Cyclotomic Character Research Articles

Mod-p Galois representations not arising from abelian varieties

It is known that any Galois representation ρ:GQ→GL(2,Fp) with determinant equal to the mod-p cyclotomic character, arises from the p-torsion of an elliptic curve over Q, if and only if p≤5. In dimension g=2, when p≤3, it is again known that any Galois representation valued in GSp(4,Fp) with cyclotomic similitude character arises from an abelian surface. In this paper, we study this question for all primes p and dimensions g≥2. When g≥2 and (g,p)≠(2,2), (2,3), (3,2), we prove the existence of a Galois representation over Q valued in GSp(2g,Fp) with cyclotomic similitude character, that cannot arise as the p-torsion representation of any g-dimensional abelian variety over Q.

Massey products in Galois cohomology and the elementary type conjecture

Let p be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to a strengthened version of Minač–Tân's Massey Vanishing Conjecture in the case of finitely generated maximal pro-p Galois groups whose pro-p cyclotomic character has torsion-free image. Consequently, the maximal pro-p Galois group of a field K containing a root of 1 of order p (and also −1 if p=2) satisfies the strong n-Massey vanishing property for every n>2 (which is equivalent to the cup-defining n-Massey product property for every n>2, as defined by Minač–Tân) in several relevant cases.

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.

Mean Values of Derivatives of $L$-functions for Cyclotomic Characters in Function Fields

We compute the mean value of derivatives of $L$-functions for cyclotomic characters in function fields. We also show the non-vanishing property of derivatives of $L$-functions for cyclotomic characters in function fields.

Reductions of Galois representations and the theta operator

Let [Formula: see text] be a prime, and let [Formula: see text] be a cuspidal eigenform of weight at least [Formula: see text] and level coprime to [Formula: see text] of finite slope [Formula: see text]. Let [Formula: see text] denote the mod [Formula: see text] Galois representation associated with [Formula: see text] and [Formula: see text] the mod [Formula: see text] cyclotomic character. Under an assumption on the weight of [Formula: see text], we prove that there exists a cuspidal eigenform [Formula: see text] of weight at least [Formula: see text] and level coprime to [Formula: see text] of slope [Formula: see text] such that [Formula: see text] up to semisimplification. The proof uses Hida–Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions of the local Galois representations associated to cusp forms with slopes in the interval [Formula: see text] were determined by Deligne, Buzzard and Gee and for slopes in [Formula: see text] by Bhattacharya, Ganguli, Ghate, Rai and Rozensztajn. We show that these reductions, in spite of their somewhat complicated behavior, are compatible with the displayed equation above. Moreover, the displayed equation above allows us to predict the shape of the reductions of a class of Galois representations attached to eigenforms of slope larger than [Formula: see text]. Finally, the methods of this paper allow us to obtain upper bounds on the radii of certain Coleman families.

Arithmetic properties encoded in the Galois module structure of [formula omitted

The power classes of a field are well-known for their ability to parameterize elementary p-abelian Galois extensions. These classical objects have recently been reexamined through the lens of their Galois module structure. Module decompositions have been computed in several cases, providing deep new insight into absolute Galois groups. The surprising result in each case is that there are far fewer isomorphism types of indecomposables than one would expect generically, with summands predominately free over associated quotient rings. Though non-free summands are the exception both in their form and prevalence, they play the critical role in controlling arithmetic conditions in the field which allow the rest of the decomposition to be so simple.Suppose m,n∈N and p is prime. In a recent paper, a surprising and elegant decomposition for pmth power classes has been computed when the underlying Galois group is a cyclic group of order pn. As with previous module decompositions, at most one non-free summand appears. Outside of a particular special case when p=2, the structure of this exceptional summand was determined by a vector a∈{−∞,0,…,n}m and a natural number d. In this paper we give field-theoretic interpretations for a and d, showing they are related to the solvability of a family of Galois embedding problems and the cyclotomic character associated to K/F.

The Neukirch–Uchida theorem with restricted ramification

Abstract Let K be a number field and S a set of primes of K. We write K S / K {K_{S}/K} for the maximal extension of K unramified outside S and G K , S {G_{K,S}} for its Galois group. In this paper, we prove the following generalization of the Neukirch–Uchida theorem under some assumptions: “For i = 1 , 2 {i=1,2} , let K i {K_{i}} be a number field and S i {S_{i}} a set of primes of K i {K_{i}} . If G K 1 , S 1 {G_{K_{1},S_{1}}} and G K 2 , S 2 {G_{K_{2},S_{2}}} are isomorphic, then K 1 {K_{1}} and K 2 {K_{2}} are isomorphic.” Here the main assumption is that the Dirichlet density of S i {S_{i}} is not zero for at least one i. A key step of the proof is to recover group-theoretically the l-adic cyclotomic character of an open subgroup of G K , S {G_{K,S}} for some prime number l.

$1$-smooth pro-$p$ groups and Bloch–Kato pro-$p$ groups

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a homomorphism of pro-$p$ groups $G\to1+p\mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$, together with the cyclotomic character, are 1-smooth. We prove that a finitely generated $p$-adic analytic pro-$p$ group is 1-smooth if, and only if, it occurs as the maximal pro-$p$ Galois group of a field containing a root of 1 of order $p$. This gives a positive answer to De Clerq-Florence's Smoothness Conjecture - which states that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated $p$-adic analytic pro-$p$ groups.

Self-duality of the lattice of transfer systems via weak factorization systems

Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a homomorphism of pro-$p$ groups $G\to1+p\mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$, together with the cyclotomic character, are 1-smooth. We prove that a finitely generated $p$-adic analytic pro-$p$ group is 1-smooth if, and only if, it occurs as the maximal pro-$p$ Galois group of a field containing a root of 1 of order $p$. This gives a positive answer to De Clerq-Florence's "Smoothness Conjecture" - which states that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated $p$-adic analytic pro-$p$ groups.

A refined lifting theorem for supersingular Galois representations

Let p≥5 be a prime number, F a finite field of characteristic p and let χ¯ be the mod-p cyclotomic character. Let ρ¯:GQ→GL2(F) be a Galois representation such that the local representation ρ¯↾GQp is flat and irreducible. Further, assume that detρ¯=χ¯. The celebrated theorem of Khare and Wintenberger asserts that if ρ¯ satisfies some natural conditions, there exists a normalized Hecke-eigencuspform f=∑n≥1anqn and a prime p|p in its field of Fourier coefficients such that the associated p-adic representation ρf,p lifts ρ¯. In this manuscript we prove a refined version of this theorem, namely, that one may control the valuation of the p-th Fourier coefficient of f. The main result is of interest from the perspective of the p-adic Langlands program.

Irreducibility of limits of Galois representations of Saito\u2013Kurokawa type

We prove (under certain assumptions) the irreducibility of the limit sigma _2 of a sequence of irreducible essentially self-dual Galois representations sigma _k: G_{{mathbf {Q}}} rightarrow {{,mathrm{GL},}}_4(overline{{mathbf {Q}}}_p) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to 1 oplus rho oplus chi with rho irreducible, two-dimensional of determinant chi , where chi is the mod p cyclotomic character. More precisely, we assume that sigma _k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as krightarrow 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to rho ) which we assume are non-zero.

Non-vanishing of $L$-functions for cyclotomic characters in function fields

Non-vanishing of $L$-functions for cyclotomic characters in function fields

Generalized Beilinson Elements and Generalized Soulé Characters

AbstractThe generalized Soulé character was introduced by H. Nakamura and Z. Wojtkowiak and is a generalization of Soulé’s cyclotomic character. In this paper, we prove that certain linear sums of generalized Soulé characters essentially coincide with the image of generalized Beilinson elements in K-groups under Soulé’s higher regulator maps. This result generalizes Huber–Wildeshaus’ theorem, which is a cyclotomic field case of our results, to an arbitrary number fields.

Deformations of Saito-Kurokawa type and the Paramodular Conjecture

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\overline{\sigma}$ with $\overline{\sigma}^{{\rm ss}}=\chi^{k-2}\oplus\rho\oplus\chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an absolutely irreducible reduction of the Galois representation $\rho_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f({\bf Q},\rho_f(1-k)\otimes{\bf Q}_p/{\bf Z}_p)$ and $H^1_f({\bf Q},\rho(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\overline{\sigma}$ then the universal deformation ring is a dvr. When $k=2$ this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable congruence of Siegel modular forms. As an example we prove the modularity of an abelian surface of conductor 731. When $k>2$, we obtain an $R^{{\rm red}}=T$ theorem showing modularity of all such deformations of $\overline{\sigma}$.

Profinite Groups with a Cyclotomic $p$-Orientation

Let p be a prime. A continuous representation \theta\colon G\to\text{GL}_1(\mathbb{Z}_p) of a profinite group G is called a cyclotomic p -orientation if for all open subgroups U\subseteq G and for all k,n\geq1 the natural maps H^k(U,\mathbb{Z}_p(k)/p^n)\to H^k(U,\mathbb{Z}_p(k)/p) are surjective. Here \mathbb{Z}_p(k) denotes the \mathbb{Z}_p -module of rank 1 with U -action induced by \theta\vert_U^k . By the Rost-Voevodsky theorem, the cyclotomic character of the absolute Galois group G_{\mathbb{K}} of a field \mathbb{K} is, indeed, a cyclotomic p -orientation of G_{\mathbb{K}} . We study profinite groups with a cyclotomic p -orientation. In particular, we show that cyclotomicity is preserved by several operations on profinite groups, and that Bloch-Kato pro- p groups with a cyclotomic p -orientation satisfy a strong form of Tits' alternative and decompose as semi-direct product over a canonical abelian closed normal subgroup.

Arithmetic of Certain ℓ-Extensions Ramified at Three Places

Let l be a regular odd prime number, k the lth cyclotomic field, k∞ the cyclotomic ℤl-extension of k, K a cyclic extension of k of degree l, and = K · k∞. Under the assumption that there are exactly three places not over l that ramify in the extension K∞/k∞ and K satisfies some additional conditions, we study the structure of the Iwasawa module Tl(K∞) of K∞ as a Galois module. In particular, we prove that Tl(K∞) is a cyclic G(K∞/k∞)-module and the Galois group Γ = G(K∞/K) acts on Tl(K∞) as $$\sqrt \chi $$, where $$\chi :\Gamma \to \mathbb{Z}_\ell^ \times $$ is the cyclotomic character.

On the sections of universal hyperelliptic curves

In this paper, we will give an algebraic proof for determining the sections for the universal pointed hyperelliptic curves CHg,n/k→Hg,n/k, when g≥3 and the image of the ℓ-adic cyclotomic character Gk→Z× is infinite. Furthermore, we will study the nonabelian phenomena associated to the universal hyperelliptic curves. For example, we will show that the section conjecture holds for CHg/k→Hg/k and the unipotent analogue of the conjecture holds for the pointed cases. This work is an extension of Hain's original work on the rational points of universal curves to the hyperelliptic case.

CM fields of Dihedral type and the Colmez conjecture

In this paper, we consider some CM fields which we call of dihedral type and compute the Artin L-functions associated to all CM types of these CM fields. As a consequence of this calculation, we see that the Colmez conjecture in this case is very closely related to understanding the log derivatives of certain Hecke characters of real quadratic fields. Recall that the ‘abelian case’ of the Colmez conjecture, proved by Colmez himself, amounts to understanding the log derivatives of Hecke characters of $$\mathbb {Q}$$ (cyclotomic characters). In this paper, we also prove that the Colmez conjecture holds for ‘unitary CM types of signature $$(n-1, 1)$$ ’ and holds on average for ‘unitary CM types of a fixed CM number field of signature $$(n-r, r)$$ ’.

Distinguishing Galois representations by their normalized traces

Suppose ρ1 and ρ2 are two pure ℓ-adic degree n representations of the absolute Galois group of a number field K of weights k1 and k2 respectively, having equal normalized Frobenius traces Tr(ρ1(σv))/Nvk1/2 and Tr(ρ2(σv))/Nvk2/2 at a set of primes v of K with positive upper density. Assume further that the algebraic monodromy group of ρ1 is connected and ρ1 is absolutely irreducible. We prove that ρ1 and ρ2 are twists of each other by a power of the ℓ-adic cyclotomic character times a character of finite order. As a corollary, we deduce a theorem of Murty and Pujahari proving a refinement of the strong multiplicity one theorem for normalized eigenvalues of newforms.

ON THE THETA OPERATOR FOR MODULAR FORMS MODULO PRIME POWERS

We consider the classical theta operator on modular forms modulo and level prime to , where is a prime greater than three. Our main result is that mod will map forms of weight to forms of weight and that this weight is optimal in certain cases when is at least two. Thus, the natural expectation that mod should map to weight is shown to be false. The primary motivation for this study is that application of the operator on eigenforms mod corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the -operator mod gives an explicit weight bound on the twist of a modular mod Galois representation by the cyclotomic character.