Abstract : The Hasse-Weil zeta function of a regular proper flat scheme over the integers is expected to extend meromorphically to the whole complex plane and satisfy a functional equation. The local epsilon factors of vanishing cycles are the local factors of the constant term in the functional equation. For their absolute values, Bloch proposed a conjecture, called Bloch's conductor formula, which describes them in terms of the Euler characteristics of a certain (complex of) coherent sheaf. In this talk, under the assumption that the non-smooth locus is isolated and that the residue characteristic is odd, I explain that the coherent sheaf appearing in the Bloch's conjecture is naturally endowed with a quadratic form and I would like to propose a conjecture that describes the local epsilon factors themselves in terms of the quadratic form. The conjecture holds true in the following cases: 1) for non-degenerate quadratic singularities, 2) for finite extensions of local fields, or 3) in the positive characteristic case.

Keywords : local epsilon factor; vanishing cycles; quadratic form