## Eta invariants, signature defects of cusps, and values of L-functions.(English)Zbl 0531.58048

The neighbourhood of a cusp on a Hilbert modular variety over a totally real field F, $$[F:{\mathbb{Q}}]=n=2k$$, is of the form W/G, where $W=W(d)=\{z\in {\mathbb{C}}^ n| Im (z_ j)>0,\quad \prod Im (z_ j)\geq d\},\quad d>0$ and $$G=G(M,V)$$ denotes the following group of automorphisms of $$W:z\mapsto \epsilon z+\mu \in W$$ with $$\epsilon\in V$$, $$\mu\in M$$. Here M is a lattice in F and V a totally positive unit group in F (both of maximal rank) with $$VM=M$$. The components of $$\epsilon z+\mu$$ are $$\epsilon^{(j)}z_ j+\mu^{(j)}$$ with the conjugate numbers $$\epsilon^{(j)}$$, $$\mu^{(j)}$$ of $$\epsilon$$,$$\mu$$. The boundary $$X=\partial W/G$$ of W/G is an n-torus bundle over the (n-1)- torus $${\mathbb{R}}^{n-1}/V$$ with fiber $${\mathbb{R}}^ n/M$$. To the torus bundle X one can associate as follows a rational number $$\sigma$$ (X), called the signature defect of X. The coordinates $$x_ 1,y_ 1,...,x_ n,y_ n (z_ j=x_ j+iy_ j)$$ on $${\mathbb{C}}^ n$$ induce a flat connection $$\nabla^ L$$ and a Riemannian metric g on the tangent bundle of X. All Pontryagin and Stiefel-Whitney numbers of X vanish, so there is a compact oriented manifold Y with $$\partial Y=X$$. Let $$L_ k(p_ 1,...,p_ k)\in H^{4k}(Y,X)$$ be the Hirzebruch L-polynomial in the relative Pontryagin classes $$p_ j$$ of Y. The signature defect is then defined as the difference $\sigma(X):=L_ k(p_ 1,...,p_ k)[Y,X]- sign(Y).$ Applying the signature theorem and the Novikov additivity of the signature one sees that $$\sigma$$ (X) does not depend on the particular choice of Y. In the paper under review the authors prove the equation $$\sigma(X)=L(0)$$, where L(0) is the value of the L-function L(s) at $$s=0$$, given for $$Re(s)>1$$ by $$L(s)=\sum N(\mu)| N(\mu)|^{- 1-s}$$, where $$\mu\neq 0$$ runs over a system of representatives for $$M^*/V$$ with the dual lattice $$M^*$$ of M with respect to the trace in F. This equation has been proved in the case $$n=2$$ and then in full generality conjectured by F. Hirzebruch [Enseignement Math. 19, 183-281 (1973; Zbl 0285.14007)]. Motivated by this conjecture Atiyah, Patodi and Singer introduced about ten years ago the spectral $$\eta$$- invariant and extended with the help of this invariant the Hirzebruch signature theorem to compact oriented Riemannian manifolds with boundary [M. F. Atiyah, V. K. Patodi and I. M. Singer, Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008)]. Let A be the elliptic self-adjoint operator $$A=\pm(*d-d*)$$ acting on forms of even degree, $$A:\Gamma(\wedge^{ev}T^*X)\to \Gamma(\wedge^{ev}T^*X)$$. Denote by $$\lambda$$ the eigenvalues of A. The $$\eta$$-invariant $$\eta_ A(0)$$ of A is defined as the value at $$s=0$$ of the meromorphic function $$\eta_ A(s)$$ given for $$Re(s)>>0$$ by $$\eta_ A(s)=\sum \lambda | \lambda |^{-1-s}$$. Note that $$\eta_ A(0)$$ depends on the chosen connection on TX. In the case of the Levi-Cività connection on TX (with respect to the metric g) the equation $$\eta_ A(0)=\sigma(X)$$ has been proved in the cited paper. Here the authors prove first that $$\eta_ A(0)$$ does not change by replacing the Levi-Cività connection by the flat connection $$\nabla^ L$$. In the second step the authors prove the equality $$\eta_ A(0)=L(0)$$, where $$\eta_ A$$ is defined with respect to the flat connection $$\nabla^ L$$. This part constitutes in fact the bulk of the paper. The basic idea here is to unwind the operator A along the fibers of X. In this way one gets for $$Re(s)>>0$$ the formula $\eta_ A(s)=\sum N(\mu)| N(\mu)|^{-1-s/n}\eta(h(\mu),s),\quad 0\neq \mu \in M^*/V$ with $$h=h(\mu)=| N(\mu)|^{-1/n}$$ and the $$\eta$$-function $$\eta$$ (h,s) of certain family of operators $$B_ h:L^ 2({\mathbb{R}}^{n-1};\wedge^{ev}{\mathbb{R}}^{2n- 1})\circlearrowright$$. This gives the representation $\eta_ A(s)=\eta(\alpha,s)L(s/n)+\gamma(s),$
$\gamma(s)=\sum N(\mu)| N(\mu)|^{-1-s/n}(\eta(h(\mu),s)-\eta(\alpha,s))$ with a suitable constant $$\alpha>h(\mu)$$ for all $$\mu$$. The crucial point in the proof is to show $$\gamma(0)=0$$ and $$\eta(\alpha,0)=1$$. Here the authors have to cope with substantial difficulties. To this end they use the heat- equation method and the Feynman-Kac representation of the heat kernel to estimate the integral representation of $$\eta$$ ($$h(\mu)$$,s)-$$\eta$$ ($$\alpha$$,s). The essential point, however, seems to be algebraic in nature, namely the vanishing of certain traces (cf. Lemma 10.3).
Reviewer: R.Sczech

### MSC:

 58J35 Heat and other parabolic equation methods for PDEs on manifolds 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 14J25 Special surfaces 14J17 Singularities of surfaces or higher-dimensional varieties 14B05 Singularities in algebraic geometry 14G25 Global ground fields in algebraic geometry 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 57R20 Characteristic classes and numbers in differential topology 11R52 Quaternion and other division algebras: arithmetic, zeta functions 11R80 Totally real fields 53C05 Connections (general theory)

### Citations:

Zbl 0285.14007; Zbl 0297.58008
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