Cyclotomic Extension Research Articles

A generalization of Kummer theory to Hopf-Galois extensions

We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an H-Galois extension L/K is H-Kummer if L can be generated by adjoining to K a finite set S of eigenvectors for the action of the Hopf algebra H on L. This extends the classical Kummer condition for the classical Galois structure. With this new perspective, we shall characterize a class of H-Kummer extensions L/K as radical extensions that are linearly disjoint with the n-th cyclotomic extension of K. This result generalizes the description of Kummer Galois extensions as radical extensions of a field containing the n-th roots of the unity. The main tool is the construction of a product Hopf-Galois structure on the compositum of almost classically Galois extensions L1/K, L2/K such that L1∩M2=L2∩M1=K, where Mi is a field such that LiMi=L˜i, the normal closure of Li/K. When L/K is an extension of number or p-adic fields, we shall derive criteria on the freeness of the ring of integers OL over its associated order in an almost classically Galois structure on L/K.

Primes and conductors of characters

If G is a finite p-solvable group, P is a Sylow p-subgroup of G and χ∈Irr(G) is an irreducible complex character of degree not divisible by p, we prove that the field of values Q(χP) of the restriction of χ to P has index not divisible by p in the cyclotomic extension Q(e2πipf), where pf is the p-part of the conductor of χ. This proves the p-solvable case of the main conjecture in [4].

The Chromatic Fourier Transform

Abstract We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$ , as well as a certain duality for the $E_n$ -(co)homology of $\pi $ -finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$ . We use this theory to generalize said duality in three different directions. First, we extend it from $\mathbb {Z}$ -module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of $E_n$ . Second, we lift it to the telescopic setting by replacing $E_n$ with $T(n)$ -local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg–Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal $\infty $ -categories of local systems of $K(n)$ -local $E_n$ -modules [-12pc] and relate it to (semiadditive) redshift phenomena. The Great Wave off Kanagawa, Katsushika Hokusai.

Birational anabelian Grothendieck Conjecture for curves over arbitrary cyclotomic extension fields of number fields

In anabelian geometry, various strong/desired forms of Grothendieck Conjecture-type results for hyperbolic curves over relatively small arithmetic fields — for instance, finite fields, number fields, or p-adic local fields — have been obtained by many researchers, especially by A. Tamagawa and S. Mochizuki. Let us recall that, in their proofs, the Weil Conjecture or p-adic Hodge theory plays an essential role. Therefore, to obtain such Grothendieck Conjecture-type results, it appears that the condition that the cyclotomic characters of the absolute Galois groups of the base fields are highly nontrivial is indispensable. On the other hand, in an author's recent joint work with Y. Hoshi and S. Mochizuki, we introduced the notion of TKND-AVKF-field [concerning the divisible subgroups of the groups of rational points of semi-abelian varieties] and obtained the semi-absolute version of the Grothendieck Conjecture for higher dimensional (≥2) configuration spaces associated to hyperbolic curves of genus 0 over TKND-AVKF-fields contained in the algebraic closure of the field of rational numbers. For instance, every [possibly, infinite] cyclotomic extension field of a number field is such a TKND-AVKF-field. In particular, this Grothendieck Conjecture-type result suggests that the condition that the cyclotomic character of the absolute Galois group of the base field under consideration is sufficiently nontrivial is, in fact, not indispensable for strong/desired form of anabelian phenomena. In the present paper, to pose another evidence for this observation, we prove the relative birational version of the Grothendieck Conjecture for smooth curves over TKND-AVKF-fields with a certain mild condition that every cyclotomic extension field of a number field satisfies. From the viewpoint of the condition on base fields, this result may be regarded as a partial generalization of F. Pop and S. Mochizuki's results on the birational version of the Grothendieck Conjecture for smooth curves.

ON THE ESSENTIAL TORSION FINITENESS OF ABELIAN VARIETIES OVER TORSION FIELDS

AbstractThe classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety A over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety B. Assuming the Mumford–Tate conjecture, and up to a finite extension of the base field K, we give a necessary and sufficient condition for the finiteness of $A(K_B)_{\mathrm tors}$ in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension at most 3, or when both have complex multiplication.

Fourier coefficients of Hilbert modular forms at cusps

The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform f and a matrix sigma in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of f||_ksigma .

Irreducible Radical Field Extensions

This paper is devoted to irreducible radical field extensions, i.e., extensions that can be obtained by adjunction of roots of irreducible binomials. We find a criterion for cyclotomic extensions to be irreducible radical. This criterion developes the Gauss–Wantzel theorem about constructible polygons. We also prove that any normal solvable extension of some field K is irreducible radical until K has all roots of unity. This generalizes Abel’s theorem, which fills the gap in the uncomplete Ruffini proof of the impossibility theorem for the general equation of degree five or higher. Finally, we prove that the root set of any irreducible solvable polynomial coincides with the value set of some radical formula using irreducible radical extensions.

Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

Abstract Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of K. For G a finitely generated subgroup of K ×, we consider the cyclotomic-Kummer extensions K ( ζ n t , G n ) / K ( ζ n t ) K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right) for all positive integers n and t, and we describe an explicit finite procedure to compute at once the degree of all these extensions.

Lubin–Tate theory and overconvergent Hilbert modular forms of low weight

Let K be a finite extension of ℚp and let Γ be the Galois group of the cyclotomic extension of K. Fontaine’s theory gives a classification of p-adic representations of Ga( $$\overline K /K$$ ) in terms of (φ, Γ)-modules. A useful aspect of this classification is Berger’s dictionary which expresses invariants coming from p-adic Hodge theory in terms of these (φ, Γ)-modules. In this article, we use the theory of locally analytic vectors to generalize this dictionary to the setting where Γ is the Galois group of a Lubin—Tate extension of K. As an application, we show that if F is a totally real number field and v is a place of F lying above p, then the p-adic representation of Gal( $${\overline F _v}/{F_v}$$ ) associated to a finite slope overconvergent Hilbert eigenform which is Fv-analytic up to a twist is Lubin—Tate trianguline. Furthermore, we determine a triangulation in terms of a Hecke eigenvalue at v. This generalizes results in the case F = ℚ obtained previously by Chenevier, Colmez and Kisin.

Mertens’ theorem for Chebotarev sets

We generalize Mertens’ product theorem to Chebotarev sets of prime ideals in Galois extensions of number fields. Using work of Rosen, we extend an argument of Williams from cyclotomic extensions to this more general case. Additionally, we compute these products for Cheboratev sets in abelian extensions, [Formula: see text] sextic extensions, and sets of primes represented by some quadratic forms.

Double exponential lower bounds for possible solutions in the Second Case of the Fermat Last Theorem

In a recent paper, the first author provided some lower bounds to possible solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have in fact been proved not to have any unknown solutions, these improved bounds are interesting in the context of a new effective abc inequality announced in the paper [MFHMP] and developed from Mochizuki's [Mo] [IUTT-IV, Theorem A]. In this paper we provide a strengthening of the lower bound for FLT2 given in the preceding paper. This improvement was necessary in order to take advantage of the best upper bounds for primes p for which it was verified on a computer that FLT2 has no solutions, when proving Fermat's Last Theorem with the effective abc inequality.

Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all primes above [Formula: see text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic [Formula: see text]-extension of a finite extension [Formula: see text] of [Formula: see text] where [Formula: see text] is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell–Weil ranks of [Formula: see text] over a subextension of the cyclotomic [Formula: see text]-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the [Formula: see text]-parts of the Tate–Shafarevich groups of [Formula: see text] over these extensions.

Volume and symplectic structure for ℓ-adic local systems

We introduce a notion of volume for an ℓ-adic local system over an algebraic curve and, under some conditions, give a symplectic form on the rigid analytic deformation space of the corresponding geometric local system. These constructions can be viewed as arithmetic analogues of the volume and the Chern-Simons invariants of a representation of the fundamental group of a 3-manifold which fibers over the circle and of the symplectic form on the character varieties of a Riemann surface. We show that the absolute Galois group acts on the deformation space by conformal symplectomorphisms which extend to an ℓ-adic analytic flow. We also prove that the locus of local systems which are arithmetic over a cyclotomic extension is the critical set of a collection of rigid analytic functions. The vanishing cycles of these functions give additional invariants.

Infinite families of cyclotomic function fields with any prescribed class group rank

We prove the existence of the maximal real subfields of cyclotomic extensions over the rational function field k = F q ( T ) whose class groups can have arbitrarily large ℓ n -rank, where F q is the finite field of prime power order q . We prove this in a constructive way: we explicitly construct infinite families of the maximal real subfields k ( Λ ) + of cyclotomic function fields k ( Λ ) whose ideal class groups have arbitrary ℓ n -rank for n = 1, 2, and 3, where ℓ is a prime divisor of q − 1 . We also obtain a tower of cyclotomic function fields K i whose maximal real subfields have ideal class groups of ℓ n -ranks getting increased as the number of the finite places of k which are ramified in K i get increased for i ≥ 1 . Our main idea is to use the Kummer extensions over k which are subfields of k ( Λ ) + , where the infinite prime ∞ of k splits completely. In fact, we construct the maximal real subfields k ( Λ ) + of cyclotomic function fields whose class groups contain the class groups of our Kummer extensions over k . We demonstrate our results by presenting some examples calculated by MAGMA at the end.

Using decomposition groups to prove theorems about quadratic residues

In this text we elaborate on the modern viewpoint of the quadratic reciprocity law via methods of alge- braic number theory and class field theory. We present original short and simple proofs of so called addi- tional quadratic reciprocity laws and of the multiplicativity of the Legendre symbol using decompositon groups of primes in quadratic and cyclotomic extensions of Q.

Cohomology of algebraic varieties over the maximal cyclotomic extension of a global field

Cohomology of algebraic varieties over the maximal cyclotomic extension of a global field

The torsion subgroup of the elliptic curve $Y^2 = X^3 + AX$ over the maximal abelian extension of $\mathbb{Q}$

We determine explicitly the structure of the torsion group over the maximal abelian extension of $\mathbb{Q}$ and over the maximal $p$-cyclotomic extensions of $\mathbb{Q}$ for the family of rational elliptic curves given by $Y^2 = X^3 + AX$, where $A$ is an integer.

On the Mordell-Weil ranks of supersingular abelian varieties in cyclotomic extensions

Let F F be a number field unramified at an odd prime p p and let F ∞ F_\infty be the Z p \mathbf {Z}_p -cyclotomic extension of F F . Let A A be an abelian variety defined over F F with good supersingular reduction at all primes of F F above p p . Büyükboduk and the first named author have defined modified Selmer groups associated to A A over F ∞ F_\infty . Assuming that the Pontryagin dual of these Selmer groups is a torsion Z p [ [ Gal ( F ∞ / F ) ] ] \mathbf {Z}_p[[\textrm {Gal}(F_\infty /F)]] -module, we give an explicit sufficient condition for the rank of the Mordell-Weil group A ( F n ) A(F_n) to be bounded as n n varies.

Constructing certain special analytic Galois extensions

For every prime p≥5 for which a certain condition on the class group Cl(Q(μp)) is satisfied, we construct a p-adic analytic Galois extension of the infinite cyclotomic extension Q(μp∞) with some special ramification properties. In greater detail, this extension is unramified at primes above p and tamely ramified above finitely many rational primes and its Galois group over Q(μp∞) is isomorphic to a finite index subgroup of SL2(Zp) which contains the principal congruence subgroup. For the primes 107, 139, 271 and 379 such extensions were first constructed by Ohtani and Blondeau. The strategy for producing these special extensions at an abundant number of primes is through lifting two-dimensional reducible Galois representations which are diagonal when restricted to p.

Fourier expansions at cusps

In this article, we study the fields generated by the Fourier coefficients of modular forms at arbitrary cusps. We prove that these fields are contained in certain cyclotomic extensions of the field generated by the Fourier coefficients at infinity. We also show that this bound is tight in the case of newforms with trivial Nebentypus. The main tool is a result of Shimura on the interplay between the actions of $$\mathrm {GL}_2^+(\mathbb {Q})$$ and $${{\,\mathrm{Aut}\,}}(\mathbb {C})$$ on modular forms.