Non-existence of Certain Galois Representations with a Uniform Tame Inertia Weight

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R=T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Bockle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvea-Mazur. We generalize Bockle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under assumptions necessary to apply a small $R = \mathbb{T}$ theorem and an assumption on the local mod $p$ representation, we prove that every irreducible component of the universal polarized deformation space contains an automorphic point. When combined with work of Chenevier, this implies new results on the Zariski density of automorphic points in polarized deformation space in dimension three.

We construct a projective Galois representation attached to an abelian L-surface with quaternionic multiplication, describing the Galois action on its Tate module. We prove that such representation characterizes the Galois action on the isogeny class of the abelian L-surface, seen as a set of points of certain Shimura curves.

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

Mod-p Galois representations not arising from abelian varieties

Stable reduction of modular curves.- On p-adic families of automorphic forms.- ?-curves and abelian varieties of GL2-type from dihedral genus 2 curves.- The old subvariety of J0(NM).- Irreducibility of Galois actions on level 1 Siegel cusp forms.- On elliptic K-curves.- ?-curves and Galois representations.- On the local behaviour of ordinary modular Galois representations.- Arithmetic of ?-curves.- Serre's conjecture for mod 7 Galois representations.- Pairings in the arithmetic of elliptic curves.- Explicit parametrizations of ordinary and supersingular regions of X0(Pn).- Elliptic ?-curves with complex multiplication.- Abelian varieties over ? with large endomorphism algebras and their simple components over ?.- Abelian varieties over ? and modular forms.- Shimura curves embedded in Igusa's threefold.- Shafarevich-Tate groups of nonsquare order.

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

We study the growth of the rank of elliptic curves and, more generally, Abelian varieties with respect to finite extensions of number fields. First, we show that if \(A\) is an Abelian variety over a number field \(K\) and \(L/K\) is a finite Galois extension such that \({{\mathrm{Gal}}}(L/K)\) does not have an index 2 subgroup, then \({{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)\) can never be 1. We show that \({{\mathrm{rk}}}A(L)-{{\mathrm{rk}}}A(K)\) is either 0 or \(\ge p-1\), where \(p\) is the smallest prime divisor of \(\# {{\mathrm{Gal}}}(L/K)\), and we obtain more precise results when \({{\mathrm{Gal}}}(L/K)\) is alternating, \({{\mathrm{SL}}}_2(\mathbb {F}_p)\) or \({{\mathrm{PSL}}}_2(\mathbb {F}_p)\) for \(p>2\). This implies a restriction on \({{\mathrm{rk}}}E(K(E[p]))-{{\mathrm{rk}}}E(K(\zeta _p))\) when \(E/K\) is an elliptic curve whose mod \(p\) Galois representation is surjective. We obtain similar results for the growth of the rank over certain non-Galois extensions. Second, we show that for every \(n\ge 2\) there exists an elliptic curve \(E_n\) over a number field \(K_n\) such that \(\mathbb {Q}\otimes {{\mathrm{End}}}_\mathbb {Q}{{\mathrm{Res}}}_{K_n/\mathbb {Q}} E_n\) contains a number field of degree \(2^n\). We ask whether every elliptic curve \(E/K\) has infinite rank over \(K\mathbb {Q}(2)\), where \(\mathbb {Q}(2)\) is the compositum of all quadratic extensions of \(\mathbb {Q}\). We show that if the answer is yes, then for any \(n\ge 2\), there exists an elliptic curve \(E_n\) over a number field \(K_n\) admitting infinitely many quadratic twists whose rank is a positive multiple of \(2^n\).

Relative Galois Module Structure and Steinitz Classes of Dihedral Extensions of Degree 8

We construct infinitely ramified Galois representations ρ such that the al(ρ)'s have distributions in contrast to the statements of Sato–Tate, Lang–Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalizing a result of Ramakrishna.

We establish new cases of quadratic number fields [Formula: see text] unramified away from a prime [Formula: see text] and [Formula: see text] whose absolute Galois group has no irreducible two-dimensional continuous Galois representations in [Formula: see text]. Our work builds on methods of Moon–Taguchi and Şengün and the usual analytic techniques of Odlyzko and Poitou where we note one of the new conditional cases arises via a correction of Poitou’s estimate. The results here seem optimal in that it seems these methods alone will yield no further cases either due to prohibitive computational issues or a failure of the analytic obstructions.

This article deals with universal deformations of dihedral representations with a particular focus on the question when the universal deformation is dihedral. Results are obtained in three settings: (1) representation theory, (2) algebraic number theory, (3) modularity. As to (1), we prove that the universal deformation is dihedral if all infinitesimal deformations are dihedral. Concerning (2) in the setting of Galois representations of number fields, we give sufficient conditions to ensure that the universal deformation relatively unramified outside a finite set of primes is dihedral, and discuss in how far these conditions are necessary. As side-results, we obtain cases of the unramified Fontaine–Mazur conjecture, and in many cases positively answer a question of Greenberg and Coleman on the splitting behaviour at p of p-adic Galois representations attached to newforms. As to (3), we prove a modularity theorem of the form ‘ $$R=\mathbb {T}$$ ’ for parallel weight one Hilbert modular forms for cases when the minimal universal deformation is dihedral.

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.

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent ℚ-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without nontrivial endomorphisms (Mumford’s example, not of PEL type). We give new results for abelian varieties not of PEL type.