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We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over
$\mathbb{Q}$
of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use
$R=T$
theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from
$\text{GSp}(4)$
to
$U(4)$
. In that case we obtain a corollary for abelian surfaces.
Let
$F/\mathbf{Q}$
be a totally real field and
$K/F$
a complex multiplication (CM) quadratic extension. Let
$f$
be a cuspidal Hilbert modular new form over
$F$
. Let
${\it\lambda}$
be a Hecke character over
$K$
such that the Rankin–Selberg convolution
$f$
with the
${\it\theta}$
-series associated with
${\it\lambda}$
is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg
$L$
-values
$L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$
, as
${\it\chi}$
varies over the class group characters of
$K$
. Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters
${\it\chi}$
such that
$L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$
increases with the absolute value of the discriminant of
$K$
. We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg
$L$
-values. The Waldspurger formula plays an underlying role.
Let
$p\geqslant 5$
be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup
${\rm\Gamma}(L)$
of
$\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$
for a principal ideal
$(L)\neq 0$
of
$\mathbb{Z}_{p}[[T]]$
for the canonical ‘weight’ variable
$t=1+T$
. If
$L\notin {\rm\Lambda}^{\times }$
, the power series
$L$
is proven to be a factor of the Kubota–Leopoldt
$p$
-adic
$L$
-function or of the square of the anticyclotomic Katz
$p$
-adic
$L$
-function or a power of
$(t^{p^{m}}-1)$
.
We prove vanishing of the μ-invariant of the p-adic Katz L-function in N. M. Katz [p-adic L-functions for CM fields, Invent. Math. 49 (1978), 199–297].
By
Haruzo Hida, Department of Mathematics UCLA Los Angeles, Ca 90095-1555 U.S.A. hida@math.ucla.edu
Edited by
David Burns, King's College London,Kevin Buzzard, Imperial College of Science, Technology and Medicine, London,Jan Nekovář, Université de Paris VI (Pierre et Marie Curie)
An exact control theorem is proven for nearly ordinary $p$-adic automorphic forms on symplectic and unitary groups over
totally real fields if the algebraic group is split at $p$. In particular, a given nearly ordinary holomorphic Hecke
eigenform can be lifted to a family of holomorphic Hecke eigenforms indexed by weights of the standard maximal split
torus of the group. Their $q$-expansion coefficients are Iwasawa functions on the Iwasawa algebra of ${\mathbb{Z}}_p$-points of
the split torus. The method is applicable to any reductive algebraic groups yielding Shimura varieties of PEL type
under mild assumptions on the existence of integral toroidal compactification of the variety. Even in the Hilbert
modular case, the result contains something new: freeness of the universal nearly ordinary Hecke algebra over the
Iwasawa algebra, which eluded my scrutiny when I studied general theory of the $p$-adic Hecke algebra in the 1980s.
The purpose of this chapter is to identify the GL(2)-Hecke algebras with universal deformation rings with certain additional structures. This fact was first conjectured by B. Mazur and now is a theorem of Wiles in many cases (see Subsection 3.2.7 for a description of the present knowledge to date: October 1999), which is one of the key points of his proof of Fermat's last theorem. In this chapter, we will prove the theorem in a typical case (which covers the case when the weight is bigger than or equal to 2), assuming the knowledge of the modular two-dimensional Galois representations, control theorems of Hecke algebras and the Poitou–Tate duality theorem on Galois cohomology. We will come back later to the duality theorems used here and give a full exposition of them in Chapter 4. As for modular Galois representations and control theorems, we content ourselves only by describing the precise result necessary for the proof and giving some indication of further reading (see Theorem 3.15, Corollary 3.19 and Theorem 3.26). These two results left untouched here will be covered in my forthcoming book [GMF].
Modular Forms on Adele Groups ofGL(2)
We first recall a general theory of elliptic modular forms in the language of adeles.
In the past few years (1995–98), I have given several advanced graduate courses at UCLA in order to provide a comprehensive account of the proof by Wiles (and Taylor) of the identification of certain Hecke algebras with universal deformation rings of Galois representations. Assuming a good knowledge of Class field theory, I started with an overview of the theory of automorphic forms on linear algebraic groups, specifically, GL(n) over number fields. Since second year graduate students often lack knowledge of representation theory of profinite groups, necessary to carry out the task, I went on to describe basic representation theory, the theory of pseudo-representations and their deformation. To reach this point, I had already covered almost a one-year course. Then I continued to give a sketch of the rationality and the control theorems of the space of elliptic modular forms, which is the basis of the definition of the Hecke algebra. In the meantime, K. Fujiwara and F. Diamond independently gave, in 1996, a substantial simplification of the proof of Wiles, which I incorporated in my course. After having proved the theorem, assuming many things, I came back to the material I used in the proof, in particular the duality theorems (due to Poitou and Tate) of Galois cohomology groups. Thus the first chapters follow faithfully my series of courses; so, logically the reader might have to jump around between chapters.
It is difficult to provide a brief summary of techniques used in modern number theory. Traditionally, mathematical research has been classified by the method mathematicians exploit to study their research areas, except possibly for number theory. For example, algebraists study mathematical questions related to abstract algebraic systems in a purely algebraic way (only allowing axioms defining their algebraic systems), differential geometers study manifolds via infinitesimal analysis, and algebraic geometers study geometry of algebraic varieties (and its siblings) via commutative algebras and category theory. There are no central techniques which distinguish number theory from other subjects, or rather, number theorists exploit any techniques available to hand to solve problems specific to number theory. In this sense, number theory is a discipline in mathematics which cannot be classified by methodology from the above traditional viewpoint but is just a web of rather specific problems (or conjectures) tightly and subtly knit to each other. We just study numbers, those simple ones, like integers, rational numbers, algebraic numbers, real and complex numbers and p-adic numbers, and that is it.
What has emerged from our rather long history is that we continue to study at least two aspects of these numbers: the numbers of the base field and the numbers of its extensions. For example, the quadratic reciprocity lawdescribes in a simple way how rational primes decompose as a product of prime ideals in a quadratic extension only using data from rational integers.
This book provides a comprehensive account of a key (and perhaps the most important) theory upon which the Taylor–Wiles proof of Fermat's last theorem is based. The book begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and results on elliptic modular forms, including a substantial simplification of the Taylor–Wiles proof by Fujiwara and Diamond. It contains a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology groups. The final chapter presents a proof of a non-abelian class number formula and includes several new results from the author. The book will be of interest to graduate students and researchers in number theory (including algebraic and analytic number theorists) and arithmetic algebraic geometry.
Here we shall give a detailed exposition of a general theory of Group representations, pseudo-representations and their deformation. These results will be used later. The reader who knows the theory well can skip this chapter.
Group Representations
A representation of degree n of a group G is a group homomorphism of G into the group of invertible n × n matrices GLn(A) with coefficients in a commutative ring A. When the structure of the group is very complicated or when the group is very large, such as the absolute Galois group over ℚ, it is often easier to study representations rather than the group itself. In this section, we study the basic properties of group representations.
Coefficient rings
Any ring in this section is commutative with the identity 1 = 1A. If we refer to an algebra R, then R may not be commutative. A ring A is called local if there is only one maximal ideal mA in A. An A-module M is artinian (resp. noetherian) if the set of A-submodules of M satisfies the descending (resp. ascending) chain condition. If A itself as an A-module is artinian (resp. noetherian), we just call A artinian (resp. noetherian). For an artinian A-module M, if M ⊃ M1 ⊃ ⊃ Mn = {0} is the maximal descending chain of, A-submodules, the number n is called the length of M and is written as ℓA(M). If A is an artinian ring, A is noetherian (Akizuki's theorem, [CRT] Theorem 1.3.2).