Mutually Isospectral Riemann Surfaces

✍ Scribed by Robert Brooks; Ruth Gornet; William H. Gustafson

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 327 KB
• Volume: 138
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1998.1750

✦ Synopsis

In this paper, we address the following question: Given a natural number g, how many Riemann surfaces S 1 , ..., S k of genus g can there be such that S 1 , ..., S k all share the same spectrum of the Laplacian?

It was shown by Buser in [Bu] that there is an upper bound N(g) to the size of such isospectral sets, depending only on the genus. More precisely, he gave the following upper estimate for N(g):

The problem of finding a lower bound for N( g) was addressed by R. Tse in [Tse1, 2], where he showed that: Theorem 0.2 ([Tse1, 2]). There exists a sequence g i Ä and a constant c such that N(g i ) cg i .

In this paper, we will exhibit a constant c and a sequence g i Ä such that

In particular, the number of isospectral, nonisometric Riemann surfaces of genus g grows faster than polynomially in g. Our construction will give a value of c of approximately 1Â(4 log(2)).

More precisely, we have:

Theorem 0.3. For each natural number n>2 and prime p, the number N(g) of mutually isospectral Riemann surfaces of genus g=1+(n&1) p 2n

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