On formal groups and Tate cohomology in local fields

Ellerbrock, Nils; Nickel, Andreas

doi:10.4064/aa170509-5-12

Cited by 3 publications

(1 citation statement)

References 6 publications

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“…We set and write for the inertia group. By [EN18, Lemma 2.3], it suffices to show that and for all , where denotes Tate cohomology. Because M is a (finite) p -group and , we get .…”

Section: The Cohomology Ofmentioning

confidence: 99%

The epsilon constant conjecture for higher dimensional unramified twists of (1)

Bley¹,

Cobbe

2021

Can. J. Math.-J. Can. Math.

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Let / be a finite Galois extension of -adic number fields and let nr : −→ Gl (Z ) be an -dimensional unramified representation of the absolute Galois group which is the restriction of an unramified representation nr Q : Q −→ Gl (Z ). In this paper we consider the Gal( / )-equivariant local -conjecture for the -adic representation = Z (1) ( nr ). For example, if is an abelian variety of dimension defined over Q with good ordinary reduction, then the Tate module =ˆassociated to the formal groupˆof is a -adic representation of this form. We prove the conjecture for all tame extensions / and a certain family of weakly and wildly ramified extensions / . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

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“…We set and write for the inertia group. By [EN18, Lemma 2.3], it suffices to show that and for all , where denotes Tate cohomology. Because M is a (finite) p -group and , we get .…”

Section: The Cohomology Ofmentioning

confidence: 99%

The epsilon constant conjecture for higher dimensional unramified twists of (1)

Bley¹,

Cobbe

2021

Can. J. Math.-J. Can. Math.

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On the cohomology of Kobayashi’s plus/minus norm groups and applications

Lim¹

2021

Math. Proc. Camb. Phil. Soc.

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The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed Selmer groups.

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On Refined Conjectures of Birch and Swinnerton-Dyer type for Hasse–Weil–Artin 𝐿-Series

Burns,

Macias Castillo

2024

Memoirs of the AMS

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We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse–Weil–Artin L L -series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult Archimedean considerations related to refinements of the conjecture of Deligne and Gross and delicate p p -adic congruence relations that involve the bi-extension height pairing of Mazur and Tate and are related to key aspects of noncommutative Iwasawa theory. In important special cases we provide strong evidence, both theoretical and numerical, in support of the conjectures.

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On formal groups and Tate cohomology in local fields

Cited by 3 publications

References 6 publications

The epsilon constant conjecture for higher dimensional unramified twists of (1)

The epsilon constant conjecture for higher dimensional unramified twists of (1)

On the cohomology of Kobayashi’s plus/minus norm groups and applications

On Refined Conjectures of Birch and Swinnerton-Dyer type for Hasse–Weil–Artin 𝐿-Series

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