We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz’s
$2$
-variable
$p$
-adic
$L$
-functions) and algebraic objects (two ‘everywhere unramified’ Iwasawa modules) involving codimension two cycles in a
$2$
-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field
$K$
(where an odd prime
$p$
splits) of an elliptic curve
$E$
, defined over
$\mathbb{Q}$
, with good supersingular reduction at
$p$
. On the analytic side, we consider eight pairs of
$2$
-variable
$p$
-adic
$L$
-functions in this setup (four of the
$2$
-variable
$p$
-adic
$L$
-functions have been constructed by Loeffler and a fifth
$2$
-variable
$p$
-adic
$L$
-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the
$\mathbb{Z}_{p}^{2}$
-extension of
$K$
. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.
Let
$p$
be an odd prime number and
$E$
an elliptic curve defined over a number field
$F$
with good reduction at every prime of
$F$
above
$p$
. We compute the Euler characteristics of the signed Selmer groups of
$E$
over the cyclotomic
$\mathbb{Z}_{p}$
-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above
$p$
and mixed signs in the definition of the signed Selmer groups.
We study the variation of
$\unicode[STIX]{x1D707}$
-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the
$p$
-adic zeta function. This lower bound forces these
$\unicode[STIX]{x1D707}$
-invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When
$U_{p}-1$
generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the
$p$
-adic
$L$
-function is simply a power of
$p$
up to a unit (i.e.
$\unicode[STIX]{x1D706}=0$
). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.
J. Bellaïche and M. Dimitrov showed that the
$p$
-adic eigencurve is smooth but not étale over the weight space at
$p$
-regular theta series attached to a character of a real quadratic field
$F$
in which
$p$
splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over
$F$
at the overconvergent cuspidal Eisenstein points, being the base change lift for
$\text{GL}(2)_{/F}$
of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.
Let
$E$
be an elliptic curve over
$\mathbb{Q}$
without complex multiplication. Let
$p\geq 5$
be a prime in
$\mathbb{Q}$
and suppose that
$E$
has good ordinary reduction at
$p$
. We study the dual Selmer group of
$E$
over the compositum of the
$\text{GL}_{2}$
extension and the anticyclotomic
$\mathbb{Z}_{p}$
-extension of an imaginary quadratic extension as an Iwasawa module.
Let
$p$
be a prime and let
$G$
be a finite group. By a celebrated theorem of Swan, two finitely generated projective
$\mathbb{Z}_{p}[G]$
-modules
$P$
and
$P^{\prime }$
are isomorphic if and only if
$\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P$
and
$\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P^{\prime }$
are isomorphic as
$\mathbb{Q}_{p}[G]$
-modules. We prove an Iwasawa-theoretic analogue of this result and apply this to the Iwasawa theory of local and global fields. We thereby determine the structure of natural Iwasawa modules up to (pseudo-)isomorphism.
We investigate the Galois structures of
$p$
-adic cohomology groups of general
$p$
-adic representations over finite extensions of number fields. We show, in particular, that as the field extensions vary over natural families the Galois modules formed by these cohomology groups always decompose as the direct sum of a projective module and a complementary module of bounded
$p$
-rank. We use this result to derive new (upper and lower) bounds on the changes in ranks of Selmer groups over extensions of number fields and descriptions of the explicit Galois structures of natural arithmetic modules.
We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.
Let K be a totally real number field of degree r. Let K∞ denote the cyclotomic -extension of K, and let L∞ be a finite extension of K∞, abelian over K. The goal of this paper is to compare the characteristic ideal of the χ-quotient of the projective limit of the narrow class groups to the χ-quotient of the projective limit of the rth exterior power of totally positive units modulo a subgroup of Rubin–Stark units, for some
$\overline{\mathbb{Q}_{2}}$
-irreducible characters χ of Gal(L∞/K∞).
We construct the
$\unicode[STIX]{x1D6EC}$
-adic crystalline and Dieudonné analogues of Hida’s ordinary
$\unicode[STIX]{x1D6EC}$
-adic étale cohomology, and employ integral
$p$
-adic Hodge theory to prove
$\unicode[STIX]{x1D6EC}$
-adic comparison isomorphisms between these cohomologies and the
$\unicode[STIX]{x1D6EC}$
-adic de Rham cohomology studied in Cais [The geometry of Hida families I:
$\unicode[STIX]{x1D6EC}$
-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s
$\unicode[STIX]{x1D6EC}$
-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$
-modules attached to Hida’s ordinary
$\unicode[STIX]{x1D6EC}$
-adic étale cohomology by Dee [
$\unicode[STIX]{x1D6F7}$
–
$\unicode[STIX]{x1D6E4}$
modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable
$\unicode[STIX]{x1D6EC}$
-adic duality theorems for each of the cohomologies we construct.
We describe an algorithm for finding the coefficients of
$F(X)$
modulo powers of
$p$
, where
$p\neq 2$
is a prime number and
$F(X)$
is the power series associated to the zeta function of Kubota and Leopoldt. We next calculate the 5-adic and 7-adic
${\it\lambda}$
-invariants attached to those cubic extensions
$K/\mathbb{Q}$
with cyclic Galois group
${\mathcal{A}}_{3}$
(up to field discriminant
${<}10^{7}$
), and also tabulate the class number of
$K(e^{2{\it\pi}i/p})$
for
$p=5$
and
$p=7$
. If the
${\it\lambda}$
-invariant is greater than zero, we then determine all the zeros for the corresponding branches of the
$p$
-adic
$L$
-function and deduce
${\rm\Lambda}$
-monogeneity for the class group tower over the cyclotomic
$\mathbb{Z}_{p}$
-extension of
$K$
.
We study the growth of
$\unicode[STIX]{x0428}$
and
$p^{\infty }$
-Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the ‘positive
${\it\mu}$
-invariant’ setting in the Iwasawa theory of elliptic curves. The towers we consider are
$p$
-adic and
$l$
-adic Lie extensions for
$l\neq p$
, in particular cyclotomic and other
$\mathbb{Z}_{l}$
-extensions.
Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension
$K/\mathbf{Q}$
. The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let
$p$
be an odd prime split in
$K/\mathbf{Q}$
. We prove the non-triviality of the
$p$
-adic Abel–Jacobi image of generalised Heegner cycles modulo
$p$
over the
$\mathbf{Z}_{p}$
-anticyclotomic extension of
$K$
. The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the
$\mathbf{Z}_{p}$
-anticyclotomic extension of
$K$
.
Let
$A$
be an abelian variety over a global field
$K$
of characteristic
$p\geqslant 0$
. If
$A$
has nontrivial (respectively full)
$K$
-rational
$l$
-torsion for a prime
$l\neq p$
, we exploit the fppf cohomological interpretation of the
$l$
-Selmer group
$\text{Sel}_{l}\,A$
to bound
$\#\text{Sel}_{l}\,A$
from below (respectively above) in terms of the cardinality of the
$l$
-torsion subgroup of the ideal class group of
$K$
. Applied over families of finite extensions of
$K$
, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of
$l$
-ranks of class groups of quadratic extensions of every
$K$
containing a fixed finite field
$\mathbb{F}_{p^{n}}$
(depending on
$l$
). For number fields, it suggests a new approach to the Iwasawa
${\it\mu}=0$
conjecture through inequalities, valid when
$A(K)[l]\neq 0$
, between Iwasawa invariants governing the growth of Selmer groups and class groups in a
$\mathbb{Z}_{l}$
-extension.
For the
$(d+1)$
-dimensional Lie group
$G=\mathbb{Z}_{p}^{\times }\ltimes \mathbb{Z}_{p}^{\oplus d}$
, we determine through the use of
$p$
-power congruences a necessary and sufficient set of conditions whereby a collection of abelian
$L$
-functions arises from an element in
$K_{1}(\mathbb{Z}_{p}\unicode[STIX]{x27E6}G\unicode[STIX]{x27E7})$
. If
$E$
is a semistable elliptic curve over
$\mathbb{Q}$
, these abelian
$L$
-functions already exist; therefore, one can obtain many new families of higher order
$p$
-adic congruences. The first layer congruences are then verified computationally in a variety of cases.
We propose a fast method of calculating the
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$
-part of the class numbers in certain non-cyclotomic
$\mathbb{Z}_p$
-extensions of an imaginary quadratic field using elliptic units constructed by Siegel functions. We carried out practical calculations for
$p=3$
and determined
$\lambda $
-invariants of such
$\mathbb{Z}_3$
-extensions which were not known in our previous paper.
When the branch character has root number
$- 1$
, the corresponding anticyclotomic Katz
$p$
-adic
$L$
-function vanishes identically. For this case, we determine the
$\mu $
-invariant of the cyclotomic derivative of the Katz
$p$
-adic
$L$
-function. The result proves, as an application, the non-vanishing
of the anticyclotomic regulator of a self-dual CM modular form with root number
$- 1$
. The result also plays a crucial role in the recent work of Hsieh
on the Eisenstein ideal approach to a one-sided divisibility of the CM main
conjecture.
Using the
$\ell $
-invariant constructed in our previous paper we prove a
Mazur–Tate–Teitelbaum-style formula for derivatives of
$p$
-adic
$L$
-functions of modular forms at trivial zeros. The novelty of this
result is to cover the near-central point case. In the central point case our formula
coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens
and by Kato, Kurihara and Tsuji at the end of the 1990s.
Let π(f) be a nearly ordinary automorphic representation of the multiplicative group of an indefinite quaternion algebra B over a totally real field F with associated Galois representation ρf. Let K be a totally complex quadratic extension of F embedding in B. Using families of CM points on towers of Shimura curves attached to B and K, we construct an Euler system for ρf. We prove that it extends to p-adic families of Galois representations coming from Hida theory and dihedral ℤdp-extensions. When this Euler system is non-trivial, we prove divisibilities of characteristic ideals for the main conjecture in dihedral and modular Iwasawa theory.