The first eigenvalue problem and tensor products of zeta functions.(English)Zbl 1065.11035

The paper under review is concerned with Selberg’s eigenvalue conjecture for the lowest positive eigenvalue $$\lambda_1$$ of the Laplacian on a congruence subgroup of $$\text{SL}_2(\mathbb{Z})$$ or $$\text{SL}_2({\mathcal O}_\alpha)$$, where $${\mathcal O}_\alpha$$ is the ring of integers of an imaginary quadratic number field of discriminant $$d<0$$. The author introduces the following assumption on the integer $$n\geq 2$$ and a cuspidal automorphic representation $$\pi$$ of $$\text{GL}_2(\mathbb{A}_k)$$ with a number field $$k$$:
There is a cuspidal automorphic representation $$\pi_n=\text{Sym}^{n-1}(\pi)$$ of $$\text{GL}_n(\mathbb{A}_k)$$ whose $$L$$-function is equal to $$L(s,\text{Sym}^{n-1}(\pi))$$.
This assumption is known to hold for $$n\geq 5$$. If this assumption is true for all $$n$$ then Selberg’s conjecture holds true. Moreover, if this assumption holds for some integer $$n\geq 2$$ the author gives lower bounds for $$\lambda_1$$ depending on $$n$$ which give the best known lower bounds in the two-dimensional case (for $$n=4$$ or 5, respectively). For $$n=5$$ this bound yields in the three-dimensional case (i.e. for congruence subgroups of $$\text{SL}_2({\mathcal O}_d)): \lambda_1\geq \frac{160}{169}=0.946\dots$$ which is remarkably close to Selberg’s conjecture $$\lambda_1\geq 1$$.
In the last section the author comments on the general principle that tensor products of zeta-functions will be useful for solving problems of the type of the Riemann hypothesis, and he introduces a possibly useful candidate.

MSC:

 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
Full Text:

References:

 [1] Deligne, P.: La Conjecture de Weil. I. Inst. Hautes Êtudes Sci. Publ. Math., 43 , 273-307 (1974). · Zbl 0287.14001 [2] Kim, H.: Functoriality of the exterior square of $$GL_4$$ and the symmetric fourth power of $$GL_2$$. J. Amer. Math. Soc., 16 , 139-183 (2003). · Zbl 1018.11024 [3] Kurokawa, N.: Multiple zeta functions: an example. Adv. Stud. Pure Math., 21 , 219-226 (1992). · Zbl 0795.11037 [4] Kim, H., and Shahidi, F.: Functorial products for $$GL_3\times GL_3$$ and the symmetric cube for $$GL_2$$. Ann. of Math., 155 , 837-893 (2002). · Zbl 1040.11036 [5] Kim, H., and Sarnak, P.: Refined estimates towards the Ramanujan and Selberg Conjectures. J. Amer. Math. Soc., 16 , 175-181 (2003). · Zbl 1018.11024 [6] Kurokawa, N., and Koyama, S.: Multiple sine functions. Forum Math., 15 , 839-876 (2003). · Zbl 1065.11065 [7] Koyama, S., and Kurokawa, N.: Multiple zeta functions: the double sine function and the signatured double Poisson summation formula. Compositio Mathematica. · Zbl 1135.11329 [8] Koyama, S., and Kurokawa, N.: Multiple Euler products (2004). (Preprint). · Zbl 1061.11047 [9] Langlands, R.: Problems in the theory of automorphic forms. Lecture Notes in Math., vol. 170, Springer, Berlin, pp. 18-61 (1970). · Zbl 0225.14022 [10] Luo, W., Rudnick, Z., and Sarnak, P.: On Selberg’s eigenvalue conjecture. Geom. Funct. Anal., 5 , 387-401 (1995). · Zbl 0844.11038 [11] Luo, W., Rudnick, Z., and Sarnak, P.: On the generalized Ramanijan conjectures for $$GL(n)$$. Proc. Sympos. Pure Math., 66-2 , Amer. Math. Soc., Providence, RI, pp. 301-310 (1999). · Zbl 0965.11023 [12] Sarnak, P.: The arithmetic and geometry of some hyperbolic three manifolds. Acta Math., 151 , 253-295 (1983). · Zbl 0527.10022 [13] Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. London Math. Soc., 31 , 79-98 (1975). · Zbl 0311.10029 [14] Shahidi, F.: Automorphic $$L$$-functions: a survey. Automorphic forms, Shimura varieties and $$L$$-functions Vol. I. Perspect. Math., vol. 10, Academic Press, Boston, pp. 415-437 (1990). · Zbl 0715.11067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.