top"> R.P. Langlands, "Problems in the theory of automorphic forms" , ''Lectures in Modern Analysis and Applications'' , ''Lecture Notes in Mathematics'' , '''170''' , Springer (1970) pp. 18–86 {{MR|0302614}} {{ZBL|0225.14022}} </TD></TR>
 +
<TR><TD valign="top">[a9]</TD> <TD valign="top"> H. Maass, "Nichtanalytishe Automorphe Funktionen" ''Math. Ann.'' , '''121''' (1949) pp. 141–183 {{MR|}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a10]</TD> <TD valign="top"> P. Sarnak, "Selberg's eigenvalue conjecture" ''Notices Amer. Math. Soc.'' , '''42''' : 4 (1995) pp. 1272–1277 {{MR|}} {{ZBL|}} </TD></TR>
 +
<TR><TD valign="top">[a11]</TD> <TD valign="top"> I. Satake, "Spherical functions and Ramanujan's conjecture" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''IX''' , Amer. Math. Soc. (1966) pp. 258–264 {{MR|211955}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> A. Selberg, "On the estimation of Fourier coefficients of modular forms" , ''Proc. Symp. Pure Math.'' , '''VIII''' , Amer. Math. Soc. (1965) pp. 1–15 {{MR|0182610}} {{ZBL|0142.33903}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> A. Weil, "On some exponential sums" ''Proc. Nat. Acad. Sci.'' , '''34''' (1948) pp. 204–207 {{MR|0027006}} {{ZBL|0032.26102}} </TD></TR>
 +
</table>

Latest revision as of 06:13, 1 April 2023


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

[a1] P. Deligne, "La conjecture de Weil I" Publ. Math. IHES , 43 (1974) pp. 273–307 MR0340258 Zbl 0314.14007 Zbl 0287.14001
[a2] I. Gel'fand, M. Graev, I. Piatetski-Shapiro, "Representation theory and automorphic functions" , W.B. Saunders (1969) MR233772 Zbl 0718.11022 Zbl 0138.07201 Zbl 0136.07301 Zbl 0121.30601
[a3] S. Gelbart, H. Jacquet, "A relation between automorphic representations of $GL(2)$ and $GL(3)$" Ann. Sci. École Norm. Sup. , 11 (1978) pp. 471–552 MR0533066 Zbl 0406.10022
[a4] D. Hejhal, "The Selberg trace formula II" , Lecture Notes in Mathematics , 1001 , Springer (1983) Zbl 0543.10020
[a5] H. Iwaniec, "Selberg's lower bound for the first eigenvalue of congruence groups" , Number Theory, Trace Formula, Discrete Groups , Acad. Press (1989) pp. 371–375 MR993327
[a6] H. Kim, F. Shahidi, "Functorial products for $GL(2) \times GL(3)$ and functorial symmetric cube for $GL(2)$" C.R. Acad. Sci. Paris , 331 : 8 (2000) pp. 599–604 MR1799096 Zbl 1002.11044
[a7] W. Luo, Z. Rudnick, P. Sarnak, "On Selberg's eigenvalue conjecture" Geom. Funct. Anal. , 5 (1995) pp. 387–401 MR1334872
[a8] R.P. Langlands, "Problems in the theory of automorphic forms" , Lectures in Modern Analysis and Applications , Lecture Notes in Mathematics , 170 , Springer (1970) pp. 18–86 MR0302614 Zbl 0225.14022
[a9] H. Maass, "Nichtanalytishe Automorphe Funktionen" Math. Ann. , 121 (1949) pp. 141–183
[a10] P. Sarnak, "Selberg's eigenvalue conjecture" Notices Amer. Math. Soc. , 42 : 4 (1995) pp. 1272–1277
[a11] I. Satake, "Spherical functions and Ramanujan's conjecture" , Algebraic Groups and Discontinuous Subgroups , Proc. Symp. Pure Math. , IX , Amer. Math. Soc. (1966) pp. 258–264 MR211955
[a12] A. Selberg, "On the estimation of Fourier coefficients of modular forms" , Proc. Symp. Pure Math. , VIII , Amer. Math. Soc. (1965) pp. 1–15 MR0182610 Zbl 0142.33903
[a13] A. Weil, "On some exponential sums" Proc. Nat. Acad. Sci. , 34 (1948) pp. 204–207 MR0027006 Zbl 0032.26102
How to Cite This Entry:
Selberg conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Selberg_conjecture&oldid=52534
This article was adapted from an original article by S. Gelbart (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article