Two-dimensional Galois Representations Research Articles

Components of moduli stacks of two-dimensional Galois representations

Abstract In the article [CEGS20b], we introduced various moduli stacks of two-dimensional tamely potentially Barsotti–Tate representations of the absolute Galois group of a p-adic local field, as well as related moduli stacks of Breuil–Kisin modules with descent data. We study the irreducible components of these stacks, establishing, in particular, that the components of the former are naturally indexed by certain Serre weights.

Explicit Serre weights for two-dimensional Galois representations over a ramified base

Given a totally real number field F F and a mod p p Galois representation ρ : G F → G L 2 ( F ¯ p ) \rho \colon G_F\to \mathrm {GL}_2(\bar {\mathbf {F}}_p) , we propose an explicit definition of the set of Serre weights W ( ρ ) W(\rho ) attached to ρ \rho . We prove that our explicit definition is equivalent to previous definitions available in the literature. As a consequence we obtain an explicit Serre’s modularity conjecture for Hilbert modular forms over totally real number fields. Our work generalises previous work of Dembélé–Diamond–Roberts and Calegari–Emerton–Gee–Mavrides which together give explicit and equivalent sets of weights when p p is unramified in F F .

The Fontaine-Mazur conjecture in the residually reducible case

We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations overQ\mathbb {Q}when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a Taylor-Wiles patching argument for the completed homology in this case. As a key input, we generalize the work of Skinner-Wiles in the ordinary case. In addition, we also treat the residually irreducible case at the end of the paper. Combining with people’s earlier work, we can prove the Fontaine-Mazur conjecture completely in the regular case whenp≥5p\geq 5.

Reducible Galois representations and arithmetic homology for GL(4)

We prove that a sum of two odd irreducible two-dimensional Galois representations with squarefree relatively prime Serre conductors is attached to a Hecke eigenclass in the homology of a subgroup of GL(4,ℤ), with the level, nebentype, and coefficient module of the homology predicted by a generalization of Serre’s conjecture to higher dimensions. To do this we prove along the way that any Hecke eigenclass in the homology of a congruence subgroup of a maximal parabolic subgroup of GL(n,ℚ) has a reducible Galois representation attached, where the dimensions of the components correspond to the type of the parabolic subgroup. Our main new tool is a resolution of ℤ by GL(n,ℚ)-modules consisting of sums of Steinberg modules for all subspaces of ℚ n .

Explicit Serre weights for two-dimensional Galois representations

We prove the explicit version of the Buzzard–Diamond–Jarvis conjecture formulated by Dembeleet al.(Serre weights and wild ramification in two-dimensional Galois representations, Preprint (2016),arXiv:1603.07708[math.NT]). More precisely, we prove that it is equivalent to the original Buzzard–Diamond–Jarvis conjecture, which was proved for odd primes (under a mild Taylor–Wiles hypothesis) in earlier work of the third author and coauthors.

On two-dimensional Galois representations with squarefree conductor

On two-dimensional Galois representations with squarefree conductor

Deformations of Galois representations and exceptional monodromy

For any simple algebraic group $G$ of exceptional type, we construct geometric $\ell$-adic Galois representations with algebraic monodromy group equal to $G$, in particular producing the first such examples in types $\mathrm{F}_4$ and $\mathrm{E}_6$. To do this, we extend to general reductive groups Ravi Ramakrishna's techniques for lifting odd two-dimensional Galois representations to geometric $\ell$-adic representations.

On class groups of imaginary quadratic fields

In this paper, it is proved that one can find imaginary quadratic fields with class number not divisible by a specified prime l and with certain specified splitting conditions at a finite number of primes. Such existence theorems are useful in the arithmetic of elliptic curves and, potentially, also in certain lifting problems for reducible two-dimensional Galois representations. The methods used are a blend of geometry and the theory of modular forms, especially the trace formula.

Galois representations attached to automorphic forms on over fields

AbstractIn this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on${\rm GL}_2$over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑n=1∞a(n)qn denote the unique weight 16 normalized cuspidal eigenform on SL2(Z). In the early 1970s, Serre and Swinnerton-Dyer conjectured thata(p)2p−15≡0,1,2,4(mod59),when p≠59 is prime. This was proved in 1983 by Haberland. We describe a general method for proving congruences for the coefficients of eigenforms which arise from odd octahedral complex two-dimensional Galois representations, of which this congruence is the prototypical example. In particular, we prove all such congruences for the coefficients of eta-product newforms.

On the semi-simplicity of the $U_p$ -operator on modular forms

For and positive integers, let denote the -vector space of cuspidal modular forms of level and weight . This vector space is equipped with the usual Hecke operators , . If we need to consider several levels or weights at the same time, we will denote this by , or ! . If " is a prime number dividing , our $# is also known under the name %&# . One of our main results can be stated very easily: if ('*) and ",+ does not divide , then the operator $# is semi-simple. We can prove the same result for weight /. , under the assumption that certain crystalline Frobenius elements are semi-simple. Milne has shown in [11, 0 2] that this semi-simplicity is implied by Tate’s conjecture claiming that for 1 projective and smooth over a finite field of characteristic " , and 23 54 , 687:9 =@? AB C1D FEHGJILKNMO equals the order of PQ C1R SB at 2 . Ulmer proved in [17] that $# is semi-simple, for 5'T. and ",U not dividing , under the assumption of the Birch-Swinnerton-Dyer conjecture for elliptic curves over function fields in characteristic " . His method is quite different from ours: assuming that $# is not semi-simple, he really shows that the Birch-Swinnerton-Dyer conjecture does not hold for an explicitly given elliptic curve. The structure of our proof is as follows. Using the theory of newforms, the problem is shown to be equivalent to the problem of showing that, for a normalized newform V of weight , prime-to-" level and character W , the polynomial X UZY\[ #]X<^\W8 _" `" !ba,c has no double root. This polynomial happens to be the characteristic polynomial of the Frobenius element at " in the two-dimensional Galois representations associated to V ; it is also the characteristic polynomial of the crystalline Frobenius asociated to V . We show that this crystalline Frobenius cannot be a scalar. In Sections 2 and 3 we prove the results concerning these Frobenius elements for 'd) and e f) , respectively. Section 2 is quite elementary, whereas in Section 3 we use a lot of the machinary for comparing " -adic etale and crystalline cohomology. In Section 4 we give some applications: the Ramanujan inequality is a strict inequality in certain cases, certain Hecke algebras are reduced, hence have non-zero discriminant. Section 5 gives some results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras. g partially supported by the Institut Universitaire de France