Cyclotomic Character Research Articles

Level raising for p-adic Hilbert modular forms

This paper generalises previous work of the author to the setting of overconvergent p-adic automorphic forms for a definite quaternion algebra over a totally real field. We prove results which are analogues of classical ‘level raising’ results in the theory of mod p modular forms. Roughly speaking, we show that an overconvergent eigenform of finite slope whose associated local Galois representation at some auxiliary prime l - p is (a twist of) a direct sum of trivial and cyclotomic characters lies in a family of eigenforms whose local Galois representation at l is generically (a twist of) a ramified extension of trivial by cyclotomic. We give some explicit examples of p-adic automorphic forms to which our results apply, and give a general family of examples whose existence would follow from counterexamples to the Leopoldt conjecture for totally real fields. These results also play a technical role in other work of the author on the problem of local–global compatibility at Steinberg places for Hilbert modular forms of partial weight one.

On some anabelian properties of arithmetic curves

In this paper we generalize an argument of Neukirch from birational anabelian geometry to the case of arithmetic curves. In contrast to the function field case, it seems to be more complicated to describe the position of decomposition groups of points at the boundary of the scheme \({{\rm Spec}\, \mathcal{O}_{K, S}}\), where K is a number field and S a set of primes of K, intrinsically in terms of the fundamental group. We prove that it is equivalent to give the following pieces of information additionally to the fundamental group \({\pi_1({\rm Spec}\, \mathcal{O}_{K, S})}\) : the location of decomposition groups of boundary points inside it, the p-part of the cyclotomic character, the number of points on the boundary of all finite étale covers, etc. Under a certain finiteness hypothesis on Tate–Shafarevich groups with divisible coefficients, one can reconstruct all these quantities simply from the fundamental group.

Hecke Operators and the Stable Homology of GL(n)

Let R be a field of any characteristic and A a principal ideal domain. We make a conjecture that asserts that any Hecke operator T acts punctually on any Hecke eigenclass in the stable homology with trivial coefficients R of a principal congruence subgroup Γ in GL(n, A), i.e., as multiplication by the number of single cosets contained in T. In the case where A = ℤ, this conjecture implies that the Galois representation consisting of the sum of the 0th, 1st, ⋅, n − 1st powers of the cyclotomic character is attached to any stable Hecke eigenclass. When A = ℤ and Γ =GL(n, ℤ), this conjecture was already made by Calegari and Venkatesh. We obtain partial results giving evidence for the conjecture. These results imply that some part of the Hecke algebra acts punctually. If the characteristic of R is 0, they show that the entire Hecke algebra acts punctually, which was already known in a completely different way using a result of A. Borel.

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori–Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate–Shafarevich group of an elliptic curve in the Z p 2 -extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.

The sign of Galois representations attached to automorphic forms for unitary groups

AbstractLet K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

Uniform boundedness of p-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves. More precisely, we prove the following. Denote by Γ k the absolute Galois group of k. For an abelian variety A over k and a character $\chi:\Gamma_{k}\rightarrow\mathbb{Z}_{p}^{\ast}$ , define A[p ∞](χ) to be the module of p-primary torsion of $A(\overline{k})$ on which Γ k acts as χ-multiplication. Assume that χ does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p ∞](χ) is always finite, but the exponent of A[p ∞](χ) may depend on A, a priori. Our main result is about the uniform boundedness of A[p ∞](χ) when A varies in a 1-dimensional family. More precisely, if S is a curve over k and A is an abelian scheme over S, then there exists an integer N:=N(A,S,k,p,χ), such that A s [p ∞](χ)⊂A s [p N ] holds for any s∈S(k). This arithmetic result is obtained as a corollary of the following geometric result on the p-primary torsion of abelian varieties over function fields of curves, combined with Mordell’s conjecture (Faltings’ theorem). Let K be the function field of a curve over an algebraically closed field of characteristic 0 and A an abelian variety over K. Assume for simplicity that A contains no nontrivial isotrivial subvariety. Then, for any c≥0, there exists an integer N:=N(c,A,S,k,p)≥0 such that A[p ∞](K′)⊂A[p N ] for all finite extension K′/K with K′ of genus ≤c. A key ingredient of the proof of this geometric result is a certain result on the number of points on reduction modulo p n of p-adic analytic homogeneous spaces. Our uniform boundedness result when χ is the trivial character gives the uniform boundedness for the k-rational p-primary torsion in the fibres A s , s∈S(k) alluded to above. When χ is the p-adic cyclotomic character, together with certain descent methods, it also yields a proof of the 1-dimensional case of (a generalized variant of) the modular tower conjecture, which was, actually, the original motivation for this work. This is a conjecture arising from the regular inverse Galois problem, whose original form was posed by M. Fried in the early 1990s.

On the p-adic L-function of Hilbert modular forms at supersingular primes

In this paper, I discuss the construction of the p-adic L-function attached to a Hilbert modular form f, supersingular or ordinary, which turns out to be the non-archimedean Mellin transform of an h-admissible measure. And h is explicitly given. As a special case, when the Fourier coefficient of f at p|p is zero, plus/minus p-adic L-functions are furthermore defined as bounded functions, and they interpolate special values of L(f,χ,s) for cyclotomic characters χ. This can be used to formulate Iwasawa main conjecture for supersingular elliptic curve defined over a totally real field.

On the Cuspidality of Pullbacks of Siegel Eisenstein Series and Applications to the Bloch-Kato Conjecture

Let k > 9 be an even integer and p a prime with p > 2k −2. Let f be a newform of weight 2k −2 and level so that f is ordinary at p and is irreducible. Under some additional hypotheses, we prove that , where S is the Pontryagin dual of the Selmer group associated to with ϵ the p-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito–Kurokawa lift of f and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain the lower bound on the Selmer group.

Compatible families of elliptic type

In axiomatizing their study of Frobenius distributions [5], Lang and Trotter introduce the notion of an adelic Galois representation of elliptic type, and they ask in passing whether every such representation arises from an elliptic curve (see pp. 5 and 19 of [5]). Formulated in the language of `-adic representations [7], their question is as follows. Put G = Gal(Q/Q), let p denote a prime, and write σp for any Frobenius element at a prime ideal of Q over p. Let {ρ`} be a two-dimensional strictly compatible family of integral `-adic representations ofG with exceptional set S, and for p / ∈ S put a(p) = tr ρ`(σp) with any ` 6= p. Also, let ω` : G→ Z` denote the `-adic cyclotomic character. Although ρ` is a priori a map into GL(2,Q`), after a conjugation in GL(2,Q`) we may regard it as a map G→ GL(2,Z`).

Galois theory and integral models of Λ-rings

We show that any Λ-ring, in the sense of Riemann-Roch theory, which is finite etale over the rational numbers and has an integral model as a Λ-ring is contained in a product of cyclotomic fields. In fact, we show that the category of such Λ-rings is described in a Galois-theoretic way in terms of the monoid of pro-finite integers under multiplication and the cyclotomic character. We also study the maximality of these integral models and give a more precise, integral version of the result above. These results reveal an interesting relation between Λ-rings and the class field theory.

On mod 3 Galois representations with conductor 4

Let GQ be the absolute Galois group Gal(Q=Q) of Q. Let Fp be an algebraic closure of the flnite fleldFp of p elements. In this paper, we prove the non-existence of certain mod 3 Galois representation: Theorem 1. There exist no irreducible representations ‰ : GQ ! GL2(F3) with N(‰) dividing 4. Here, N(‰) = Q p-3 p n p(‰) is the Artin conductor of ‰ outside 3 ([6], x1.2; the deflnition of the exponent np(‰) will be recalled below). This proves a special case of Serre’s conjecture ([6]). Indeed, the conjecture predicts that such a representation, up to twist by a power of the mod 3 cyclotomic character, come from a cuspidal eigenform of level 4 and weight • 4, but there are no such forms. Such a result may serve as the flrst step of an inductive proof of Serre’s conjecture for N(‰) = 4 if Khare’s proof in the case of N(‰) = 1 ([3]) can be extended. Serre’s conjecture is known to be true if the image Im(‰) of ‰ is solvable ([4], Thm. 4). So, it remains for us to prove the Theorem 1 in the following two cases: (i) Im(‰) is non-solvable, (ii) ‰ is even and Im(‰) is solvable.

Existence of Nonelliptic mod l Galois Representations for Every l > 5

For l = 3 and 5 it is known that every odd, irreducible, two-dimensional representation of Gal(/Q) with values in F l and determinant equal to the cyclotomic character must “come from” the l-torsion points of an elliptic curve defined over Q. We prove, by giving concrete counter-examples, that this result is false for every prime l > 5.

Congruences Between Selmer Groups

In this paper we study when two congruent $l$-adic Galois representations have congruent Selmer groups. We obtain results for representations from cyclotomic characters, Hecke characters and adjoints of modular forms.

Demuškin Groups with Group Actions and Applications to Deformations of Galois Representations

We determine the universal deformation ring, in the sense of Mazur, of a residual representation \(\bar \rho :G_K \to {GL}_{2} (k)\), where k is a finite field of characteristic p and K is a local field of residue characteristic p. As one might hope for, but is not proven in the global case, the deformation ring is a complete intersection, flat over W(k), with the exact number of equations given by the dimension of \(H^2 (G_K ,{ad}_{\bar \rho })\). We then go on to determine the ordinary locus inside the deformation space and, using ideas of Mazur, apply this to compare the universal and the universal ordinary deformation spaces. Provided that the universal ring for ordinary deformations with fixed determinant is finite flat over W(k), as was shown in many cases by Diamond, Fujiwara, Taylor–Wiles and Wiles, we show that the corresponding universal deformation ring – with no restriction of ordinariness or fixed determinant – is a complete intersection, finite flat over W(k) of the dimension conjectured by Mazur, provided that the restriction of \((\bar \rho )\) to the inertia subgroup is different from the inverse cyclotomic character.

Torsion of abelian varieties, Weil classes and cyclotomic extensions

Let K⊂C be a field finitely generated over Q, K(a)⊂C the algebraic closure of K and G(K)=Gal (K(a)/K) its Galois group. For each positive integer m we write K(μm) for the subfield of K(a) obtained by adjoining to K all mth roots of unity. For each prime [lscr ] we write K([lscr ]) for the subfield of K(a) obtained by adjoining to K all [lscr ]-power roots of unity. We write K(c) for the subfield of K(a) obtained by adjoining to K all roots of unity in K(a). Let K(ab)⊂K(a) be the maximal abelian extension of K. The field K(ab) contains K(c); if K=Q then Q(ab)=Q(c) (the Kronecker-Weber theorem). We write χ[lscr ][ratio ]G(K)→Z*[lscr ] for the cyclotomic character defining the Galois action on all [lscr ]-power roots of unity. We write χ[lscr ]= χ[lscr ] mod [lscr ][ratio ]G(K) →Z*[lscr ]→(Z/[lscr ]Z)* for the cyclotomic character defining the Galois action on the [lscr ]th roots of unity. The character χ[lscr ] identifies Gal (K([lscr ])/K) with a subgroup of Z*[lscr ]=Gal (Q([lscr ])/Q). Let μ(Z[lscr ]) be the finite cyclic group μ(Z[lscr ]) of all roots of unity in Z*[lscr ]. Its order is equal to [lscr ]−1 if [lscr ] is odd and 2 if [lscr ]=2. Let Q([lscr ])′ be the subfield of μ(Z[lscr ])-invariants in Q([lscr ]). Clearly, Gal (Q([lscr ])/Q([lscr ])′)=μ(Z[lscr ]) and Gal (Q([lscr ])′/Q)= Z*[lscr ]/μ(Z[lscr ]) is isomorphic to Z[lscr ].

Syntomic regulators and special values of p -adic L -functions

In this paper p-adic analogs of the Lichtenbaum Conjectures are proven for abelian number fields F and odd prime numbers p, which generalize Leopoldt's p-adic class number formula, and express special values of p-adic L-functions in terms of orders of K-groups and higher p-adic regulators. The approach uses syntomic regulator maps, which are the p-adic equivalent of the Beilinson regulator maps. They can be compared with etale regulators via the Fontaine-Messing map, and computations of Bloch-Kato in the case that p is unramified in F lead to results about generalized Coates-Wiles homomorphisms and cyclotomic characters.

Adjoint modular Galois representations and their Selmer groups.

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(phi) of a two-dimensional modular Galois representation phi. We start with the p-adic Galois representation phi0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(phi0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(phi0)) from the proof of Wiles of the Shimura-Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let phi denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting phi0, and the characteristic power series of the Selmer group Sel(ad(phi)) is given by a p-adic L-function interpolating L(1, ad(phik)) for weight k + 2 specialization phik of phi. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(phi) [symbol, see text] nu-1), where nu is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

On the modularity of reducible mod $l$ Galois representations

Given an odd, semisimple, reducible, 2-dimensional mod l Galois representation, we investigate the possible levels of the modular forms giving rise to it. When the representation is the direct sum of the trivial character and a power of the mod l cyclotomic character, we are able to characterize the primes that can arise as levels of the associated newforms. As an application, we determine a new explicit lower bound for the highest degree among the fields of coefficients of newforms of trivial Nebentypus and prime level. The bound is valid in a subset of the primes with natural (lower) density at least 3/4.

Crystalline lifts of two-dimensional $\mathrm{mod} \: p$ automorphic Galois representations

We show that a sufficient condition for an irreducible automorphic Galois representation $\rho: G_F\to\mathrm{GL}_2({\overline{{\bf F}}_p})$ of a totally real field $F$ to have an automorphic crystalline lift is that for each place $v$ of $F$ above $p$ the restriction $\mathrm{det}\rho|_{I_v}$ is a fixed power of the mod $p$ cyclotomic character. Moreover, we show that the only obstruction to controlling the level and character of such automorphic lifts arises for badly dihedral representations.