Abelian l-Adic Representations and Elliptic Curves Jean-Pierre Serre College de France McGill University Lecture Notes written with the collaboration of WILLEM K UYK and JOHN LABUIE I'Nc "lIo ADDISON-WESLEY PUBLISHING COMPANY, INC ��: THE ADVANCED BOOK PROGRAM +- PROC; Redwood City, California Menlo Park, California Reading, M: New York· Amsterdam· Don Mills, Ontario· Sydney Madrid Singapore· Tokyo· San Juan· Wokingharn, United Kingdom ' _"A � R_epr�n,t.a!!o�S � �/ Abelian l-A n ' " ' - '- �.: -, Originally published In 1968 b y IIIPtlc Curves ' W A Benjamin, Inc Library of Congress Cataloging-in-PubIlcation Data Serre, Jean Pierre Abelian L-adic representation s and elliptic curves (Advanced book classics series) On t.p "I" in I-adic is transcribed in lower-case script Bibliography: p Includes in dex Representations of groups Curves, Elliptic Fields, Algebraic I Title QA171.S525 1988 ll Series 512'.22 88-19268 ISBN 0-201-09384-7 Copyr ight© 1989,1968 by Addison-Wesley Publishing Company All rights reserved No part of this publication may be repro duc ed, stored in a retrieval system, or transmitted, in any form or by any means, electronic, photocopying, recording, or otherwise, withou t the pri or written permission of the publisher Manufactured in the United States of America Published simultaneously in Canada Publisher's Foreword "Advanced Book Classics" is a reprint series which has come into being as a direct result of public demand for the individual volumes in this program That was our initial criterion for launching the series Additional criteria for selection of a book's inclusion in the series include: • • • Its intrinsic value for the current scholarly buyer It is not enough for the book to have some historic significance, but rather it must have a timeless quality attached to its content, as well In a word, "uniqueness." The book's global appeal A survey of our international markets revealed that readers of these volumes comprise a boundary less , worldwide audience The copyright date and imprint status of the book Titles in the program are frequently fifteen to twenty years old Many have gone out o f prin t, some are about to go out of print Our aim is to sustain the lifespan of these very special volumes We have devised an attractive design and trim-size for the "ABC" titles, giving the series a striking appearance, while len din g the individual titles unifying id entity as part of the "Advanced Book Classics" program S ince "classic" books demand a long-lasting binding, we have made them available in hardcover at an affordable price We envision them being p urchased by individuals for reference and research use, and for personal and public libraries We also foresee their use as primary and recommended course materials for university level courses in the appropriate subject area The "Advanced Book Classics" program is not static Titles will continue to be added to the series in ensuing years as works meet the criteria for inclusion which we've imposed As the series grows, we naturally anticipate our book buying audience to grow with it We welcome your support and your suggestions concerning future volumes in the program and invite you to communicate directly with us vii Vita Jean-Pierre Serre Professor of Algebm and Geometry at the College de France, Paris, was born in Bages, France, on September 15, 1926 He gmduated from Ecole Normale Superieure, Paris, in 1948, and obtained his Ph.D from the Sorbonne in 1951 In 1954 he was awarded a Fields Medal for his work on topology (homotopy groups) and algebraic geometry (coherent sheaves) Since then, his main topics of interest have been number theory, group theory, and modular forms Professor Serre has been a frequent visitor of the United States, especially at the Institute for Advanced Study, Princeton, and Harvard University He is a foreign member of the National Academy of Sciences of the U.S.A viii Special Preface The present edition differs from the original one (published in 1968) by: • the inclusion of short notes giving references to new results; • a supplementary bibliography Otherwise, the text has been left unchanged, except for the correction of a few misprints The added bibliography does not claim to be complete Its aim is just to help the reader get acquainted with some of the many developments of the past twenty years (for those prior to 1977, see also the survey [78]) Among these developments, one may especially mention the following: l-adic representations associated to abelian varieties over number fields Deligne (cf [52]) has proved that Hodge cohomology classes behave under the action of the Galois group as if they were algebraic, thus providing a very useful substitute for the still unproved Hodge conjecture Faltings ([54], see also Szpiro [82] and Faltings-Wiistholz [56]), has proved Tate's conjecture that the map HomK (A,B) � Z, � Homa.J(T,(A), T ,(B» is an isomorphism (A and B being abelian varieties over a number field K), together with the semi-simplicity of the Galois module Q, � T/(A) and similar results for T /(A)/rr/(A) Ix Preface This book reproduces, with a few complements, a set of lectures given at McGill University, Montreal, from Sept.5 to Sepl18, 1967 It has been written in collaboration with John LABurn (Chap I, IV) and Willem KUYK (Chap II, III) To both of them, I want to express my heartiest thanks Thanks also due to the secretarial staff of the Institute for Advanced Study for its careful typing of the manuscript JEAN-PIERRE SERRE Princeton, Fall 1967 xi Special Preface x when I is large enough These results may be used to study the structure of the Galois group of the division points of A, cf [80] For instance, if dimA is odd and En ( a ) ( the only non­ trivial o ne ) follow s fro m the cor ollary to theo r em of A C on ­ ve r s ely , if E s complex multiplication by an imaginar y quadratic field F , the group Gal( K / K ) a c t s on Vp thr ough F � Q ( s e e p chap I I , ) and thi s action i s thu s s em i - simple Co n s e que ntly, the I V- 44 A B E LIA N l -A DIC R E PR E S E N T A T IO NS exact s e quenc e (a' ) =:l> ( 1,, ) split s ; thi s s hows tha t ( a ) =:l> ( b ' ) Since (a) =:l> ( b) , hence al s o that ( a l ) i s trivia l , the the o r e m is pr ove d If E s no c omplex mul tiplication, � is the Bor e l subalge b of End( V ) fo r me d b -y tho s e u e E nd( V ) - bra X P P s uch that u( X) C X ; the ine r tia al gebra � i s the subalgebra !.X o f C OR O L LAR y - b formed b y tho s e u X Le t Xx and e E nd( V ) s u c h that u ( V ) C P p x Xy be the characte r s o f Gal( K / K ) defined by the one - dimensional module s X and y Since k i s finite , Xy i s o f infinite o rder I f X i s the cha r a c t e r de fine d by the ac tion of Gal( K/ K ) on V ( IJ ) , the i somorphi s m s p X � y - A2 V - V P P ( IJ ) -1 s how that X X y = X · Hence the r e striction of X x and X X y to X X the ine r tia subgr oup of Gal(K/ K) a r e of infinite orde r T hi s show s fi r st that � i s either � o r a C artan subalgebra of � ; s inc e X X the s e c ond ca s e would imply ( b ' ) , i t i s impo s s ibl e , henc e � = �X Similarly, one s e e s fi r st that i i s c ontained in r ' the n that its X action on X i s non tri vial ; sinc e it i s an ideal in � = � ' the s e X p r operti e s imply i = r ' X -p -p Remark T he above r e sult is g ive n in [ ] p , Th , but mi s ­ stated: the algebra ! has be en wrongly defined a s fo rme d of X tho s e u such that u( X) = (in stead of u( V ) C X) p C OROLLAR Y - If E s c omplex multiplication, � is a split Cartan s ubalg ebra of End( Vp l If D is a sup plementary sub s p ace I V-45 E L LI P T I C C UR YE S t o X s table unde r Gal( K / K) , the n X and D a r e the characte r i s ti c s ub spa c e s o f � and the inertia a lgebra � is the s ubalgebra of End( V p ) fo rm e d b y tho se u e End( V ) s uch that u( D) p The p r oof i s analo gou s to the one of C o r s imple r ) = 0, u(X) C X ( and in fac t B IB LIOGRAPH Y [1] [2 ] [3 ] E ARTIN - Coll e c te d Pap e r s (e dited b y S Lang and J T ate ) , Addi s on - W e sley, 19 E ARTIN and J TATE - Cla s s field the o ry Ha rvard, 1961 M ARTIN e t A GROTHENDIECK - Cohomo10gie e ta1e d e s s ch e m a s S e m G e om a1 g , sur Yvette [4 ] [5] [6 ] (7] [8 ] [9] [1 ] W H E S , 196 / 4, Bure s BURNSIDE - The Theo ry of Group s (Sec ond Edit ) Camb r idg e Uill v P r e s s 1911 J CASSELS - Diophantine equations with special r efe re nc e to elliptic curve s J London Math Soc 41 19 6 p 193 - J CASSELS and A FROHLICH - Algeb raic Numbe r Theory Ac ademic Pr e s s 196 N CEBOTARE V - Die B e s timmung de r Dichtigkeit eine r Menge von P rimzah1en we1che zu e ine r geg ebenen Substitutionskla s s e g ehl:>ren Math Annalen 19 p 51 - 228 C CHEVALLE Y - Deux th e or�m e s d a r i thm e tiqu e J Math Soc Japan 19 51, p - 4 M DEURING - Die Typen d e r Multiplikato renring e ellipti sche r Funktione nk�rp e r Abh Math Sem Hamburg 14 1941, p 197 -27 J IGUSA - Fibre sy stems of Jac obian va rietie s Ame r • • ' J Ill o f Ma th s , , , p - 9 ; 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II ibid 60 , p 63-90, Springer-Verlag 977 [82] L SZPIRO-Seminaire sur les pinceaux arithmetiques: la conjec ture de Mordell, Asterisque 27, S M.F., 985 [83] M WALDSCHMIDT Transcendance et exponentielles en plusieurs vari a b le s Invent Math 63 (198 1), p 97- 27 - , [84] A WILES-On ordinary } -adic representations associated to modular forms, preprint, Princeton, 987 [85] J.-P WINTENBERGER-Groupes algebriques associes a certaines represenla­ tions p-adiques, Amer J Math 108 ( 1986), p 1425- 1466 [86] Y G ZARHIN-Abelian varieties, l-adic representations and SLl, MaLh US S R lzv ( 80), p 275-288 [87] Y G ZARHIN-Abelian varieties, I-adic representations and Lie algebras Rank independence on I, Invent Math 55 ( 979), p 165- 76 [88] Y G ZARHIN-Weights of simple Lie algebras in the c oho m ol og y of alge­ brai c varieties, Math USSR Izv 24 ( 985) , p 245-28 INDEX Admis s ib l e ( cha c t e r ) : A lmo s t loca l ly a lgebra i c Ani s o t ro pic ( to rus ) : I I I A : 111 I I A Arithmet i c ( subgroup ) : I I A As s oc i a t ed ( a l gebra i c morph i s m • a lgebra i c repres enta tion ) : • with a loca l ly • 111 , 111 Aut ( V ) : No ta tions C, C : m " 11 Cebo ta rev ' s theo rem 2 Character g rou p ( o f a torus ) ll1 A c ( ep ) : C 11 • K : 111 = c : l I I A Exer K/ E Compa t ib le ( repres ent a t ions ) : Comp l ex mu l t i p li ca t ion : 2.3, 11 2.8, IV Conducto r ( o f a loca l ly a lgebra i c repres enta t ion ) : C oo , c l1l D : : 11 ll Decompo s i t ion group : De f ined over k ( repres enta tion Dens ity ( o f a s et o f p la c es ) : E et : 11 : II E l l i pt i c curve E IP : IV IV • • • ) : 2 11 111 I nd ex - Em : I I • : IV A q E q uid is tribu t ion : E '" I A E : IV v Ex c e p t i o na l s e t ( o f a s tr i c t ly compa t ib l e s y s t em ) � " J � ' �t : I11 A : 11 F ' f : I!.2.3 v v Fro b enius e l ement : • Fro benius endomo rph i s m : IV A rE Gt Gt IV IV �t GLy 11 2.8, IV IV , IV App • : No ta t i ons , Om : II 1 Good reduct ion ( o f an e l liptic curve ) : Gro s s encha rakter o f type ( A ) : o He i ght : 1V A Hod ge- Ta t e d ecompo s i t ion Hod ge- Tate modu le : 1, m : Id e le : : 111 111 11 II Id e l e c las s es 1t : IV 11 11 1V A pp Inert ia group : 1.2 Inte gra l ( repres enta t ion ) : 2.3 Is o geny , i s o genous curves 1V j : IV • K, K s : No ta t io ns t-ad ic repres enta t ion ( o f a fi eld ) 1.1 1.2.3 Ind ex - 1.2.3 A - ad ic repres enta t io n ( o f a fie ld ) La t t i c e : 1 - L- func t ion : 2.5 Loca l ly a l gebra ic ( repres entat ion ) : 111 , 111 , 111 4, 111 Modu lar invariant ( o f a n e l l i pt ic curve ) a Modu lus ( o f • Ra tiona l ( repres enta t ion ) • • • • ) : 1V : , 1V : 11 S a farevi c ( theo rem o f • • • ) : 1V : 11 Stric t ly compa t i b l e ( sys t em o f repres entations ) Supp( m ) : 1 Tate ' s e l l i pt ic curv es : 1V A.1 1 Ta t e ' s theorem : 1 , 1 A g : 1 ql T r, ( \.l) : T : 11.2 m Torus : 1 Transve c t ion : T Um 111 : 11 • Reduct ion ( o f an e l l i p t i c curve ) : Sm IV loca l ly a lgebra ic repres enta t i o n ) : Mu l t i p l i ca t ive type ( grou p o f ) S Neron- Ogg- a fa revi c ( c r i t erion o f Re � ( H ) : = 1V 11 � / Q ( Gm/ K ) : , U : 1 Uni formly d i s t ribu ted ( s equenc e ) : I A Unrami f i ed ( repres enta tion ) : V r, ( 1-1) : I Weiers tra s s form ( o f an e l l i pt i c curve ) : 1V v ,m • • 1.2.3, 1nd ex - 'XE : 1II.A 'XL : X( T ) , X ( T ) m 11 Y, yO , Y - , y+ : I: K 1 • 11 , II.A [...]... In Locally Algebraic Abelian Representations §1 The local case III-I 1.1 Definitions III-I 1.2 Alternative definition of "locally algebraic" via Hodge-Tate modules III-5 §2 The global case III-7 2.1 Definitions III-7 2.2 Modulus of a locally algebraic abelian representation III-9 2.3 Back to S'" III-I2 A mild generalization The functionfield case III-16 25 §3 The case of a composite of quadratic fields... p ' of K c OITlpatible with p? [ n o : easy co u nter-exam ples] 4 Let p, p ' be rational i, i' -adic r ep re s entation s of K which are c OITlpatible and s eITli - s iITlpl e ( i ) If p is abelian ( i e , if IITl( p ) i s abelian) , i s it true th_a t p ' is abelian? ( W e shall s ee in c hapte r III that thi s is true at lea st if p is "loc ally algebraic " ) [yes: th is follows fro m [63].]... tanc e , K K GL y (K) = Aut (V ) • Abelian l- Adic Representations and Elliptic Curves CHAPTER I £ -ADIC REPRESENTATIONS l 1 1 THE NOTION OF AN 1 -ADIC RE PRESENTATION Definition Let K be a field , and let K s be a s epa rable alg ebraic clo ­ Gal(K / K) be the Galo i s g roup of the exten s ion s K / K The g ro up G, with the Krull top ology, is c ompa ct and totally s dis c onne c ted Let 1 b e a... is abelian (see ch II, 2 5 ) 0 P 1S L- functions attached to rational representations Let K be a number field and let P = (P ) be a strictly com­ L patible system of rational l- adic representations, with exceptional set S If v ¢ S, denote by Pv, ( T ) the rational polynomial P det (l - Fv, P T ) , for any L � P ; by assumption, this polynomial v 1 does not depend on the choice of L Let s be a complex... w v l /k The kernel of G -> Gal(.t /k ) is the inertia group I of w v w w v w The quotient group D /I is a finite cyclic group generated by w w Nv the Frobenius elelTIent F ; we have F(x') = X, for all X, � 1 w w The valuation w (resp v) is called unralTIified if I = {l} AllTIost w all places of K ar e unralTIified D If L is an arbitrary algebraic extension of Q , � L to be the pr ojective lilTIit... and P v, Pl (b) P ( T ) = Pv l (T ) g v , S u Sl u 51' v, P , P ' 1 (a) Let S 1 1-12 ABELIAN l -ADIC RE PRESENTATIONS Whe n a s y s te ITl ( P ) IS s t r i c tly cOITlpatible , the r e i s a s mall­ I e st finite s et S having p rope rti e s ( a ) and (b) above We call it the exc eptional s et of the systeITl ExaITlple s The s y s teITl s of i -adic repr e s entations g iven in exaITlple s 1, 2,... unraITlified A sOITlewhat s iITlila r que stion i s : I s any c OITlpatibl e sy steITl strictly c OITlpatible? 2 Can any rational i -adic r ep re s entation be obtained (by ten sor p r oduc t s , dir ect SUITlS, etc ) froITl one s c OITling froITl i -adic c ohoITlology ? 3 Given a rationa l i -adic r ep r e s entati on p o f K, and a p riITle i' , doe s the r e exist a rational i' -adic r ep r e s e... of places From this, one gets an ( cf.2.1 ) Im ( pz ) then irreducibility theorem The determination of the Lie algebra of follows, using the properties of abelian representations given in chap II, Ill; one has to know that Pz ' if abelian, is locally algebraic, but this is a consequence of the result of Tate given in chap III The variation of 1 m ( P L) with L is dealt with in § 3 Similar results... w The restriction and let w e: � � i by abuse K of language, we also say that v is the restriction of w to K, and of w to K is an integral lTIultiple of an elelTIent v we wr ite wi v ("w divides v") pletion of L (resp Let L (resp K ) be the COITIv w K) with respect to w (resp v) We have w :: Gal(Lw /Kv ) The gr oup D w is lTIapped hOlTIolTIorphically onto the Galois group Gal (l /k ) of the corresponding... characteristic polynolTIial (which is inde­ pendent of 1) is a consequence of Weil's results on endolTIorphisms of abelian varieties (d [4 0 ] and [1 2 ] , chap VII) The rationality of ABELIAN 1 -ADIC REPRESENTATIONS 1-10 the cohomology representations is a well-known open question K, and assume that p' be a prime DEFINITION - Let l' p, p' � l' -adic representation of ar e rational Then p, p' are said