Iwasawa Theory Research Articles

On prime ideals of noetherian skew power series rings

We study prime ideals in skew power series rings T:= R[[y; τ, δ]], for suitably conditioned complete right Noetherian rings R, automorphisms τ of R, and τ-derivations δ of R. Such rings were introduced by Venjakob, motivated by issues in noncommutative Iwasawa theory. Our main results concern “Cutting Down” and “Lying Over.” In particular, assuming that τ extends to a compatible automorphsim of T, we prove: If I is an ideal of R, then there exists a τ-prime ideal P of T contracting to I if and only if I is a τ-δ-prime ideal of R. Consequently, under the more specialized assumption that δ = τ − id (a basic feature of the Iwasawa-theoretic context), we can conclude: If I is an ideal of R, then there exists a prime ideal P of T contracting to I if and only if I is a τ-prime ideal of R. Our approach depends essentially on two key ingredients: First, the algebras considered are Zariskian (in the sense of Li and Van Oystaeyen), and so the ideals are all topologically closed. Second, topological arguments can be used to apply previous results of Goodearl and the author on skew polynomial rings.

Iwasawa theory for elliptic curves at supersingular primes: A pair of main conjectures

We extend Kobayashiʼs formulation of Iwasawa theory for elliptic curves at supersingular primes to include the case ap≠0, where ap is the trace of Frobenius. To do this, we algebraically construct p-adic L-functions Lp♯ and Lp♭ with the good growth properties of the classical Pollack p-adic L-functions that in fact match them exactly when ap=0 and p is odd. We then generalize Kobayashiʼs methods to define two Selmer groups Sel♯ and Sel♭ and formulate a main conjecture, stating that each characteristic ideal of the duals of these Selmer groups is generated by our p-adic L-functions Lp♯ and Lp♭. We then use results by Kato to prove a divisibility statement. For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Y7gPQsBZo6s.

ON THE TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER ?

Abstract. Let E be an elliptic curve over Q. Using Iwasawa theory,we give what seems to be the rst general upper bound for the order ofvanishing of the p -adic L -function at s = 0, and the Z p -corank of theTate-Shafarevich group for all ffitly large good ordinary primes p . 1. IntroductionLet E beanellipticcurvede nedover Q. WerecallthattheTate-Shafarevichgroup of E= Q is de ned byX( E= Q) = Ker( H 1 (Q ;E (Q)) ! ∏ v H 1 (Q v ;E (Q v ))) ; where v runs over all places of Q, and Q v is the completion of Q at v . Let p be a prime number. It is well-known that the p -primary subgroup of X( E= Q)has a nite Z p -corank, and we denote this corank by t p . It is conjectured that t p = 0 for every prime p , but this is unknown when the complex L -functionhas a zero of order at least 2 at s = 1. In principle, arguments from Galoiscohomology give an upper bound for t p , but the estimate is so bad that noone has ever written it down. In this paper, we will use p -adic argumentsfrom Iwasawa main conjecture, combined with a theorem in [1] on the non-vanishing of twisted complex

The commutativity of the Galois groups of the maximal unramified pro-p-extensions over the cyclotomic Zp-extensions

Let p be an odd prime number. For the cyclotomic Z p -extension F ∞ of a finite algebraic number field F , we denote by L ˜ ( F ∞ ) the maximal unramified pro- p -extension of F ∞ . In this paper, using Iwasawa theory and the theory of central class fields, we give necessary conditions for Gal ( L ˜ ( F ∞ ) / F ∞ ) to be abelian, and give sufficient conditions in certain special cases.

Coleman maps and thep-adic regulator

This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(\Gamma) the algebra of Qp-valued distributions on \Gamma = Gal(Qp(\mu (p^\infty) / Qp). Our first result determines the H(\Gamma)-elementary divisors of the quotient of D_{cris}(V) \otimes H(\Gamma) by the H(\Gamma)-submodule generated by (\phi * N(V))^{\psi = 0}, where N(V) is the Wach module of V. By comparing the determinant of this map with that of L_V (which can be computed via Perrin-Riou's explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when V arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato's main conjecture in terms of cotorsion Selmer groups and bounded p-adic L-functions.

IWASAWA THEORY FOR THE SYMMETRIC SQUARE OF A CM MODULAR FORM AT INERT PRIMES

AbstractLet f be a modular form with complex multiplication (CM) and p an odd prime that is inert in the CM field. We construct two p-adic L-functions for the symmetric square of f, one of which has the same interpolating properties as the one constructed by Delbourgo and Dabrowski (A. Dabrowski and D. Delbourgo, S-adic L-functions attached to the symmetric square of a newform, Proc. Lond. Math. Soc. 74(3) (1997), 559–611), whereas the other one has a similar interpolating properties but corresponds to a different eigenvalue of the Frobenius. The symmetry between these two p-adic L-functions allows us to define the plus and minus p-adic L-functions à la Pollack (R. Pollack, on the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118(3) (2003), 523–558). We also define the plus and minus p-Selmer groups analogous to the ones defined by Kobayashi (S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152(1) (2003), 1–36). We explain how to relate these two sets of objects via a main conjecture.

Refined Iwasawa theory and Kolyvagin systems of Gauss sum type

In this paper, we establish a refinement of the usual Iwasawa main conjecture for the ideal class groups of CM-fields over a totally real field, using higher Fitting ideals.

Algebraische Zahlentheorie

The workshop brought together researchers from Europe, Japan and the US, who reported on various recent developments in algebraic number theory and related fields. Dominant topics were Shimura varieties, automorphic forms and Iwasawa theory.

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori–Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate–Shafarevich group of an elliptic curve in the Z p 2 -extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.

NON-ABELIAN CONGRUENCES BETWEEN SPECIAL VALUES OF L-FUNCTIONS OF ELLIPTIC CURVES: THE CM CASE

In this work we prove congruences between special values of L-functions of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analog for elliptic curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive. The proof is based on the fact that the critical values of elliptic curves with CM, or what amounts to the same, the critical values of Grössencharacters, can be expressed as values of Hilbert–Eisenstein series at CM points. We believe that our strategy can be generalized to provide congruences for a large class of L-values.

Numerical Evidence Toward a 2-adic Equivariant “Main Conjecture”

We test a conjectural nonabelian refinement of the classical 2-adic Main Conjecture of Iwasawa theory. In the first part, we show how, in the special case that we study, the validity of this refinement is equivalent to a congruence condition on the coefficients of some power series. Then, in the second part, we explain how to compute the first coefficients of this power series and thus numerically check the conjecture in that setting.

On the tamagawa number conjecture for hecke characters

AbstractIn this paper, we prove the weak p‐part of the Tamagawa number conjecture in all non‐critical cases for the motives associated to Hecke characters of the form $\varphi ^a\overline{\varphi }^b$ where φ is the Hecke character of a CM elliptic curve E defined over an imaginary quadratic field K, under certain restrictions which originate mainly from the Iwasawa theory of imaginary quadratic fields. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Iwasawa theory for modular forms at supersingular primes

AbstractWe generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases

Fix an odd primepp. LetGGbe a compactpp-adic Lie group containing a closed, normal, pro-ppsubgroupHHwhich is abelian and such thatG/HG/His isomorphic to the additive group ofpp-adic integersZp\mathbb {Z}_p. First we assume thatHHis finite and compute the Whitehead group of the Iwasawa algebra,λ(G)\lambda (G), ofGG. We also prove some results about certain localisation ofλ(G)\lambda (G)needed in Iwasawa theory. LetFFbe a totally real number field and letF∞F_{\infty }be an admissiblepp-adic Lie extension ofFFwith Galois groupGG. The computation of the Whitehead groups are used to show that the Main Conjecture for the extensionF∞/FF_{\infty }/Fcan be deduced from certain congruences between abelianpp-adic zeta functions of Deligne and Ribet. We prove these congruences with certain assumptions onGG. This gives a proof of the Main Conjecture in many interesting cases such asZp⋊Zp\mathbb {Z}_p\rtimes \mathbb {Z}_p-extensions.

On the “main conjecture” of equivariant Iwasawa theory

We prove the “main conjecture” of equivariant Iwasawa theory whenμ=0\mu =0.

Congruences between derivatives of geometric L-functions

We prove a natural refinement of a theorem of Lichtenbaum describing the leading terms of Zeta functions of curves over finite fields in terms of Weil-etale cohomology. We then use this result to prove the validity of Chinburg’s Ω(3)-Conjecture for all abelian extensions of global function fields, to prove natural refinements and generalisations of the refined Stark conjectures formulated by, amongst others, Gross, Tate, Rubin and Popescu, to prove a variety of explicit restrictions on the Galois module structure of unit groups and divisor class groups and to describe explicitly the Fitting ideals of certain Weil-etale cohomology groups. In an Appendix coauthored with K.F. Lai and K.-S. Tan we also show that the main conjectures of geometric Iwasawa theory can be proved without using either crystalline cohomology or Drinfeld modules.

-adic Kolyvagin Systems

In this paper, we study the deformations of Kolyvagin systems that are known to exist in a wide variety of cases, by the work of B. Howard, B. Mazur, and K. Rubin for the residual Galois representations, along the cyclotomic Iwasawa algebra. We prove, under certain technical hypotheses, that a cyclotomic deformation of a Kolyvagin system exists. We also briefly discuss how our techniques could be extended to prove that one could deform Kolyvagin systems for other deformations as well. We discuss several applications of this result, particularly relation of these $\Lambda$-adic Kolyvagin systems to p-adic L-functions (in view of the conjectures of Perrin-Riou on p-adic L-functions) and applications to main conjectures; also applications to the study of Iwasawa theory of Rubin-Stark units.

Iwasawa Theory, projective modules, and modular representations

This paper shows that properties of projective modules over a group ring Zp[∆], where ∆ is a finite Galois group, can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve E. Modular representation theory for the group ∆ plays a crucial role in this study. It is necessary to make a certain assumption about the vanishing of a μ-invariant. We then study λ-invariants λE(σ), where σ varies over the absolutely irreducible representations of ∆. We show that there are non-trivial relationships between these invariants under certain hypotheses. 2000 Mathematics Subject Classification: Primary 11G05, 11R 23 Secondary 20C15, 20C20

Special values of L -functions and false Tate curve extensions

In this paper we show how the p-adic Rankin–Selberg product construction of Hida can be combined with freeness results of Hecke modules of Wiles to establish interesting congruences between particular special values of L-functions of elliptic curves. These congruences are part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension. In the appendix by Vladimir Dokchitser it is shown that these congruences, combined with results from Iwasawa theory for elliptic curves, give interesting results for the arithmetic of elliptic curves over non-abelian extensions.

The Tate-Shafarevich Group for Elliptic Curves with Complex Multiplication II

Using Iwasawa theory, we establish some numerical results, and one weak theoretical result, about the enigmatic Tate-Shafarevich group of an elliptic curve defined over the rational field, with complex multiplication. These strengthen results of this kind proven in an earlier paper of ours.