Eta invariants, signature defects of cusps, and values of L-functions.

*(English)*Zbl 0531.58048The 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).

\[ \gamma(s)=\sum N(\mu)| N(\mu)|^{-1-s/n}(\eta(h(\mu),s)-\eta(\alpha,s)) \] 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) |