Gergely Zábrádi

Characteristic elements, pairings and functional equations over the false Tate curve extension

We construct a pairing on the dual Selmer group over false Tate curve extensions of an elliptic curve with good ordinary reduction at a prime p=5. This gives a functional equation of the characteristic element which is compatible with the conjectural functional equation of the p-adic L-function. As an application we compute the characteristic elements of those modules - arising naturally in the Iwasawa-theory for elliptic curves over the false Tate curve extension - which have rank 1 over the subgroup of the Galois group fixing the cyclotomic extension of the ground field. We also show that the example of a non-principal reflexive left ideal of the Iwasawa algebra does not rule out the possibility that all torsion Iwasawa-modules are pseudo-isomorphic to the direct sum of quotients of the algebra by principal ideals.

Exactness of the reduction on étale modules

Abstract We prove the exactness of the reduction map from etale ( φ , Γ ) -modules over completed localized group rings of compact open subgroups of unipotent p-adic algebraic groups to usual etale ( φ , Γ ) -modules over Fontaines ring. This reduction map is a component of a functor from smooth p-power torsion representations of p-adic reductive groups (or more generally of Borel subgroups of these) to ( φ , Γ ) -modules. Therefore this gives evidence for this functor—which is intended as some kind of p-adic Langlands correspondence for reductive groups—to be exact. We also show that the corresponding higher Tor-functors vanish. Moreover, we give the example of the Steinberg representation as an illustration and show that it is acyclic for this functor to ( φ , Γ ) -modules whenever our reductive group is GL d + 1 ( Q p ) for some d ⩾ 1 .

Algebraic functional equations and completely faithful Selmer groups

Let E be an elliptic curve — defined over a number field K — without complex multiplication and with good ordinary reduction at all the primes above a rational prime p ≥ 5. We construct a pairing on the dual p∞-Selmer group of E over any strongly admissible p-adic Lie extension K∞/K under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group G = Gal(K∞/K). Under some mild additional hypotheses, this gives an algebraic functional equation of the conjectured p-adic L-function. As an application, we construct completely faithful Selmer groups in case the p-adic Lie extension is obtained by adjoining the p-power division points of another non-CM elliptic curve A.

Generalized Robba rings

We prove that any projective coadmissible module over the locally analytic distribution algebra of a compact p-adic Lie group is finitely generated. In particular, the category of coadmissible modules does not have enough projectives. In the Appendix a “generalized Robba ring” for uniform pro-p groups is constructed which naturally contains the locally analytic distribution algebra as a subring. The construction uses the theory of generalized microlocalization of quasi-abelian normed algebras that is also developed there. We equip this generalized Robba ring with a selfdual locally convex topology extending the topology on the distribution algebra. This is used to show some results on coadmissible modules.

Estimating the greatest common divisor of the value of two polynomials

Let

The p-adic Hodge decomposition according to Beilinson

p

On twists of modules over noncommutative Iwasawa algebras

be a fixed prime, and let

Automorphic Forms and Galois Representations: From étale P +-representations to G -equivariant sheaves on G/P

v(a)

Multivariable

stand for the exponent of

(\varphi, \Gamma)

p