The Bielefeld Algebraic and Arithmetic Geometry Oberseminar, Winter Semester 2019-2020

This semester, our Oberseminar will be about the companion conjecture. Our main reference will be
[Ca19] Cadoret, La conjecture des compagnons, Seminaire Bourbaki 1156
and the references therein. We will also use
[De12] Deligne, Finitude de l'extension de Q engendree par des traces de Frobenius en caracteristique finie
[Dr12] Drinfeld, On a conjecture of Deligne
[Ked06] Kedlaya, Fourier transform and p-adic Weil II
[Ked19] Kedlaya, Etale and crystalline companions, II, link
For more information, please contact Eike Lau.

Talks

Wednesday 23 October, 2019
14:15	V2-210	Eike Lau General introduction [Abstract] Introduction and program discussion.
16:15	V2-210	Thomas Zink Coefficient objects [Abstract] [Ca19] section 2.
Wednesday 13 November, 2019
14:15	V2-210	Fei Xie Ramification [Abstract] [Ca19] section 2.6. Tame ramification, the GrothendieckOgg-Shafarevich formula, tameness carries over to companions, semistable reduction theorem for isocrystals [Ca19] Theorem 4.2.
16:15	V2-210	Werner Hoffmann The case of Curves — Langlands correspondence [Abstract] [Ca19] section 3, in particular Corollary 3.6 and Corollary 3.9.
Wednesday 20 November, 2019
14:15	V2-210	Eike Lau Finiteness I — Quantitative version over curves. [Abstract] [De12] section 2, in particular Proposition 2.5 and Variant 2.13, or equivalently [Ca19] section 5.2.
Wednesday 11 December, 2019
14:15	V2-210	Charles Vial Finiteness II — Conclusion. [Abstract] [De12] Theorem 3.1 and Remark 3.10, or equivalently [Ca19], Theorem 5.1 and section 5.3. Use the terminology of 1-geometric skeletons as in [Ca19] section 4.
16:15	V2-210	Karsten Schroedter Existence of l-companions 1. [Abstract] Step 1: Each geometric skeleton comes from a coefficient object. This is [Ca19] Lemma 7.3 or [Dr12] Proposition 2.13. One uses a Bertini argument as in [Ca19] Theorem 6.1, which should also be explained.
Wednesday 8 January, 2020
14:15	V2-210	Ma Li Existence of l-companions 2. [Abstract] Step 2: Each 1-geometric skeleton becomes geometric over a dense open subscheme. [Ca19] Lemma 7.4 or [Dr12] Proposition 2.12.
16:15	V2-210	Michael Spiess Existence of l-companions 3. [Abstract] The third step in the proof is [Ca19] Lemma 7.5 or [Dr12] Proposition 2.14. This proves [Ca19] Theorem 7.1. Conclude with [Ca19] Corollary 7.2, and possibly also explain [Ca19] Lemma 4.1 or [Dr12] Lemma 2.7.
Wednesday 22 January, 2020
14:15	V2-210	Thomas Zink . Existence of p-companions. [Abstract] One uses elementary fibrations to prove that p-adic companions exist. [Ked19], in particular Lemma 3.1.7, Lemma 5.3.3, Lemma 6.2.2, Theorem 6.2.3.
16:15	V2-210	Jeanine van Order Field of definition. [Abstract] [Ca19] section 8

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Seminars

Conference

The Bielefeld Algebraic and Arithmetic Geometry Oberseminar, Winter Semester 2019-2020

Talks

Wednesday

23 October, 2019

Wednesday

13 November, 2019

Wednesday

20 November, 2019

Wednesday

11 December, 2019

Wednesday

8 January, 2020

Wednesday

22 January, 2020