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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
##### MathOverflow Questions:
From Gassmann-Sunada triples to isospectral manifolds
##### Keywords:
Riemann surfaces; spectrum of Laplacian; graphs; number fields
Zbl 0776.30032
Full Text:
##### References:
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