×

The Atiyah-Bott-Singer fixed point theorem and number theory. (English) Zbl 1061.58022

Yau, S.T. (ed.), Surveys in differential geometry. Papers dedicated to Atiyah, Bott, Hirzebruch and Singer . Somerville, MA: International Press (ISBN 1-57146-069-1/hbk). Surv. Differ. Geom., Suppl. J. Differ. Geom. 7, 313-326 (2000).
The first part of the paper describes the following beautiful application of the Atiyah-Bott-Singer Lefschetz formula to number theory. For a prime number \(p>3\), the corresponding principal congruence subgroup \(\Gamma (p)\) of the modular group \(\Gamma=\text{ PSL}_2({\mathbf Z})\) consists of those integral unimodular matrices which modulo \(p\) are equal to the identity. This group acts freely on the upper half-plane \(\mathbf H\), and the quotient \(\Gamma(p)\backslash\mathbf H\) is a non-compact Riemann surface, which covers the modular surface \(\Gamma\backslash\mathbf H\) finite-to-one. The Galois group of this covering is \(\text{ PSL}_2 ({\mathbf F}_p)=\Gamma/\Gamma(p)\), whose order is \(N=p(p^2-1)/2\). The Riemann surface \(\Gamma(p)\backslash{\mathbf H}\) can be compactified by adding \(N/p\) cusps, obtaining a compact Riemann surface \(X(p)\). Moreover there is an induced action of \(\text{ PSL}_2 ({\mathbf F}_p)\) on \(X(p)\) with quotient \(S^2\), and the author focuses on the induced representation of \(\text{ PSL}_2 ({\mathbf F}_p)\) on the finite-dimensional space \(H^{0,1}(X(p))\) of holomorphic \(1\)-forms on \(X(p)\).
According to Frobenius and Shur, when \(p\equiv3\bmod4\), among the irreducible representations of \(\text{ PSL}_2 ({\mathbf F}_p)\), there are two conjugate representations of \(\text{ PSL}_2 (\mathbf F_p)\) of degree \((p-1)/2\), denoted by \(\chi^+\) and \(\chi^-\). Let \(m\) and \(n\) denote the multiplicities of \(\chi^+\) and \(\chi^-\) in the above representation on \(H^{0,1}(X(p))\). Then the Theorem of Hecke asserts that \(m-n\) equals the class number \(h(-p)\) of the field \({\mathbf Q}({\sqrt{-p}})\) of discriminant \(-p\).
The author shows that this theorem also follows from the Atiyah-Bott-Singer Lefschetz formula for the Dolbeault operator, applied to the \(p\)-periodic transformation \(z \mapsto z+1\) of \(X(p)\); moreover \(m+n\) is calculated too.
The second part of the paper is a preliminary report on a joint work of the author with D. Zagier, generalizing the above ideas to the case of a compact connected Kählerian surface.
For the entire collection see [Zbl 1044.53002].

MSC:

58J20 Index theory and related fixed-point theorems on manifolds
11F11 Holomorphic modular forms of integral weight
PDFBibTeX XMLCite