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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.
We give an algebraic proof of a class number formula for dihedral extensions of number fields of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and allows one to recover similar formulas which have appeared in the literature. As a corollary of our main result, we obtain explicit bounds on the (finitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.
Let
$K$
be a two-dimensional global field of characteristic
$\neq 2$
and let
$V$
be a divisorial set of places of
$K$
. We show that for a given
$n\geqslant 5$
, the set of
$K$
-isomorphism classes of spinor groups
$G=\operatorname{Spin}_{n}(q)$
of nondegenerate
$n$
-dimensional quadratic forms over
$K$
that have good reduction at all
$v\in V$
is finite. This result yields some other finiteness properties, such as the finiteness of the genus
$\mathbf{gen}_{K}(G)$
and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups
$H^{i}(K,\unicode[STIX]{x1D707}_{2})_{V}$
for
$i\geqslant 1$
established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type
$\mathsf{G}_{2}$
.
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 number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.
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 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.
We derive a formula for Greenberg’s L-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight ≥4, under some technical assumptions. This requires a ‘sufficiently rich’ Galois deformation of the symmetric cube, which we obtain from the symmetric cube lift to GSp(4)/Q of Ramakrishnan–Shahidi and the Hida theory of this group developed by Tilouine–Urban. The L-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg’s L-invariant of all symmetric powers by using appropriate functorial transfers and Hida theory on higher rank groups.
We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into GLn. Existing theorems require that the residual representation have ‘big’ image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor–Wiles method which allows one to weaken this hypothesis.
Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.
We formulate an explicit conjecture for the leading term at s=1 of the equivariant Dedekind zeta-function that is associated to a Galois extension of number fields. We show that this conjecture refines well-known conjectures of Stark and Chinburg, and we use the functional equation of the zeta-function to compare it to a natural conjecture for the leading term at s=0.
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