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 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 latter projectivity results to derive a Kida formula for the signed Selmer groups, and study the integrality property of the characteristic element attached to the signed Selmer groups.

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

Let $N/K$ be a finite Galois extension of $p$-adic number fields and let $\rho^\mathrm{nr} : G_K \to \mathrm{Gl}_r(\mathbb Z_p)$ be an $r$-dimensional unramified representation of the absolute Galois group $G_K$ which is the restriction of an unramified representation $\rho^\mathrm{nr}_{\mathbb Q_p} : G_{\mathbb Q_p} \to \mathrm{Gl}_r(\mathbb Z_p)$. In this paper we consider the $\mathrm{Gal}(N/K)$-equivariant local $\epsilon$-conjecture for the $p$-adic representation $T = \mathbb Z_p^r(1)(\rho^\mathrm{nr})$. For example, if $A$ is an abelian variety of dimension $r$ defined over $\mathbb Q_p$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of $A$ is a $p$-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$. This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

The epsilon constant conjecture for higher dimensional unramified twists of $\mathbb Z_p^r(1)$

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

The epsilon constant conjecture for higher dimensional unramified twists of

We study the action of the Galois group $G$ of a finite extension $K/k$ of number fields on the points on an elliptic curve $E$. For an odd prime $p$, we aim to determine the structure of the $p$-adic completion of the Mordell-Weil group $E(K)$ as a $\mathbb{Z}_p[G]$-module only using information of $E$ over $k$ and the completions of $K$.

Mordell-Weil group as Galois modules

Let $L/K$ be a Galois extension of local fields of characteristic $0$ with Galois group $G$. If $\mathcal{F}$ is a formal group over the ring of integers in $K$, one can associate to $\mathcal F$ and each positive integer $n$ a $G$-module $F_L^n$ which as a set is the $n$-th power of the maximal ideal of the ring of integers in $L$. We give explicit necessary and sufficient conditions under which $F_L^n$ is a cohomologically trivial $G$-module. This has applications to elliptic curves over local fields and to ray class groups of number fields.

On formal groups and Tate cohomology in local fields

. To each Galois extension L/K of number ﬁelds with Galois group G and each integer r ≤ 0 one can associate Stickelberger elements in the centre of the rational group ring Q [ G ] in terms of values of Artin L -series at r . We show that the denominators of their coeﬃcients are bounded by the cardinality of the commutator subgroup G ′ of G whenever G is nilpotent. Moreover, we show that, after multiplication by | G ′ | and away from 2-primary parts, they annihilate the class group of L if r = 0 and higher Quillen K -groups of the ring of integers in L if r < 0. This generalizes recent progress on conjectures of Brumer and of Coates and Sinnott from abelian to nilpotent extensions. For arbitrary G we show that the denominators remain bounded along the cyclotomic Z p -tower of L for every odd prime p . This allows us to give an aﬃrmative answer to a question of Greenberg and of Gross on the behaviour of p -adic Artin L -series at zero.

Nils Ellerbrock

Papers

On formal groups and Tate cohomology in local fields

On formal groups and Tate cohomology in local fields

Integrality of Stickelberger elements and annihilation of natural Galois modules