Fine Selmer groups and ideal class groups

Letfbe an elliptic modular form andpan odd prime that is coprime to the level off. We study the link between divisors of the characteristic ideal of thep-primary fine Selmer group offover the cyclotomic$\mathbb {Z}_p$extension of$\mathbb {Q}$and the greatest common divisor of signed Selmer groups attached tofdefined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction atpto the context of modular forms.

We extend and refine the theory of “organizing modules” of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups, and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (nonabelian) Galois extensions of number fields.

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the $$\mu $$ -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

For a given Coleman family of modular forms, we construct a formal model and prove the existence of a family of Galois representations associated to the Coleman family. As an application, we study the variations of Iwasawa λ-and µ-invariants of dual fine (strict) Selmer groups over the cyclotomic Zp-extension of Q in Coleman families of modular forms. This generalizes an earlier work of Jha and Sujatha for Hida families. Résumé. Nous construisons un modèle formel pour toute famille de Coleman de formes modulaires, et nous montrons l'existence d'une famille de représentations galoisiennes qui lui est attachée. Gràce à cette construction nous étudions la variation des invariants d'Iwasawa λ et µ attachés aux duaux des groupes de Selmer fins (strictes) le long de la Zp-extension cyclotomique de Q dans des familles de Coleman de formes modulaires. Nos résultats généralisent ceux obtenus par Jha et Sujatha dans le cas de familles de Hida de formes modulaires ordinaires.

The primary purpose of this chapter is the analysis of the group Wq(R) in the case of an arithmetic Dedekind domain R. The characterization of the elements of Br(R)2 as the Brauer classes of Clifford algebras of quadratic spaces over R of rank 4 and trivial Arf invariant is an important step along the way. The quadratic spaces which it provides make it possible to describe Ker Arf by use of the total signature. In the special case where R is the ring of integers in a number field, the quadratic group Qu(R) is closely related to the ideal class group, and this connection together with the description of Ker Arf implies that Wq(R) ≅ Cl(R)2 ⊕ G, where C1(R) is the ideal class group of R and G is a free Abelian group of rank with r the number of real embeddings of the number field. An additional focus is the comparison of the number theory of Wq(R) with that of W(R) and the structure of the quotient W(R)/Wq(R).

We consider certain Massey products in the cohomology of a Galois extension of fields with coefficients in p-power roots of unity. We prove formulas for these products both in general and in the special case that the Galois extension in question is the maximal extension of a number field unramified outside a set of primes S including those above p and any archimedean places. We then consider those ℤp-Kummer extensions L∞ of the maximal p-cyclotomic extension K∞ of a number field K that are unramified outside S. We show that Massey products describe the structure of a certain "decomposition-free" quotient of a graded piece of the maximal unramified abelian pro-p extension of L∞ in which all primes above those in S split completely, with the grading arising from the augmentation filtration on the group ring of the Galois group of L∞/K∞. We explicitly describe examples of the maximal unramified abelian pro-p extensions of unramified outside p Kummer extensions of the cyclotomic field of all p-power roots of unity, for irregular primes p.

We determine all the simplest cubic fields whose ideal class groups have exponent dividing 3, thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number 1 and the determination by D. Byeon of all all the simplest cubic fields with class number 3. We prove that there are 23 simplest cubic fields with ideal class groups of exponent 3 (and 8 simplest cubic fields with ideal class groups of exponent 1, i.e. with class number one).

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be big, so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space. The main results of the thesis are the following two theorems. Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds. Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense. Besides this we also established the following results: (1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds. (2) MT holds for Ribet-type abelian varieties. (3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds. (4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I. (5) For some abelian varieties either MT or the Hodge conjecture holds.

The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra’s numerical criterion. In this paper, we relate L(1,pi ,{{,textrm{Ad},}}^circ ) to the congruence ideals for cohomological cuspidal automorphic representations pi of textrm{GL}_n over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of L(1,pi ,{{,textrm{Ad},}}^circ ). This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of textrm{GL}_n, its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.

IfHis a monoid anda = u1···uk ∈ Hwith atoms (irreducible elements)u1, … ,uk, thenkis a length ofa, the set of lengths ofais denoted byⅬ(a), and ℒ(H) = {Ⅼ(a) |a ∈ H} is the system of sets of lengths ofH. LetRbe a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisorsR•can be written as a product of atoms. We show that ifRis bounded and every stably free rightR-ideal is free, then there exists a transfer homomorphism fromR•to the monoidBof zero-sum sequences over a subsetGmax(R) of the ideal class groupG(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoidsR•andBcoincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free rightR-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford–Tate group, $$\ell $$ -adic monodromy groups, and the Sato–Tate group. Assuming the Mumford–Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree.

The purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

Results are given for a class of square {0,1}-matrices which provide information about the 4-rank of the ideal class group of certain quadratic number fields.

The class group of a number field K measures how far its ring of integers is from having unique factorization into irreducible elements. It is the quotient of the group of all fractional ideals of K by the subgroups of principal fractional ideals. It is well known from class field theory that the ideal class group is also the Galois group of the maximal unramified abelian extension of K.

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.