On Selberg’s eigenvalue conjecture. (English) Zbl 0844.11038
Suppose that \(\Gamma\) is a congruence subgroup of \(\text{SL} (2, \mathbb{Z})\). Suppose that \(0= \lambda_0< \lambda_1\leq \dots\) are the eigenvalues of the non-Euclidean Laplacian on \(L^2 (\Gamma \setminus H)\), where \(H\) is the Poincaré upper half plane. Selberg conjectured in 1965 that \(\lambda_1\geq 0.25\) and proved that \(\lambda_1\geq 0.1875\). The present paper shows that \(\lambda_1\geq 0.21\). The Selberg conjecture can be viewed as the archimedean analogue of the Ramanujan conjecture bounding Fourier coefficients of Maass wave forms. More progress has been made for the non-archimedean conjectures. The aim of this paper is to “restore the balance and establish in part for the archimedean place what is known at the finite places”. Applications are presented towards the Linnik-Selberg conjecture on the cancellation in sums of Kloosterman sums and to the error term in the prime geodesic theorem for congruence subgroups. For the latter an error term \(O(x^{3/4})\) was known; see P. Sarnak [Ph.D. Thesis (Stanford, 1980)]. The present paper replaces \(.75\) with \(.7\). The natural conjecture here is an error term with exponent \(.5+ \varepsilon\), for any \(\varepsilon>0\).
The proof of the main theorem makes use of various ingredients. Ingredient \(\# 1\) is the Gelbart-Jacquet lift; see S. Gelbart and H. Jacquet [Ann. Sci. Éc. Norm. Supér., IV. Sér. 11, 471-542 (1978; Zbl 0406.10022)]. Ingredient \(\# 2\) is the Rankin-Selberg theory of \(L\)-functions for the general linear group as in H. Jacquet, I. I. Piatetskii-Shapiro and J. A. Shalika [Am. J. Math. 105, 367-464 (1983; Zbl 0525.22018)], F. Shahidi [Am. J. Math. 103, 297-355 (1981; Zbl 0467.12013)], and C. Moeglin and J.-L. Waldspurger [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, 605-674 (1989; Zbl 0696.10023)]. Another ingredient is Deligne’s bounds on hyper-Kloosterman sums [see P. Deligne, Lect. Notes Math. 569 (1977; Zbl 0345.00010)]. And a final ingredient is an approximate functional equation of a twisted \(L\)-function \(L(s, \chi)= \sum^\infty_{n=1} {{b(n) \chi(n)} \over {n^s}}\), where \(\chi\) is a primitive, non-trivial Dirichlet character \(\text{mod } q\) and the \(b(n)\) are the coefficients of the Dirichlet series for \(L(s, \pi\times \widetilde {\pi})\), where \(\pi\) is a cuspidal automorphic representation of \(\text{GL} (m)\) over \(\mathbb{Q}\). This approximate equation is needed to show that these twisted \(L\)-functions do not vanish at a given point. This gives the main Theorem because of the form of the gamma factor for the twisted Rankin-Selberg \(L\)-function.
The authors note that the full Selberg conjecture (or Ramanujan conjecture) would follow if one knew that for any irreducible cuspidal automorphic representation \(\pi\) and any \(\beta\) with \(\text{Re} (\beta)>0\), there is an even Dirichlet character such that \(L(\beta, \pi\otimes \chi)\neq 0\). Many people have investigated such questions (e.g., G. Shimura, H. Iwaniec, and L. Barthel and D. Ramakrishnan).
It is possible to obtain the estimates in this paper when the field of rational numbers is replaced with a number field. This will appear in a future paper.
The proof of the main theorem makes use of various ingredients. Ingredient \(\# 1\) is the Gelbart-Jacquet lift; see S. Gelbart and H. Jacquet [Ann. Sci. Éc. Norm. Supér., IV. Sér. 11, 471-542 (1978; Zbl 0406.10022)]. Ingredient \(\# 2\) is the Rankin-Selberg theory of \(L\)-functions for the general linear group as in H. Jacquet, I. I. Piatetskii-Shapiro and J. A. Shalika [Am. J. Math. 105, 367-464 (1983; Zbl 0525.22018)], F. Shahidi [Am. J. Math. 103, 297-355 (1981; Zbl 0467.12013)], and C. Moeglin and J.-L. Waldspurger [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, 605-674 (1989; Zbl 0696.10023)]. Another ingredient is Deligne’s bounds on hyper-Kloosterman sums [see P. Deligne, Lect. Notes Math. 569 (1977; Zbl 0345.00010)]. And a final ingredient is an approximate functional equation of a twisted \(L\)-function \(L(s, \chi)= \sum^\infty_{n=1} {{b(n) \chi(n)} \over {n^s}}\), where \(\chi\) is a primitive, non-trivial Dirichlet character \(\text{mod } q\) and the \(b(n)\) are the coefficients of the Dirichlet series for \(L(s, \pi\times \widetilde {\pi})\), where \(\pi\) is a cuspidal automorphic representation of \(\text{GL} (m)\) over \(\mathbb{Q}\). This approximate equation is needed to show that these twisted \(L\)-functions do not vanish at a given point. This gives the main Theorem because of the form of the gamma factor for the twisted Rankin-Selberg \(L\)-function.
The authors note that the full Selberg conjecture (or Ramanujan conjecture) would follow if one knew that for any irreducible cuspidal automorphic representation \(\pi\) and any \(\beta\) with \(\text{Re} (\beta)>0\), there is an even Dirichlet character such that \(L(\beta, \pi\otimes \chi)\neq 0\). Many people have investigated such questions (e.g., G. Shimura, H. Iwaniec, and L. Barthel and D. Ramakrishnan).
It is possible to obtain the estimates in this paper when the field of rational numbers is replaced with a number field. This will appear in a future paper.
Reviewer: A.A.Terras (La Jolla)
MSC:
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 11L05 | Gauss and Kloosterman sums; generalizations |
Keywords:
Selberg’s eigenvalue conjecture; eigenvalues of the non-Euclidean Laplacian for congruence subgroups; sums of Kloosterman sums; prime geodesic theorem; error term; Gelbart-Jacquet lift; Rankin-Selberg theory of \(L\)-functions; bounds on hyper-Kloosterman sums; approximate functional equation; twisted Rankin-Selberg \(L\)-function; Ramanujan conjectureReferences:
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