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The weight in a Serretype conjecture for tame ndimensional Galois representations
, 2006
"... Abstract. We formulate a Serretype conjecture for ndimensional Galois representations that are tamely ramified at p. The weights are predicted using a representationtheoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. ..."
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Abstract. We formulate a Serretype conjecture for ndimensional Galois representations that are tamely ramified at p. The weights are predicted using a representationtheoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters. 1.
Rigid CalabiYau Threefolds Over Q Are Modular: A Footnote to Serre
"... Abstract. The proof of Serre’s conjecture on Galois representations over finite fields allows us to show, using a trick due to Serre himself, that all rigid CalabiYau threefolds defined over Q are modular. The safest general characterization of the European philosophical tradition is that it consis ..."
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Abstract. The proof of Serre’s conjecture on Galois representations over finite fields allows us to show, using a trick due to Serre himself, that all rigid CalabiYau threefolds defined over Q are modular. The safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato. Alfred North Whitehead, Process and Reality In the mid1980s, J.P. Serre conjectured in [11] that all absolutely irreducible odd twodimensional representations of GQ = Gal(Q/Q) over a finite field come from modular forms of prescribed weight, level, and character. This has now been proved by C. Khare and J.P. Wintenberger; see [6, 7]. Because this result can be seen as a generalization of Artin Reciprocity to the GL2 case (over Q), we will refer to it as “Serre Reciprocity.” Already in [11], Serre showed how, given a compatible system of ℓadic Galois representations and bounds on the weight and level of the predicted modular forms in characteristic ℓ, one can use Serre Reciprocity to obtain results in characteristic zero. We refer to this as “Serre’s method ” and describe it in Section 1 below. The goal of this paper is to use Serre’s method to show that certain geometric Galois representations are modular. Specifically, we show that the representation obtained from the third étale cohomology of a rigid CalabiYau threefold defined over Q comes from a modular form of weight 4 on Γ0(N). The proof is an immediate application of Serre’s method; it can, in fact, be read off directly from [11, Section 4.8], which
Iterated Endomorphisms of Abelian Algebraic Groups
"... Abstract. Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] has a natural tree structure. Using this data, we construct an “arboreal ” Galois representation ω whose image combines that of the usual ℓadic repres ..."
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Abstract. Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] has a natural tree structure. Using this data, we construct an “arboreal ” Galois representation ω whose image combines that of the usual ℓadic representation and the Galois group of a certain Kummertype extension. For several classes of A, we give a simple characterization of when ω is surjective. The image of ω also encodes information about the density of primes p in K such that the order of the reduction mod p of α is prime to ℓ. We compute this density in the general case for several A of interest. For example, if F is a number field, A/F is an elliptic curve with surjective 2adic representation and α ∈ A(F), with α ̸ ∈ 2A(F(A[4])), then the density of primes p with α mod p having odd order is 11/21. 1.
WEIGHTS IN GENERALIZATIONS OF SERRE’S CONJECTURE AND THE MOD p LOCAL LANGLANDS CORRESPONDENCE
"... Abstract. In this mostly expository article we give a survey of some of the generalizations of Serre’s conjecture and results towards them that have been obtained in recent years. We also discuss recent progress towards a mod p local Langlands correspondence for padic fields and its connections wit ..."
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Abstract. In this mostly expository article we give a survey of some of the generalizations of Serre’s conjecture and results towards them that have been obtained in recent years. We also discuss recent progress towards a mod p local Langlands correspondence for padic fields and its connections with Serre’s conjecture. A theorem describing the structure of some mod p Hecke algebras for GLn is proved. 1.
Explicit Level Lowering for 2Dimensional Modular Galois Representations
, 2010
"... Let f be a normalized eigenform of level Nℓα for some positive integer α and some odd prime ℓ satisfying gcd(ℓ, n) = 1. A construction of Deligne, Shimura, et. al., attaches an ℓadic continuous twodimensional Galois representation to f. The Refined Conjecture of Serre states that such a represent ..."
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Let f be a normalized eigenform of level Nℓα for some positive integer α and some odd prime ℓ satisfying gcd(ℓ, n) = 1. A construction of Deligne, Shimura, et. al., attaches an ℓadic continuous twodimensional Galois representation to f. The Refined Conjecture of Serre states that such a representation should in fact arise from a normalized eigenform of level prime to ℓ. In this thesis we present a proof of Ribet which allows us to “strip ” these powers of ℓ from the level while still retaining the original Galois representation, i.e., the residual of our new representation arising from level N will remain isomorphic to the residual of our original representation arising from level Nℓα.
Galois extension K/Q.
, 2009
"... In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective ..."
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In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare’s and Wintenberger’s notion of gooddihedral primes. Particular care is taken in order to exclude nontrivial inner twists.
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Galois Theory of Iterated Endomorphisms
"... Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime `, the set of all preimages of α under some iterate of [`] generates an extension of F that contains all `power torsion points as well as a Kummertype extension. We analyze the Galois group of this extension, and for s ..."
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Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime `, the set of all preimages of α under some iterate of [`] generates an extension of F that contains all `power torsion points as well as a Kummertype extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (α mod p) is prime to `. We compute this density in the general case for several classes of A, including elliptic curves and onedimensional tori. For example, if F is a number field, A/F is an elliptic curve with surjective 2adic representation and α ∈ A(F) with α 6 ∈ 2A(F (A[4])), then the density of p with (α mod p) having odd order is 11/21. 1.