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On local epsilon factors of the vanishing cycles of isolated singularities

By Daichi Takeuchi

Appears in collection : Franco-Asian Summer School on Arithmetic Geometry in Luminy / Ecole d'été franco-asiatique sur la géométrie arithmétique à Luminy

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.

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Citation data

  • DOI 10.24350/CIRM.V.19928403
  • Cite this video Takeuchi Daichi (5/31/22). On local epsilon factors of the vanishing cycles of isolated singularities. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19928403
  • URL https://dx.doi.org/10.24350/CIRM.V.19928403

Bibliography

  • TAKEUCHI, Daichi. Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic. arXiv preprint arXiv:2010.11022, 2020. - https://doi.org/10.48550/arXiv.2010.11022

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