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Showing 1 to 10 of 10 for “"Iwasawa theory"”.
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Iwasawa theory of p-adic Lie extensions
The Iwasawa theory of p-adic Lie groups investigates arithmetic objects above infinite field extensions of a number field k whose Galois group is a p-adic analytic group. The most prominent example (due to Serre) is produced by adjoining the p-torsion points of an elliptic curve defined over k …
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Iwasawa theory for modular forms at supersingular primes
… \pm-Selmer groups which are cotorsion over the Iwasawa algebra \Lambda=Zp[[Zp]]. We begin by studying the p-adic Hodge theory for the p-adic representation associated to f in the case when a_p=0. It allows us to give analogous definitions of Kobayashi's \pm-Coleman maps and \pm-Selmer groups. …
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Iwasawa theory for tensor products of Hilbert modular forms
The main conjecture of Iwasawa theory bridges two seemingly disjoint areas of mathematics: arithmetic and analysis. In particular, it provides a deep connection between the p-adic L-function which interpolates critical values of the complex L-series, and the Selmer group which is an important …
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Iwasawa theory over solvable three-dimensional p-adic Lie extensions
Iwasawa theory is a powerful tool which describes the mysterious relationship between arithmetic objects (motives) and the special values of L-functions. A precise form of this relationship is neatly encoded in the so-called "Iwasawa Main Conjecture". Classically the Main Conjecture (as formulated …
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Non-commutative Iwasawa theory of elliptic curves at primes of multiplicative reduction
… reduction. This thesis studies the Iwasawa theory of E over certain false Tate curve extensions F[infinity], with Galois group G = Gal(F[infinity]/Q). I show how the p[infinity]-Selmer group of E over F[infinity] controls the p[infinity]-Selmer rank growth within the false Tate curve …
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On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication
… of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of …
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First explicit reciprocity law for unitary Friedberg—Jacquet periods
… to the Bloch--Kato conjecture and to Iwasawa theory. This thesis studies the case of Galois representations attached to automorphic representations on a totally definite unitary group U(2r) over a CM field which are distinguished by the subgroup U(r) x U(r). We prove a new ``first …
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L-invariants and congruences for Galois representations of dimension 3, 4, and 8
… plays a central role in modern number theory. In this thesis we consider representations arising from tensor products of the two-dimensional representations attached to modular forms by Deligne. In particular, we shall study the Iwasawa theory of the adjoint representation, as well as …
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On Greenberg's question: an algebraic and computational approach
… equivalent number fields share the same Iwasawa invariants. In this dissertation it is shown that the problem naturally breaks up into four cases, depending on properties of Galois groups. This analysis is then used to give a positive answer to Greenberg’s question in some nontrivial …
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On Selmer groups and factoring p-adic L-functions
… result on the algebraic side involving classical Iwasawa modules, as predicted by the main conjectures for imaginary quadratic fields and Q. Our methods are inspired by this work of Greenberg. One key technical input to our methods is studying the behavior of Selmer groups under specialization.