Miscellany on traces in ℓ-adic cohomology: a survey

Illusie, Luc

doi:10.1007/s11537-006-0504-3

Cited by 23 publications

(7 citation statements)

References 37 publications

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“…The proof of this result depends heavily on the following results of Serre about congruence properties of elliptic curves as stated in his letter (for a general context, see [4]). …”

Section: Serre Congruence Properties and The Kkp Questionmentioning

confidence: 97%

On some congruence properties of elliptic curves

Qiu

2010

Arch. Math.

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In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of congruence properties of elliptic curves are also presented and discussed.Comment: 11 pages, The title is changed. Thanks to a result of J.-P. Serre from his letter on June 15, 2009 to the author, a complete solution for an open question of Kim, Koo and Park is obtained in this fifth revised version. Some related questions and results are also presented and discusse

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“…The proof of this result depends heavily on the following results of Serre about congruence properties of elliptic curves as stated in his letter (for a general context, see [4]). …”

Section: Serre Congruence Properties and The Kkp Questionmentioning

confidence: 97%

On some congruence properties of elliptic curves

Qiu

2010

Arch. Math.

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“…Let be quasi-finite over and let . If there exists an integer such that is compatible for all , then the same holds for all by [Zhe09, Proposition 1.15] (a consequence of Grothendieck’s arithmetic local monodromy theorem [ST68, Appendix] and a rationality lemma [Ill06, Lemma 8.1]).…”

Section: Compatible Systems Along the Boundarymentioning

confidence: 99%

Compatible systems and ramification

Zheng

2019

Compositio Math.

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We show that compatible systems of ℓ-adic sheaves on a scheme of finite type over the ring of integers of a local field are compatible along the boundary up to stratification. This extends a theorem of Deligne on curves over a finite field. As an application, we deduce the equicharacteristic case of classical conjectures on ℓ-independence for proper smooth varieties over complete discrete valuation fields. Moreover, we show that compatible systems have compatible ramification. We also prove an analogue for integrality along the boundary.

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“…This is an immediate consequence of Fujiwara’s trace formula (see [Fuj97, Corollary 5.4.5]; cf. [Ill06, § 3.5(b)]). ◻…”

Section: A Zeta Function Associated To a Self Mapmentioning

confidence: 99%

“…The result in [Ill06, § 3.5(b)] implies that the above power series is actually in and is independent of . The trace–determinant relation [Del74a, 1.5.3] implies that is a rational function in .…”

Section: A Zeta Function Associated To a Self Mapmentioning

confidence: 99%