The GL_2 main conjecture for elliptic curves without complex multiplication

Abstract

The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex L-functions, typified by the conjecture of Birch and Swinnerton-Dyer and its generalizations. Our goal in the present paper is to develop algebraic techniques which enable us to formulate a precise version of such a main conjecture for motives over a large class of p-adic Lie extensions of number fields. The paper ends by formulating and briefly discussing the main conjecture for an elliptic curve E over the rationals Q over the field generated by the coordinates of its p-power division points, where p is a prime greater than 3 of good ordinary reduction for E.

39 pages

References in corpus (2)

• Equivariant Bloch-Kato conjecture and non-abelian Iwasawa Main Conjecture
• A noncommutative Weierstrass preparation theorem and applications to Iwasawa theory

Cited by in corpus (65)

• Iwasawa theory and p-adic L-functions over ${\mathbf{Z}}_p^2$-extensions
• Computations in non-commutative Iwasawa theory
• On the Selmer groups of abelian varieties over function fields of characteristic p>0
• Local epsilon isomorphisms
• Equivariant Iwasawa theory and non-abelian Stark-type conjectures
• Fine Selmer groups of congruent Galois representations
• Iwasawa theory of totally real fields for certain non-commutative $p$-extensions
• Analogies between Knots and Primes, 3-Manifolds and Number Rings
• A remark on the ${\mathfrak{M}}_H(G)$-conjecture and Akashi series
• On the growth of Mordell-Weil ranks in $p$-adic Lie extensions
• Poitou-Tate duality over extensions of global fields
• Exactness of the reduction on étale modules
• Growth of Sha in towers for isogenous curves
• On Kato's local epsilon-isomorphism Conjecture for rank one Iwasawa modules
• The canonical fractional Galois ideal at s=0
• The main conjecture of Iwasawa theory for totally real fields
• Akashi series of Selmer groups
• Localisations and Completions of Skew Power Series Rings
• Proof of the Main Conjecture of Noncommutative Iwasawa Theory for Totally Real Number Fields in Certain Cases
• Congruences between abelian pseudomeasures, II
• Algebraic functional equations and completely faithful Selmer groups
• On refined conjectures of Birch and Swinnerton-Dyer type for Hasse-Weil-Artin L-series
• On the p-adic Stark conjecture at s=1 and applications
• Functoriality of the canonical fractional Galois ideal
• On the non-abelian Brumer-Stark conjecture and the equivariant Iwasawa main conjecture
• Fitting invariants in equivariant Iwasawa theory
• On the growth of even $K$-groups of rings of integers in $p$-adic Lie extensions
• Fine Selmer groups of congruent $p$-adic Galois representations
• A note on asymptotic class number upper bounds in $p$-adic Lie extensions
• On a noncommutative Iwasawa main conjecture for varieties over finite fields
• Control theorem and functional equation of Selmer groups over $p$-adic Lie extensions
• K_1 of some Iwasawa algebras
• Desingularization of complex multiple zeta-functions, fundamentals of $p$-adic multiple $L$-functions, and evaluation of their special values
• Non-commutative p-adic L-functions for supersingular primes
• Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p
• (ϕ,Γ)-modules over noncommutative overconvergent and Robba rings
• Non-vanishing of Artin-twisted L-functions of Elliptic Curves
• Some remarks on Kida's formula when $\mu \ne 0$
• Uniformizer of the False Tate Curve Extension of ${\mathbb{Q}}_p$
• On non-commutative Euler systems, I: preliminaries on `det' and `Fit'
• On the cohomology of Kobayashi's plus/minus norm groups and applications
• Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p
• An equivariant Iwasawa main conjecture for local fields
• Comparing the Selmer group of a $p$-adic representation and the Selmer group of the Tate dual of the representation
• On twists of modules over non-commutative Iwasawa algebras
• On the cohomology of plus/minus Selmer groups of supersingular elliptic curves in weakly ramified base fields
• Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions, III
• On completely faithful Selmer groups of elliptic curves and Hida deformations
• $K_0$-invariance of the completely faithful property of Iwasawa modules
• ${\mathfrak{M}}_H(G)$-property and congruence of Galois representations
• Akashi series, characteristic elements and congruence of Galois representations
• Congruences for Convolutions of Hilbert Modular Forms
• K_0 and the dimension filtration for p-torsion Iwasawa modules
• Reflexive Ideals in Iwasawa Algebras
• On Galois groups of unramified pro-p extensions
• Non-commutative Iwasawa theory of abelian varieties over global function fields
• Centres of skewfields and completely faithful Iwasawa modules
• On the transfer congruence between $p$-adic Hecke $L$-functions
• On a Noncommutative Iwasawa Main Conjecture for Function Fields
• On the vanishing of $S{K}_1$ of certain Iwasawa algebras
• On order of vanishing of characteristic elements
• On the non-commutative Main Conjecture for elliptic curves with complex multiplication
• Iwasawa theory of fine Selmer groups over global fields
• Kato's Local epsilon conjecture: $l\ne p$ case
• Tamagawa number conjecture for a $p$-adic family of uniform $F$-crystals