FREE ELECTRONIC LIBRARY - Dissertations, online materials

Pages:   || 2 | 3 | 4 | 5 |

«To my teacher, Herv´ Jacquet e Introduction Let F be a number field, and (ρ, V ) a continuous, n-dimensional representation of the absolute Galois ...»

-- [ Page 1 ] --




To my teacher, Herv´ Jacquet



Let F be a number field, and (ρ, V ) a continuous, n-dimensional representation

of the absolute Galois group Gal(F /F ) on a finite-dimensional C-vector space V.

Denote by L(s, ρ) the associated L-function, which is known to be meromorphic

with a functional equation. Artin’s conjecture predicts that L(s, ρ) is holomorphic everywhere except possibly at s = 1, where its order of pole is the multiplicity of the trivial representation in V. The modularity conjecture of Langlands for such representations ([La3]), often called the strong Artin conjecture, asserts that there should be an associated (isobaric) automorphic form π = π∞ ⊗πf on GL(n)/F such that L(s, ρ) = L(s, πf ). Since L(s, πf ) possesses the requisite properties ([JS]), the modularity conjecture implies the Artin conjecture.

For any field k, let GO(n, k) denote the subgroup of GL(n, k) consisting of orthogonal similitudes, i.e., matrices g such that t gg = λg I, with λg ∈ k ∗.

We will say that a k-representation (ρ, V ) is of GO(n)-type iff dim(V ) = n and it factors as σ ρ = [Gal(F /F ) −→ GO(n, k) ⊂ GL(V )].

In this article we prove Theorem A Let F be a number field and let (ρ, V ) be a continuous, 4-dimensional C-representation of Gal(F /F ) whose image is solvable and lies in GO(4, C). Then ρ is modular, i.e., L(s, ρ) = L(s, πf ) for some isobaric automorphic representation π = π∞ ⊗ πf of GL(4, AF ). Moreover, π is cuspidal iff ρ is irreducible.

Among the finite groups G showing up as the images of such ρ are those fitting into an exact sequence 1 → C → H × H → G → {±1} → 1, where H is any finite solvable group in GL(2, C) with scalar subgroup C, the embe- ding of C in H × H is given by x → (x, x−1 ), and the action of {±1} on (H × H)/C is induced by the permutation of the two factors of H × H. This is due to the well known fact (see section 1) that GO(4, C) contains a subgroup of index 2 which is a quotient of GL(2, C)× GL(2, C) by C∗. Of particular interest are the examples where H is a central extension of S4 or A4 (cf. section 8).

PARTIALLY SUPPORTED BY THE NSF GRANT DMS-9801328 1 2 Dinakar Ramakrishnan One can ask if this helps furnish new examples of Artin’s conjecture, and the answer is yes.

Corollary B Let F be a number field, and let ρ, ρ be continuous C-representations of Gal(F /F ) of solvable GO(4)-type. Then Artin’s conjecture holds for ρ ⊗ ρ.

We will show in section 8 that in fact there is, for each F, a doubly infinite family of such examples where the representations ρ ⊗ ρ are irreducible and primitive (i.e., not induced) of dimension 16. Primitivity is important because Artin Lfunctions are inductive, and one wants to make sure that these examples do not come by induction from known (solvable) cases in low dimensions. We will then show (in secton 9) that, given any ρ as in Theorem A with corresponding extension K/F, the strong Dedekind conjecture holds for certain non-normal extensions N/F contained in K/F, for instance when [N : F ] = 3a, a ≥ 1 (see Theorem 9.3); the assertion is that the ratio ζN (s)/ζF (s) is the standard L-function of an isobaric automorphic form η on GL(3a − 1)/F. It implies that for any cusp form π on GL(n)/F, the formally defined Euler product L(s, πN ) (see the discussion before Corollary 9.4) admits a meromorphic continuation and functional equation, and more importantly, it is divisible by L(s, π). It should be noted that we do not know if πN is automorphic. For a curious consequence of this result see Remark 9.6.

It has been known for a long time, thanks to the results of Artin and Hecke, that monomial representations of Gal(F /F ), i.e., those induced by one-dimensional representations of Gal(F /K) with K/F finite, satisfy Artin’s conjecture. (In fact this holds for any multiple of a monomial representation.) But the strong Artin conjecture is still open for these except when K/F is normal and solvable ([AC]) and when [K : Q] = 3 ([JPSS1]. The work [AC] of Arthur and Clozel implies that the strong conjecture holds for ρ with nilpotent image, in fact whenever ρ is accessible ([C]), i.e., a positive integral linear combination of representations induced from linear characters of open, subnormal subgroups. It should however be noted that Dade has shown ([Da]) that all accessible representations of solvable groups are monomial.

The odd dimensional orthogonal representations with solvable image are simpler than the even dimensional ones. Indeed we have Proposition C Let ρ be a continuous, irreducible, solvable C-representation of Gal(F /F ) of GO(n)-type with n odd. Then ρ is monomial and hence satisfies the Artin conjecture. If ρ is in addition self-dual, it must be induced by a quadratic character.

By the groundbreaking work of Langlands in the seventies ([La1]), and the ensuing theorem of Tunnell in 1980 ([Tu]), one knows that the strong Artin conjecture holds for all two-dimensional representations σ with solvable image. The raison d’etre for this article is the desire to find irreducible, solvable, but non-monomial, even primitive, examples in higher dimensions satisfying the Artin cojecture. We were led to this problem a few years ago by a remark of J.-P.Serre. One could look at the symmetric powers of a two-dimensional solvable σ, but they are all in the linear span of monomial representations and 1-dimensional twists of σ. The facts that the symmetric square of σ is modular (by Gelbart-Jacquet [GeJ]) and monomial (for σ solvable) were used crucially in [La1] and [Tu].

Solvable Artin representations of GO(4)-type 3 In the non-solvable direction, which is orthogonal (no pun intended) to the one pursued in this paper, there has been some spectacular progress recently. For odd 2-dimensionals ρ of Gal(Q/Q) with projectivization ρ of A5 -type, a theorem of Buzzard, Dickinson, Shepherd-Barron and Taylor ([BDST]) establishes the modularity conjecture assuming the following: (i) ρ is unramified at 2 and 5, and (ii) ρ(Frob2 ) has order 3. In a sequel ([T]), this is shown with different ramification conditions, namely that under ρ (i’) the inertia group at 3 has odd order, and (ii’) the decomposition group at 5 is unramified at 2. See also [BS] for some explicit examples.

Some positive examples were given earlier in [Bu], and then in [Fr]. Moreover, the very recent theorems of Kim and Shahidi ([KSh]), and Kim ([K]), establishing the automorphy of the symmetric cube, and the symmetric fourth power, on GL(2), establishes the strong Artin conjecture for sym3 (σ), and sym4 (σ), for all the σ proved modular by [BDST].

If σ, σ are Galois representations which are modular, then one can deduce the Artin conjecture for σ⊗σ by applying the Rankin-Selberg theory on GL(m)×GL(n) developed in the works of Jacquet, Piatetski-Shapiro, Shalika ([JPSS2], [JS]) and of Shahidi ([Sh1,2]); see also [MW]. This explains why the strong form of Artin’s conjecture is really a bit stronger than the original conjecture that Artin made, at least given what one knows today.

If σ, σ are 2-dimensional representations which are modular, then the strong Artin conjecture for σ ⊗ σ follows from the main theorem of [Ra1], hence the Artin conjecture holds, by the remark above, for 4-fold tensor products of such representations. Now let K/F be a quadratic extension with non-trivial automorphism θ, and let σ θ denote the θ-twisted representation defined by x → σ(θxθ−1 ), where θ is ˜˜ ˜ any lift to Gal(F /F ). (The equivalence class of σ θ is independent of the choice of θ.) ˜ Given any irreducible, non-monomial 2-dimensional representation σ of Gal(F /K) which is not isomorphic to any one-dimensional twist of σ θ (see section 3), there is an irreducible 4-dimensional representation As(σ) of Gal(F /F ) whose restriction to Gal(F /K) is isomorphic to σ ⊗ σ θ. When σ is of solvable type, one can combine the theorem of Langlands-Tunnell with that of Asai ([HLR]) to deduce the Artin conjecture for As(σ).

One of the main steps in our proof of Theorem A is that even modularity can be established for any Asai representation As(σ) (in the solvable case). To begin, there exists, by Langlands-Tunnell, a cusp form π on GL(2)/K such that L(s, σ) = L(s, π). It follows that L(s, σ ⊗ σ θ ) equals the Rankin-Selberg L-function L(s, π × (π ◦ θ)). By [Ra1], there exists an automorphic form π (π ◦ θ) on GL(4)/K such that L(s, σ ⊗ σ θ ) = L(s, π (π ◦ θ)). Since π (π ◦ θ) is θ-invariant, one can now deduce the existence of a cusp form Π on GL(4)/F whose base change to K is π (π ◦ θ), but even when π (π ◦ θ) is cuspidal, Π is unique only up to twisting by the quadratic character δ, say, of the idele class group of F corresponding to K/F by class field theory. But it is not at all clear that Π, or Π ⊗ δ, should correspond precisely to As(σ), with an identity of the corresponding L-functions. (It is easy to see that the local factors agree at half the places.) Put another way, one can construct an irreducible admissible representation As(π) of GL(4, AF ) which has the same local factors as does As(σ). But the problem is that there is no simple reason why As(π) should be automorphic, even when π is dihedral. (Recall that π is dihedral, or of CM type, iff it is associated to a representation σ of the global Weil group WK induced by a character χ of a quadratic extension M of K, which 4 Dinakar Ramakrishnan

–  –  –

Then we show, still in section 3, how to deduce Theorem A modulo Theorem D;

in fact we need Theorem D only in the (crucial) case when (iii) occurs. When (i) or (ii) occurs, the modularity already follows from Theorem M of [Ra1] and base change ([AC]), with the desired Π being in case (i) (resp. (ii)) the Rankin-Selberg F product π(τ ) ⊗ π(τ ) (resp. the automorphic induction IL (π(η))); here τ → π(τ ) the Langlands-Tunnell map on solvable 2-dimensional Galois representations.

The proof of Theorem D is accomplished in sections 4 through 7. The approach

is similar to, but somewhat more subtle than, the proof of the existence of :

A(GL(2)/F )×A(GL(2)/F ) → A(GL(4)/F ) in [Ra1]. (For any n ≥ 1, A(GL(n)/F ) denotes the set of isomorphism classes of isobaric automorphic representations of GL(n, AF ).) In section 4 we show why it suffices to have the requisite properties at almost all places. Then in section 5 the distinguished case, i.e., when π ◦ θ is an abelian twist of π, is treated separately. In the general situation, i.e., when π (π ◦ θ) is cuspidal, we crucially use the converse theorem for GL(4) due to Cogdell and Piatetski-Shapiro ([CoPS1], which requires knowledge of the niceness of the twisted L-functions L(s, As(π) × π ) for automorphic forms π of GL(2)/F for a suitable class of π. Many properties of certain closely related functions, to be denoted L1 (s, As(π) × π ), were established by Piatetski-Shapiro and Rallis ([PSR]), and by Ikeda ([Ik1,2]), via an integral representation, which we use. There is another possible approach to studying L(s, As(π) × π ) via the Langlands-Shahidi method [Sh1], which yields another family of closely related L-functions, denoted L2 (s, As(π) × π ), with boundedness properties established recently in [GeSh], but we will not use it and our argument here is hewed to make use of the integral representation.

For almost all finite places v, the local factors L(s, As(πv )×πv ) and L1 (s, As(πv )× πv ) agree. But a thorny problem arises however, due to our inability to identify the bad and archimedean factors. In fact, when Fv = R, one does not even have a computation of the corresponding L1 -factor when π, π are unramified. Luckily, things simplify quite a bit under suitable, solvable base changes K/F with K totally complex, and after constructing the base-changed candidates As(π)K for an infinitude of such K, we descend to F as in sections 3.6, 3.7 of [Ra1]. We also have to control the intersection of the ramification loci of As(π), π and K/F.

One of the reasons why we have to work with L(s, As(π) × π ) is that we know how its local ε-factors behave under twisting by a highly ramified character, and this is not apriori the case with Lj (s, As(π) × π ) for j = 1 or 2. Indeed this will present a difficulty for the lifting of (generic) automorphic forms from GO(2n) to GL(2n), and for large n one cannot, as of yet, make use of base change and descent as we do here. For the lifting from GO(2n + 1) to GL(2n), see [CoKPSSh].

We would like to thank R.P. Langlands and the Institute for Advanced Study, Princeton, for their hospitality during the year 1999-2000, and the American Institute of Mathematics, Palo Alto, for support during the month of August in

1999. This project was partially funded by the NSF and AMIAS. We also want to acknowledge with thanks some helpful conversations with Michael Aschbacher and David Wales concerning the representations of finite groups. The main result here was presented at a conference on Automorphic Forms in Luminy, France (May 1999), and then at the IAS, Princeton (April 2000), and ICTP, Trieste, Italy (August 2000), where we received some helpful feedback. We would like to thank a number of people, including Jim Arthur, Don Blasius, Kevin Buzzard, Jim Cogdell, 6 Dinakar Ramakrishnan Bill Duke, Farshid Hajir, Henry Kim, Jeff Lagarias, Nick Katz, Dipendra Prasad, A.Raghuram, Peter Sarnak, Jean-Pierre Serre, Freydoon Shahidi and Joe Shalika who have shown interest and/or made comments.

–  –  –

Here we collect some basic facts, which we will need.

Let k be a field of characteristic different from 2, with separable algebraic closure k. If V is a finite dimensional vector space with a non-degenerate, symmetric bilinear form B, the associated orthogonal similitude group is (1.1) GO(V, B) := {g ∈ GL(V ) | B(gv, gw) = λ(g)B(v, w), with λ(g) ∈ k ∗, ∀v, w ∈ V }.

The character λ : GO(V, B) → k ∗, g → λ(g), is the similitude factor. The kernel of λ is the orthogonal group O(V, B), whose elements necessarily have determinant ±1, and the kernel of det is the special orthogonal group SO(V, B).

Pages:   || 2 | 3 | 4 | 5 |

Similar works:

«The Rule of the Community of Solitude Article I Of Identity (1) We are to be known formally as the “Community of Solitude, Camaldolese, abbreviated as CoS Cam. (2) In adopting this identity, we recognize that we are a Community, comprised of several members; we are devoted to seeking Solitude in our individual and common life; and we seek to emulate the spirit of St Romuald, founder of the Camaldolese Benedictines. (3) Our purpose is to provide to our members a guided experience in...»

«Profiles of Chinese Language Programs in Victorian Schools Jane Orton Julie Tee Julia Gong Jess McCulloch Yuanlin Zhao David McRae Chinese Teacher Training Centre The University of Melbourne June 2012 Copyright © The University of Melbourne, 2012 Profiles of Chinese Language Programs in Victorian Schools Jane Orton Julie Tee Julia Gong Jess McCulloch Yuanlin Zhao David McRae Chinese Teacher Training Centre The University of Melbourne June 2012 Copyright © The University of Melbourne, 2012...»

«OECD Innovative Learning Environment Project Universe Case Austria Goethe-Gymnasium This academic secondary school uses different self-developed forms of assessment to document student progress and to monitor educational quality, including cooperation with a school in St. Petersburg for external evaluation, (repeated) screenings of the students’ reading skills, repeated questionnaires for first-year students to document their transfer from primary to secondary school, and surveys for...»


«Ana min al yahud I'm one of the Jews By Almog Behar 1. At that time, my tongue twisted around and with the arrival of the month of Tammuz the Arabic accent got stuck in my mouth, deep down in my throat. Just like that, as I was walking down the street, the Arabic accent of Grandfather Anwar of blessed memory came back to me and no matter how hard I tried to extricate it from myself and throw it away in one of the public trash cans I could not do it. I tried and tried to soften the glottal...»

«File: Sterio 2 Created on: 6/17/2012 6:54:00 PM Last Printed: 9/1/2012 5:02:00 PM KATYN FOREST MASSACRE: OF GENOCIDE, STATE LIES, AND SECRECY Milena Sterio* The Soviet secret police murdered thousands of Poles near the Katyn Forest, just outside the Russian city of Smolensk, in the early spring of 1940. The Soviets targeted members of the Polish intelligentsia—military officers, doctors, engineers, police officers, and teachers—which Stalin, the Soviet leader, sought to eradicate...»

«Classroom Cognitive and Meta-Cognitive Strategies for Teachers Research-Based Strategies for Problem-Solving in Mathematics K-12 Florida Department of Education, Bureau of Exceptional Education and Student Services 2010 This is one of the many publications available through the Bureau of Exceptional Education and Student Services, Florida Department of Education, designed to assist school districts, state agencies that support educational programs, and parents in the provision of special...»

«6 CHAPTER SAVING THE WORLD Don’t know much about history, Don’t know much biology. — S AM COOKE, “W ONDERF UL W ORLD” G od does have a sense of humor, no question. After watching me terrorize teachers for years, the Almighty dropped a teaching job right into my lap. And you say you don’t believe. The year was 1971, the month September, and every weekday morning at exactly six thirty a.m., Rod Stewart’s voice would blare from my clock radio: Wake up, Maggie, I think I got something...»

«Investigating Bear and Panda Ancestry Adapted from Maier, C.A. (2001) “Building Phylogenetic Trees from DNA Sequence Data: Investigating Polar Bear & Giant Panda Ancestry.” The American Biology Teacher. 63:9, Pages 642-646. We have already used some molecular databases to examine evolutionary relationships among reptiles, bats, and birds, as well as conducting our own studies of evolutionary relationships. We have used amino acid sequences for proteins to examine these questions. We will...»

«Incho Lee Teacher Education Quarterly, Winter 2011 Teaching How To Discriminate: Globalization, Prejudice, and Textbooks By Incho Lee Language education is a complex social practice that reaches beyond teaching and learning phonology, morphology, and syntax. Language is not neutral; it conveys ideas, cultures, and ideologies embedded in and related to the language, so that language education needs to be examined not only on the purely linguistic...»

«Conserving the indigenous language of Tai northerners through community participatory activities by Paweena Chumbia and Juajan Wongpolganan Abstract This article has been written in remembrance of Paweena Chumbia (1963-2011), a Thai teacher who dedicated herself to the conservation of the indigenous language of the Tai northerners in Mae Mo District, Lampang Province, Thailand. The aims of the article are twofold, namely, to describe and celebrate the work of this teacher who loved to teach the...»

«Last Updated: August 2016 Smiles for Life: A National Oral Health Curriculum The Society of Teachers of Family Medicine (STFM) Group on Oral HealthThird Edition Examination Items Note: Answers to items appear on page 20. Module 1: The Relationship of Oral to Systemic Health 1. What is the most common chronic disease of childhood? A. Asthma B. Seasonal allergies C. Dental caries D. Otitis media 2. The Patient Centered Medical Home is the ideal place for all of the following EXCEPT: A....»

<<  HOME   |    CONTACTS
2016 www.dissertation.xlibx.info - Dissertations, online materials

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.