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

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

Organizer(s) Abbes, Ahmed ; Mézard, Ariane ; Saito, Takeshi ; Zheng, Weizhe

Date(s) 30/05/2022 - 03/06/2022

linked URL https://conferences.cirm-math.fr/2534.html

00:00:00 / 00:00:00

3 5

On local epsilon factors of the vanishing cycles of isolated singularities

By Daichi Takeuchi

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.

Information about the video

Date of recording 31/05/2022
Date of publication 27/06/2022
Institutions CIRM , IHES
Licence CC BY NC ND
Language English
Audience Researchers, Graduate Students
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19928403
Cite this video Takeuchi, Daichi (31/05/2022). 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

Domain(s)

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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