Mutually isospectral Riemann surfaces. (English) Zbl 0997.53031

Brooks, Robert; Gornet, Ruth; Gustafson, William H.

Adv. Math. 138, No. 2, 306-322 (1998).

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.

Reviewer: S.Nikčević (Beograd)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
30F15	Harmonic functions on Riemann surfaces
11R21	Other number fields

Keywords:

Riemann surfaces; spectrum of Laplacian; graphs; number fields

Citations:

Zbl 0776.30032

References:

[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: 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: 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
[9]	Perlis, R., On the equation \(ζ_K(sζ_{K\) · 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: 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