Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the structure of locally potentially equivalent Galois representations
69
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Suppose $rho_1, rho_2$ are two $ell$-adic Galois representations of the absolute Galois group of a number field, such that the algebraic monodromy group of one of the representations is connected and the representations are locally potentially equivalent at a set of places of positive upper density. We classify such pairs of representations and show that up to twisting by some representation, it is given by a pair of representations one of which is trivial and the other abelian. Consequently, assuming that the first representation has connected algebraic monodromy group, we obtain that the representations are potentially equivalent, provided one of the following conditions hold: (a) the first representation is absolutely irreducible; (b) the ranks of the algebraic monodromy groups are equal; (c) the algebraic monodromy group of the second representation is also connected and (d) the commutant of the image of the second representation remains the same upon restriction to subgroups of finite index of the Galois group.
rate research
Read More
We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $rho$ is determined by an equality of an $m$-power character $gmapsto Tr(rho(g^m))$ for some natural number $m$. Using this, we extend Faltings finiteness criteria to determine the equivalence of two $ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.
For every prime number $pgeq 3$ and every integer $mgeq 1$, we prove the existence of a continuous Galois representation $rho: G_mathbb{Q} rightarrow Gl_m(mathbb{Z}_p)$ which has open image and is unramified outside ${p,infty}$ (resp. outside ${2,p,infty}$) when $pequiv 3$ mod $4$ (resp. $p equiv 1$ mod $4$).
A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.
In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semistable reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].
Let E be a CM number field, F its maximal totally real subfield, c the generator of Gal(E/F), p an odd prime totally split in E, and S a finite set of places of E containing the places above p. Let r : G_{E,S} --> GL_3(F_p^bar) be a modular, absolutely irreducible, Galois representation of type U(3), i.e. such that r^* = r^c, and let X(r) be the rigid analytic generic fiber of its universal G_{E,S}-deformation of type U(3). We show that each irreducible component of the Zariski-closure of the modular points in X(r) has dimension at least 6[F:Q]. We study an analogue of the infinite fern of Gouvea-Mazur in this context and deal with the Hilbert modular case as well. As important steps, we prove that any first order deformation of a generic enough crystalline representation of Gal(Q_p^bar/Q_p) (of any dimension) is a linear combination of trianguline deformations, and that unitary eigenvarieties (of any rank) are etale over the weight space at the non-critical classical points. As another application, we obtain a general theorem about the image of the localization at p of the p-adic Adjoint Selmer group of the p-adic Galois representations attached to any cuspidal, cohomological, automorphic representation Pi of GL_n(A_E) such that Pi^* = Pi^c (for any n).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
