Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Let K be a finite extension of Qp, let L/K be a finite abelian Galois extension of odd degree and let OL be the valuation ring of L. We define AL/K to be the unique fractional OL-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Qp contained in certain cyclotomic extensions, Erez has described integral normal bases for AL/Qp that are self-dual with respect to the trace form. Assuming K/Qp to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.     

On the equivariant Tamagawa number conjecture in tame CM-extensions

Let L/K be a finite Galois CM-extension with Galois group G. The Equivariant Tamagawa Number Conjecture (ETNC) for the pair ${(h^0({\rm Spec} (L))(0), {\mathbb Z}G)}$ naturally decomposes into p-parts, where p runs over all rational primes. If p is odd, these p-parts in turn decompose into a plus and a minus part. Let L/K be tame above p. We show that a certain ray class group of L defines an element in ${K_0({\mathbb Z}{_p}G_-, \mathbb Q_p)}$ which is determined by a corresponding Stickelberger element if and only if the minus part of the ETNC at p holds. For this we use the Lifted Root Number Conjecture for small sets of places which is equivalent to the ETNC in the number field case. For abelian G, we show that the minus part of the ETNC at p implies the Strong Brumer?CStark Conjecture at p. We prove the minus part of the ETNC at p for almost all primes p.     

Adjoint selmer groups as Iwasawa modules

We fix a primep. In this paper, starting from a given Galois representation ? having values inp-adic points of a classical groupG, we study the adjoint action of ? on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(?) of ?. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(?))_/L for each number fieldL. We scrutinize the behavior of Sel(Ad(?))_{/E ∞} as an Iwasawa module for a fixed ? _p -extensionE _∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kähler differentials of the universal nearly ordinary deformation ring of ?. WhenG=GL(2), ? is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.     

Representations and factors of restriction of scalars

Let L/K be a Galois extension of number fields and let A be an abelian variety defined over K. In this paper we establish the relation between the irreducible characters of the Galois group Gal(L/K) and the simple factors of the restriction of scalars Res_L/K(A) of A from L to K. Then we derive some equivalences of Birch and Swinnerton-Dyer conjectures.     

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of \Bbb Q_p {\Bbb Q}_p . We give a complete classification of those finite abelian extensions L/K L/K in which any ideal of the valuation ring of L is free over its associated order in \Bbb Q_p[Gal(L/K)] {\Bbb Q}_p[Gal(L/K)] . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.     

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ,1)≠0 then Selp(E/Hc)χ_⊗W=0 for all but finitely many primes p, where Selp(E/Hc) is the p-Selmer group of E over Hc and W is a suitable finite extension of Zp containing the values of χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-twisted version of the Birch and Swinnerton-Dyer conjecture for E over Hc (Bertolini and Darmon) and of the vanishing of Selp(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero.     

数域中tame 核的p-秩

Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K₂O_E if the group K₂O_E is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K₂O_E, where E/F is not necessarily abelian. In particular, in the case of G=C_n, D_n, A₄, we easily get some results on the divisibility of the p-rank of the tame kernel K₂O_E by the character table. Let E/Q be a normal extension with Galois group D_l, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K₂O_F, where f is the smallest positive integer such that p^f≡±1(mod l).     

Hyperelliptic jacobians with real multiplication

Let K be a field of characteristic p≠2, and let f(x) be a sextic polynomial irreducible over K with no repeated roots, whose Galois group is isomorphic to A₅. If the jacobian J(C) of the hyperelliptic curve C:y²=f(x) admits real multiplication over the ground field from an order of a real quadratic field D, then either its endomorphism algebra is isomorphic to D, or p>0 and J(C) is a supersingular abelian variety. The supersingular outcome cannot occur when p splits in D.     

On Galois structure of the integers in elementary abelian extensions of local number fields

Let p be an odd prime number and k a finite extension of Qp. Let K/k be a totally ramified elementary abelian Kummer extension of degree p² with Galois group G. We determine the isomorphism class of the ring of integers in K as an oG-module under some assumptions. The obtained results imply there exist extensions whose rings are ZpG-isomorphic but not oG-isomorphic, where Zp is the ring of p-adic integers. Moreover we obtain conditions that the rings of integers are free over the associated orders and give extensions whose rings are not free.     

On the Galois module structure over CM-fields

In this paper we make a contribution to the problem of the existence of a normal integral basis. Our main result is that unramified realizations of a given finite abelian group Δ as a Galois group Gal (N/K) of an extensionN of a givenCM-fieldK are invariant under the involution on the set of all realizations of Δ overK which is induced by complex conjugation onK and by inversion on Δ. We give various implications of this result. For example, we show that the tame realizations of a finite abelian group Δ of odd order over a totally real number fieldK are completely characterized by ramification and Galois module structure.     

Breuil modules for Raynaud schemes

Let p be an odd prime, and let OK be the ring of integers in a finite extension K/Qp. Breuil has classified finite flat group schemes of type (p,…,p) over OK in terms of linear-algebraic objects that have come to be known as Breuil modules. This classification can be extended to the case of finite flat vector space schemes G over OK. When G has rank one, the generic fiber of G corresponds to a Galois character, and we explicitly determine this character in terms of the Breuil module of G. Special attention is paid to Breuil modules with descent data corresponding to characters of that become finite flat over a totally ramified extension of degree pd−1; these arise in Gee's work on the weight in Serre's conjecture over totally real fields.

Video abstract

For a video summary of this paper, please visit http://www.youtube.com/watch?v=9oWYJy_XrZE.     

On a Normal Integral Basis Problem over Cyclotomic Zp-extensions, II

Let p be an odd prime number, K an imaginary abelian field with ζ_p∈K^×, and K_∞/K the cyclotomic Z_p-extension with its nth layer K_n. In the previous paper, we showed that for any n and any unramified cyclic extension L/K_n of degree p, LK_n+1/K_n+1 does have a normal integral basis (NIB) even if L/K_n has no NIB, under the assumption that p does not divide the class number of the maximal real subfield K⁺ (and some additional assumptions on K). In this paper, we show that similar but more delicate phenomena occur for a certain class of tamely ramified extensions of degree p.     

Iwasawa invariants for the False-Tate extension and congruences between modular forms

For an ordinary prime p?3, we consider the Hida family associated to modular forms of a fixed tame level, and their Selmer groups defined over certain Galois extensions of Q(μp) whose Galois group is G≅Zp?Zp. For Selmer groups defined over the cyclotomic Zp-extension of Q(μp), we show that if the μ-invariant of one member of the Hida family is zero, then so are the μ-invariants of the other members, while the λ-invariants remain the same only in a branch of the Hida family. We use these results to study the behavior of some invariants from non-commutative Iwasawa theory in the Hida family.     

Complex multiplication and parity in Iwasawa theory

We give a new, somewhat elementary method for proving parity results about Iwasawa-theoretic Selmer groups and apply our method to certain Galois representations which are not self-dual. The main result is essentially that Iwasawa's λ-invariants for these representations over dihedral -extensions are even. Our approach is a specialization argument and does not make use of Neková?'s deformation-theoretic Cassels pairing, though Neková?'s theory implies our results. Examples of the representations we consider arise naturally in the study of CM abelian varieties defined over the totally real subfield of the reflex field of the CM type. We also discuss connections with “large Selmer rank” in the sense of Mazur-Rubin and give several examples in the context of abelian varieties and modular forms.     

Sur l'indépendance l-adique de nombres algébriques

Let l a prime number and K a Galois extension over the field of rational numbers, with Galois group G. A conjecture is put forward on l-adic independence of algebraic numbers, which generalizes the classical ones of Leopoldt and Gross, and asserts that the l-adic rank of a G submodule of K^x depends only on the character of its Galois representation. When G is abelian and in some other cases, a proof is given of this conjecture by using l-adic transcendence results.     

On the cohomology of Witt vectors of p-adic integers and a conjecture of Hesselholt

Let K be a complete discrete valued field of characteristic zero with residue field kK of characteristic p>0. Let L/K be a finite Galois extension with Galois group G such that the induced extension of residue fields kL/kK is separable. Hesselholt (2004) [2] conjectured that the pro-abelian group {H¹(G,Wn(OL))}_n∈N is zero, where OL is the ring of integers of L and W(OL) is the ring of Witt vectors in OL w.r.t. the prime p. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt?s conjecture for all Galois extensions.     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

Galois realizability of groups of orders p 5 and p 6

Let p be an odd prime and k an arbitrary field of characteristic not p. We determine the obstructions for the realizability as Galois groups over k of all groups of orders p ⁵ and p ⁶ that have an abelian quotient obtained by factoring out central subgroups of order p or p ². These obstructions are decomposed as products of p-cyclic algebras, provided that k contains certain roots of unity.     

L-functions of twisted Legendre curves

Let K be a global field of char p and let Fq be the algebraic closure of Fp in K. For an elliptic curve E/K with nonconstant j-invariant, the L-function L(T,E/K) is a polynomial in 1+T⋅Z[T]. For any N>1 invertible in K and finite subgroup T⊂E(K) of order N, we compute the mod N reduction of L(T,E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic rank of E/K. We construct infinite families of curves of rank zero when q is an odd prime power such that for some odd prime ?. Our construction depends upon a construction of infinitely many twin-prime pairs (Λ,Λ−1) in Fq[Λ]×Fq[Λ]. We also construct infinitely many quadratic twists with minimal analytic rank, half of which have rank zero and half have (analytic) rank one. In both cases we bound the analytic rank by letting T≅Z/2⊕Z/2 and studying the mod-4 reduction of L(T,E/K).