P. Deligne, Les constantes des 'equations fonctionnelles des fonctions L. Modular Functions one Variable II, Proc. internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349 (1973), 501--597.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for compatible systems. However, it should be noted that, for an irreducible # adic representation # # , the equivalence between being symplectic and having the exterior square L function admit a pole is not known, though predicted by the Tate conjectures. One knows by a theorem of Langlands (see [De2] for an elegant global proof) that there is a factorization W (## # # ) # v W v (## # # ) where each local constant W v (## # # ) depends only on the restriction # v# # # v to the decomposition group D v at v. 4 DIPENDRA PRASAD AND DINAKAR RAMAKRISHNAN Now fix a place v, and ....

....(i) k v = E, ii) k # v # = E # , and (iii) the decomposition group of v # in Gal(k # k) is the whole group. Moreover, all the primes in S split completely in k # . This lemma is a consequence of Krasner s lemma and its proof will be left to the reader. A slightly weaker version can be found in [De2], page 544, as Lemme 4.13. Applying this lemma to our setup, we get a Galois extension M k of number fields with local extension L E, such that Gal( M k) Gal( L E) Then Gal( E L) and Gal( E L # ) are subgroups of Gal( M k) and let us denote their fixed fields in M by M # and M ....

[Article contains additional citation context not shown here]

: P. Deligne, Les constantes des equations fonctionnelles des fonctions L, in Modular functions of one variable II, Springer Lecture Notes 349 (1973), 501-597.

....Y pjst(s Gammat) w p (A p ; B p ) avec w p (A p ; B p ) 8 : 1 si p = 2 et A 2 j 16 (mod 32) Gamma i GammaB p p j si p 6 jB p (A p Gamma B p ) i Gamma1 p j si pjB p (A p Gamma B p ) et p 3; o u A p et B p sont donn es par la table I. Preuve. D apr es Deligne [8], le signe w de l equation fonctionnelle d une courbe elliptique quelquonque E=Q est le produit des signes locaux w p , o u p est un diviseur premier du conducteur et de w1 = Gamma1. Ces signes d ependent du type de r eduction. Si la courbe E a une r eduction multiplicative en p, alors w p = ....

P. Deligne, Les constantes des 'equations fonctionnelles des fonctions L, in Antwerp II: Modular Functions of One Variable, Lecture Notes in Math., Springer-Verlag 349 (1973), 501--597.

.... # v L(s, r(#(# v ) # v ) and #(s, #; r# ) # v #(s, r(#(# v ) # v ) where v runs over all the places of F , # v # #(# v ) the arrow giving the local Langlands correspondence for GL(n) F v , and local factors are those attached to representations of the (extended) Weil group ([De73]) To be precise, in the treatment of the non archimedean case in [De73] Deligne uses the Weil Deligne group WDFv , but it is not di#cult to see how its representations are in bijection with those of W # Fv . Also, the local # factors depend on the choice of a non trivial additive character ....

.... v ) # v ) where v runs over all the places of F , # v # #(# v ) the arrow giving the local Langlands correspondence for GL(n) F v , and local factors are those attached to representations of the (extended) Weil group ( De73] To be precise, in the treatment of the non archimedean case in [De73], Deligne uses the Weil Deligne group WDFv , but it is not di#cult to see how its representations are in bijection with those of W # Fv . Also, the local # factors depend on the choice of a non trivial additive character and the Haar measure, but we suppress this in our notation. Originally, ....

[Article contains additional citation context not shown here]

: P. Deligne, Les constantes des equations fonctionnelles des fonctions L, in Modular functions of one variable II, Springer Lecture Notes 349 (1973), 501-597.

....1 as # j has trivial central character. Put # v = # 1,v# # 2,v# # 3,v# # 4,v . By Prop. 4.3.1, we are reduced to showing that W (# v ) with the obvious definition) is 1. Being the tensor product of an even number of symplectic representations, # v is orthogonal, and by Deligne ([De]) the root number of # v : # v # 4[1] is given by its second Stie#el Whitney class w 2 ( # v ) But since Sp(2, C) is simply connected, and since the image of # v in O(4, C) factors through a 4 fold product of Sp(4, C) this image is simply connected. It follows that # v lifts to the Spin ....

: P. Deligne, Les constantes des equations fonctionnelles des fonctions L, in Modular functions of one variable II, Springer Lecture Notes 349 (1973), 501-597.

....X i ( Gamma1) i [R i f ( Omega ffl X=S Omega E; r X=S ) GM(r) Gamma1) d f (c d( Omega 1 X=S ) Delta c n (E; r) in H 2n (X) Omega Q: For n = 1, there is an analogy with the situation where S is a finite field F q , E; r) is a local system V . Then the work of Deligne [12], 13] and subsequent work by Laumon [21] S. Saito [26] T. Saito [27] show that 3.0.3 for n = 1 remains true: det X i ( Gamma1) i H i et (X; V ) Gamma1) d det V jc d( Omega 1 X ) 3.0.4) as Frobenius modules over F q . ALGEBRAIC THEORY OF CHARACTERISTIC CLASSES 9 Both 3.0.3 ....

Deligne, P.: Les constantes des 'equations fonctionnelles des fonctions L, Springer LN 349 (1973), 903-952.

....Bismut [2] prove an analogue of theorem 0. 1 using characteristic classes c n (E r ) 2 H 2n Gamma1 (X an ; C =Q(n) as defined by Chern and Simons [6] Analogously, if S is replaced by a finite field F q , E; r) by a tame representation of the fundamental group ae, then Deligne s theorem [8], 9] and subsequent work by Laumon [19] S. Saito [24] and T. Saito [25] show that for n = 1 the formula (0.14) remains true O i (detH i et (X; ae Gamma dim(ae) Delta Q ) Gamma1) i = det aejc d( Omega 1 X ) Gamma1) d (0.16) as dimension 1 Q vectorspaces with ....

....diagrams 4.10 and 4.12, this concludes the proof of the proposition. 46 SPENCER BLOCH AND H EL ENE ESNAULT Let (E; r) be any connection on X as in theorem 5.1. Let Y = D H, with H D = such that the condition 5.1 is fulfilled with E replaced with E(H) and D replaced with Y . Then, by [8] (for an algebraic version of it, see e.g. 14] the inclusion Omega X=S (log Y ) Omega E Omega X=S (log Y ) Omega E(H) is a quasi isomorphism. We may thus define Nw 1 (Rf Omega X=S (log Y ) Omega E) Nw 1 (f ( Omega X=S (log Y ) Omega E(H) Proposition 5.3. The ....

Deligne, P.: Les constantes des 'equations fonctionnelles des fonctions L, Springer LN 349 (1973), 903-952.

....the local factor is defined by i ; dx j = Z p Gammaa( Gamman( R Theta p Gamma1 (x) x) dx if is ramified; p n( p n( if is unramified; Supported in part by US NSF Grant #DMS 97 01225. 2 YE YANGBO and the local factor Ep=Q p ( is defined in Deligne [3] and Langlands [9] We could use (1) as the definition of Ep=Q p ( as in Jacquet and Langlands [6] and Arthur and Clozel [1] Then by setting = j or = 1 we get Ep=Q p ( p Gammad=2 i j; dx j Delta Delta Delta i j m Gamma1 ; dx j (2) because (1; dx) 1 and ....

P. Deligne, Les constantes des equations fonctionnelles des fonctions L, in: Modular Functions of One Variable II, Lect. Notes Math., Vol. 349, Springer, Berlin Heidelberg New York, 1973, pp. 501-597.

No context found.

P. Deligne, Les constantes des 'equations fonctionnelles des fonctions L. Modular Functions one Variable II, Proc. internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349 (1973), 501--597.

No context found.

P. Deligne, Les constantes des equations fonctionnelles des fonctions L. Modular Functions one Variable II, Proc. internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349 (1973), 501--597.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC