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We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in
${{Z}}_p$
-extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a
${{Z}}_p$
-extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different
${{Z}}_p$
-extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.
The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.
In this paper, we study the growth of fine Selmer groups in two cases. First, we study the growth of fine Selmer ranks in multiple
$\mathbb{Z}_{p}$
-extensions. We show that the growth of the fine Selmer group is unbounded in such towers. We recover a sufficient condition to prove the
$\unicode[STIX]{x1D707}=0$
conjecture for cyclotomic
$\mathbb{Z}_{p}$
-extensions. We show that in certain non-cyclotomic
$\mathbb{Z}_{p}$
-towers, the
$\unicode[STIX]{x1D707}$
-invariant of the fine Selmer group can be arbitrarily large. Second, we show that in an unramified
$p$
-class field tower, the growth of the fine Selmer group is unbounded. This tower is non-Abelian and non-
$p$
-adic analytic.
This paper completes the construction of
$p$
-adic
$L$
-functions for unitary groups. More precisely, in Harris, Li and Skinner [‘
$p$
-adic
$L$
-functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such
$p$
-adic
$L$
-functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and
$p$
-adic differential operators [Eischen, ‘A
$p$
-adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘
$p$
-adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local
$\unicode[STIX]{x1D701}$
-integrals occurring in the Euler product (including at
$p$
). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.
Fix an odd prime p. Let
$\mathcal{D}_n$
denote a non-abelian extension of a number field K such that
$K\cap\mathbb{Q}(\mu_{p^{\infty}})=\mathbb{Q}, $
and whose Galois group has the form
$ \text{Gal}\big(\mathcal{D}_n/K\big)\cong \big(\mathbb{Z}/p^{n'}\mathbb{Z}\big)^{\oplus g}\rtimes \big(\mathbb{Z}/p^n\mathbb{Z}\big)^{\times}\ $
where g > 0 and
$0 \lt n'\leq n$
. Given a modular Galois representation
$\overline{\rho}:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F})$
which is p-ordinary and also p-distinguished, we shall write
$\mathcal{H}(\overline{\rho})$
for the associated Hida family. Using Greenberg’s notion of Selmer atoms, we prove an exact formula for the algebraic λ-invariant
\begin{equation}
\lambda^{\text{alg}}_{\mathcal{D}_n}(f) \;=\; \text{the number of zeroes of }
\text{char}_{\Lambda}\big(\text{Sel}_{\mathcal{D}_n^{\text{cy}}}\big(f\big)^{\wedge}\big)
\end{equation}
at all
$f\in\mathcal{H}(\overline{\rho})$
, under the assumption
$\mu^{\text{alg}}_{K(\mu_p)}(f_0)=0$
for at least one form f0. We can then easily deduce that
$\lambda^{\text{alg}}_{\mathcal{D}_n}(f)$
is constant along branches of
$\mathcal{H}(\overline{\rho})$
, generalising a theorem of Emerton, Pollack and Weston for
$\lambda^{\text{alg}}_{\mathbb{Q}(\mu_{p})}(f)$
.
For example, if
$\mathcal{D}_{\infty}=\bigcup_{n\geq 1}\mathcal{D}_n$
has the structure of a p-adic Lie extension then our formulae include the cases where: either (i)
$\mathcal{D}_{\infty}/K$
is a g-fold false Tate tower, or (ii)
$\text{Gal}\big(\mathcal{D}_{\infty}/K(\mu_p)\big)$
has dimension ≤ 3 and is a pro-p-group.
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.
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 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.
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 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.
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.