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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.

MSC:
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
30F15 Harmonic functions on Riemann surfaces
11R21 Other number fields
Citations:
Zbl 0776.30032
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References:
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