Artin formalism and heat kernels.

*(English)*Zbl 0789.11055The Artin formalism for \(L\)-series of number fields is known to be satisfied in topology, in the context of the characteristic polynomial in linear algebra, when one has an endomorphism acting functorially on some representation spaces associated with a finite covering.

In the present paper, it is shown that Artin’s formalism is satisfied in the infinite-dimensional analogue of a one-parameter group \(\exp (-tA)\), where \(A\) is a positive self-adjoint operator. The kernel representing the operator \(\exp (-tA)\) is called the associated heat kernel and could be called the characteristic kernel, since it plays the role of the characteristic polynomial in the finite-dimensional case. It follows that any homomorphic image of the heat kernel (in a sense precisely defined) also satisfies the Artin formalism, for instance the trace of the heat kernel, the coefficients of the asymptotic expansion of this trace at the origin, the regularized determinant of the Laplacian operating on a metrized sheaf, the Selberg zeta function, etc. Other applications are given to derive systematically relations among theta functions. These include the classical Kronecker, Jacobi and Riemann relations.

In the present paper, it is shown that Artin’s formalism is satisfied in the infinite-dimensional analogue of a one-parameter group \(\exp (-tA)\), where \(A\) is a positive self-adjoint operator. The kernel representing the operator \(\exp (-tA)\) is called the associated heat kernel and could be called the characteristic kernel, since it plays the role of the characteristic polynomial in the finite-dimensional case. It follows that any homomorphic image of the heat kernel (in a sense precisely defined) also satisfies the Artin formalism, for instance the trace of the heat kernel, the coefficients of the asymptotic expansion of this trace at the origin, the regularized determinant of the Laplacian operating on a metrized sheaf, the Selberg zeta function, etc. Other applications are given to derive systematically relations among theta functions. These include the classical Kronecker, Jacobi and Riemann relations.

Reviewer: S.Lang (New Haven)

##### MSC:

11M41 | Other Dirichlet series and zeta functions |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

14H42 | Theta functions and curves; Schottky problem |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |

11R42 | Zeta functions and \(L\)-functions of number fields |

14K25 | Theta functions and abelian varieties |