zbMATH — the first resource for mathematics

Mutually isospectral Riemann surfaces. (English) Zbl 0997.53031
The authors deal with the following question: Given a natural number \(g\), how many Riemann surfaces \(S_1, \ldots, S_k\) of genus \(g\) can exist such that \(S_1, \ldots, S_k\) all share the same spectrum of the Laplacian? It was shown by P. Buser and M. Seppälä [Duke Math. J. 67, No. 1, 39-55 (1992; Zbl 0776.30032)] that there is an upper bound \(N(g)\) to the size of such isospectral sets, depending only on the genus: \(N(g) \leq e^{720g^2}.\) The problem of finding a lower bound for \(N(g)\) was addresses by R. Tse. He showed that there exists a sequence \(g_i \to \infty\) and a constant \(c\) such that \(N(g_i) \geq c \sqrt{g_i}.\)
In this paper, the authors exhibit a constant \(c\) and a sequence \(g_i \to \infty\) such that \(N(g_i) \geq g_i^{c\log(g_i)}.\) The number of isospectral, nonisometric Riemann surfaces of genus \(g\) grows faster than polynomially in \(g\). Their construction gives a value of \(c\) of approximately \(1/(4\log(2))\). More precisely, they prove that for each natural number \(n > 2\) and prime \(p\), the number \(N(g)\) of mutually isospectral Riemann sufraces of genus \(g = 1 + (n - 1)p^{2n}\) is at least \(N(g) \geq p^{n^2 - n}.\) They consider analogous problems in the setting of graphs and of number fields. They prove that for any \(n\), there are \(2^{n - 1}\) isospectral graphs that are 6-regular with \(4n\) vertices. They also prove that for natural numbers and odd primes \(p\) there exist \(N(d) = p^{n^2 - n}\) nonisomorphic number fields of degree \(d = p^{2n}\) over the rational numbers that have the same zeta function.

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
30F15 Harmonic functions on Riemann surfaces
11R21 Other number fields
Zbl 0776.30032
Full Text: DOI Link
[1] R. Brooks, A. Lubotzky, Non sunada graphs · Zbl 0926.58021
[2] Buser, P., Geometry and spectra of compact Riemann surfaces, (1992) · Zbl 0770.53001
[3] Cassels, J.W.S.; Fröhlich, A., Algebraic number theory, (1967), Academic Press San Diego · Zbl 0153.07403
[4] Cvetković, D.; Doob, M.; Gutman, I.; Torgasev, A., Recent results in the theory of the graph spectra, (1988), North-Holland Amsterdam · Zbl 0634.05054
[5] DeTurck, D.; Gordon, C.S., Isospectral deformations. I. Riemannian structures on two-step nilspaces, Comm. pure math., 40, 367-387, (1988) · Zbl 0649.53025
[6] Gordon, C.S.; Wilson, E., Isospectral deformations of compact solvmanifolds, J. differential geom., 19, 241-256, (1984) · Zbl 0523.58043
[7] Greenberg, L., Maximal Fuchsian groups, Bull. amer. math. soc., 69, 569-573, (1963) · Zbl 0115.06701
[8] G. Margulis, Discrete groups of motions of manifolds of non-positive curvature, Proceedings International Congress, Vancouver, 1974, 2, 21, 34 · Zbl 0336.57037
[9] Perlis, R., On the equationζK(sζK(s, J. number theory, 9, 342-360, (1977) · Zbl 0389.12006
[10] Reichardt, H., Konstruktion von zahlkoerpern mit gegebener galoisgruppe von primzahlpotentzordnung, J. reine angew. math., 177, 1-5, (1937) · JFM 63.0146.02
[11] Serre, J.P., Topics in Galois theory, (1992), Jones and Bartlett Boston
[12] Sunada, T., Riemannian coverings and isospectral manifolds, Ann. of math., 121, 169-186, (1985) · Zbl 0585.58047
[13] Tse, R., A lower bound for the number of isospectral surfaces of arbitrarily large genus, (1988), University of Southern California
[14] Tse, R., A lower bound for the number of isospectral surfaces, Recent developments in geometry, (1989), p. 161-164
[15] Vignéras, M.F., Exemples de sous-groupes discretes non-conjugués dePSL, C. R. acad. sci. Paris, 287, 47-49, (1978)
[16] Vignéras, M.F., Variétés riemanniennes isospectrales et non isométriques, Ann. of math., 112, 21-32, (1980) · Zbl 0445.53026
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.