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# ACCEL HONORS CALC MATH 135

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JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00 Num er 0 Xxxx xx Page 0007000 5 0394roa4vxxooooro ALGEBRAIC FAMILIES OF NONZERO ELEMENTS OF SHAFAREVICHTATE GROUPS JEANLOUIS COLLIOTTHELENE AND BJORN POONEN 1 INTRODUCTION Around 1940 Lind Lin and independently but shortly later Reichardt Re discovered that some genus 1 curves over Q such as 2y2 1 717954 violate the Hasse principle ie there exist zy E R satisfying the equation and for each prime p there exist zy E Qp satisfying the equation but there do not exist zy E Q satisfying the equation In fact even the projective nonsingular model has no rational points We address the question of constructing algebraic families of such examples Can one nd an equation in z and y whose coef cients are rational functions of a parameter t such that specializing t to any rational number results in a genus 1 curve violating the Hasse principle The answer is yes even if we impose a non triviality condition this is the content of our Theorem 12 Any genus 1 curve X is a torsor for its Jacobian E Here E is an elliptic curve a genus 1 curve with a rational point If X violates the Hasse principle then X moreover represents a nonzero element of the Shafarevich Tate group We may relax our conditions by asking for families of torsors of abelian varieties In this case we can nd a family with stronger properties one in which specializing the parameter to any number oz 6 Q of odd degree over Q results in a torsor of an abelian variety over Qa violating the Hasse principle A precise version of this result follows Throughout this paper P1 unadorned denotes the projective line over Q Theorem 11 There epists an open subscheme U of PlL containing all closed points of odd degree and there eccist smooth projective geometrically integral varieties A and X over Q equipped with dominant morphisms WA and my to PlL such that 1 AU 7TU is an abelian scheme of relative dimension 2 over U 2 XU 7T1U is an AU torsor over U 3 int241 P1hv for every local eld kw D Q archimedean or not 4 X has no zero cycle of odd degree over Q Received by the editors June 7 19991 1991 Mathematics Subject Classi cation Primary 11G10 Secondary 11G30 11G35 14H40 14J27I Key words andphrases Shafarevich Tate group BrauerManin obstruction Hasse principle cubic surface Cassels Tate pairing Lefschetz pencil Most of the research for this paper was done while the authors were both enjoying the hospitality of the Isaac Newton Institute Cambridge England The rst author is a researcher at CINIRISI The second author is partially supported by NSF grant DMS9801104 a Sloan Fellowship and a Packard Fellowshipi 1997 American Mathematical Society 2 JEANLOUIS COLLIOTTHELENE AND BJORN POONEN 5 A is a non constant family ie there eccist ui E such that the bers Au and Au are not isomorphic ouer 6 The generic ber ofA a P1 is absolutely simple ie simple ouer Remarks 1 Conditions 1 through 4 imply that for each closed point 5 E U of odd degree in particular for all s E P1Q the ber XS represents a nonzero element of the Shafarevich Tate group mAs Let E be an elliptic curve over Q and suppose that C is a genus 1 curve over Q representing an element of whose order is even Let B be a smooth projective geometrically integral variety over Q equipped with a morphism 7T5 B a P1 making ng1U a non constant abelian scheme over U U being as above Then A E gtltQ B and X C gtltQ B equipped with the morphisms to P1 obtained by composing the second projection with B a P1 satisfy 1 through Condition 6 is designed to rule out such trivial77 examples 3 The varieties A and X promised by the theorem will be smooth compacti cations over Q of PngPl and Pic Pl for a relative curve C a P1 of genus 2 For the theory of the relative Picard functor we refer the reader to BLR The key ingredient which will let us prove the non existence of rational points of X over elds of odd degree condition is the formula from PS which gives the value of Cassels Tate pairing when the torsor Picgil of Pic0 for a genus g curve over a global eld is paired with itself Using that same formula we will show that the non existence of zero cycles of degree one on X can be explained by a Brauer Manin obstruction As mentioned at the outset we also have an example of relative dimension 1 Here we control bers only above rational points not above all points of odd degree Theorem 12 There epists an open subscheme U of P1 containing P1Q and smooth projective geometrically integral varieties S and X ouer Q equipped with dominant morphisms 7T5 and my to P1 such that 1 XU a U is a proper smooth family whose bers are geometrically integral curves of genus Z 8U PicOm U so 8 is an elliptic surface ouer P1 smooth aboue U 3 7T2XQp P1Qp far allp S 00 4 XQ is empty 5 The j inuariant of EU a U is a non constant function on U 2 The proof of this begins by taking a Lefschetz pencil in a cubic surface violating the Hasse principle A suitable base change of this will give us X a P1 When the failure of the Hasse principle for the cubic surface is due to a Brauer Manin obstructionl we can construct a second family yU a U of genus 1 curves such that PicgUU 8U the analogue of 3 holds and the Cassels Tate pairing satis es Xt 34gt 13 for all t E UQ While proving the last statement we are led to prove an auxiliary result Lemma 34 which may be of independent interest if there is a Brauer Manin obstruction to the Hasse principle for a smooth cubic surface V then only one element oz 6 BrV is needed to create the obstruction 1It has been conjectured that the BrauerManin obstruction is the only obstruction to the Hasse principle for cubic surfaces In any case there are examples of cubic surfaces violating the Hasse principle because of such an obstruction See 25 and 31 belowi FAMILIES OF SHAFAREVICHTATE GROUP ELEMENTS 3 Throughout this paper etale cohomology of a scheme X with values in a sheaf f is denoted HXf The notation for the Galois cohomology of the spectrum of a eld h will simply be Hkf The cohomological Brauer group BrX of a scheme X is the group H tXGm The Shafarevich Tate group MA of an abelian variety A over a number eld h is de ned as the kernel of the natural map H1kA a H H1hvA where the product is over all places v of k and kw denotes the completion If h is a eld then h denotes an algebraic closure of k and if X is a h scheme then X X gtltk h Given an integral ie reduced and irreducible k variety we write kX for the function eld of X If Xk is geometrically integral we denote by hX the function eld of X 2 A FAMILY OF ABELIAN SURFACES In this section we prove Theorem 11 21 Minimal models of hyperelliptic curves We need a simple lemma describing the minimal model of certain hyperelliptic curves Recall DM that a stable curve over a base S is a proper at morphism C a S whose geometric bers are reduced connected 1 dimensional schemes CS such that CS has only ordinary double points and any nonsingular rational component of CS meets the other components in more than 2 points Lemma 21 Let R be a Dedekind domain with 2 6 Pi and let K be its fraction eld Let fpz E Rzz be homogeneous of even degree 2n 2 4 and assume that its discriminant generates a squarefree ideal ofR Let CK be the smooth projective model of the a ne curve de ned by y2 fp 1 over K Then the minimal proper regular model CR ofC is stable and can be obtained by glueing the a ne curves yz fx 1 and Y2 f1z over R along the open subsets where z 31 0 and z 31 0 respectively using the identi cations z z 1 and Y z y In particular its geometric bers are integral Proof The model C described is nite over P which manifests itself here as two copies of A with variables x z glued along x 31 0 and z 31 0 using the identi cation 2 x l so C is proper over R The regularity and stability follow from Corollaire 6 and Remarque 18 in Liu The integrality of the bers follows from the explicit construction and implies that C is the minimal model 22 The relative curve Let C be the smooth projective model of the curve 21 y2 7 6 x5 t5p 8t6 7 over Qt Let ftz E Qt denote the right hand side of 21 PARI shows that the discriminant At of ftp with respect to z is an irreducible polynomial of degree 30 in Qt In particular At i 0 so 0 is of genus 2 Let so 6 Ab C P1 be the closed point corresponding to At and let U P1 50 Since At is squarefree the minimal model C0 over Spec Qt is described by Lemma 21 Let T t l Then G is birational over Qt to the af ne curve 22 Y2 7 X6TX5X 87T6 via the change of variables X xt and Y ytg The minimal model C1 over Spec QlT is given by Lemma 21 again Glueing C0 and C1 gives the minimal model C a P1 Then C a P1 is stable and has good reduction over U since this can be checked locally on the base Moreover C is smooth and proper over Q 4 JEANLOUIS COLLIOT THELFNE AND BJORN POONEN 23 Local points on the bers In what follows7 we will specialize C at points in P1k where k is a eld containing Q Although t above denoted an indeterminate7 we will abuse notation below by letting t denote also the specialized value in kUoo7 and the corresponding k valued point Speck 7 P1 If t 31 00 resp t 31 07 then the ber Ct is birational to the curve over k given by the equation 21 resp 22 Lemma 22 1 ft 6 UR then C has no real divisor of degree 1 2 Ifk is a nite ertension onp for some nite prime p and ift E Uk then Ctk 31 0 Proof For tz E R7 the weighted AM GM inequality gives lzG Et6 2 tSzi 2 7t5z and z6 l 2 z5 2 7z5 6 6 6 6 Combining these shows that z6z5t5z8t67 gt0 Thus C has a dense open subscheme with no real points But C gtltQ R is smooth over R7 so CR 0 Hence C and Ct for t E UR have no real zero cycles of odd degree From now on7 we assume t E Uk where k Qp lt 007 and we let 1117 be the p adic valuation on k normalized so that 0171 1 By convention7 ift 007 then 11pt 700 lt 0 First suppose p 2 1f 02t lt 07 then Hensel7s Lemma shows that 22 has a k point with Y 07 and with X near 0 1f 02t 2 07 then 21 has a k point with z 0 From now on7 we assume p is odd Let Fq be the residue eld of k Suppose 1t 2 0 Then let fz E qu denote the reduction of the right hand side of 21 If fz is a square in FPM then equating coef cients of 6 5 4 3 shows that the reduction would have to be 7z3 z22 7 zS 11627 but equating the coef cients of z27z17z0 gives the inconsistent system 0 7564 55 7164 7 836 1256 of equations in F177 where t denotes the reduction of t Hence fz is not the square of any polynomial in FAX 1f instead vpt lt 07 then let fz 7X6 X 8 E qu be the reduction of the right hand side of 227 which again is not a square in FAX In either case no matter what the value of ipt7 write fz jz2hz in Fq with hz squarefree7 so by the previous paragraph deg h gt 02 Any Fq point on 22 hz with z nite and 31 0 gives rise to a nonsingular Fq point on the af ne curve yz fz7 which can be lifted by Hensel7s Lemma to prove Ctk 31 0 Also7 if there is a point at 00 on the projective nonsingular model of 22 hz over Fq then there will be points on Ct at 00 The number of points on 22 hz with 0 is at most 2degj 6 7 degh 6 7 2g 2 4 7 297 where g is the geometric genus of 22 Therefore by the Weil bound7 we automatically nd the desired point if ltq172mgt7lt472ggt gt0 for g 012 These hold for q 2 177 so it remains to consider the cases q 3579711713 1f q E 1 mod 47 then Ct has k points at 00 For each remaining q7 we check by hand that 2At this point we know that y2 has a multiplicity one component which is geometrically integral7 so it follows automatically as in Proposition 311 in v Y or Lemma 15 in PS that C has a krational divisor of degree 1 In fact this would suf ce for our application7 except in Section 2177 where it will be convenient to have the stronger result C106 0 FAMILIES OF SHAFAREVICHTATE GROUP ELEMENTS 5 for each possible residue class oft and for the case T E 0 in 22 there is still a nonsingular af ne point on y2 24 Components of the Picard scheme Since C is a proper smooth surface over Q it is projective over Q and hence also projective over P1 Moreover the geometric bers of C a P1 are integral Hence we may apply 931 from BLR to deduce that 1 The relative Picard functor PicCp1 is represented by a separated and locally smooth Pl scheme PicCp1 Hnez PicgPi where PngPl denotes the open and closed subscheme of PicCp1 consisting of all line bundles of degree n 3 For each n E Z PngPl is a torsor for PicgPi and is quasi projective 3 Let A PicgPi and let X Pic Pl For any Pl scheme 7T Z a P1 let ZU denote 7T 1U Then AU is an abelian U scheme by 944 from BLR Resolution of singularities gives us Pl schemes WA A a P1 and my X a P1 such that 1 A and X are smooth and projective over Q 2 AU 2 AU and XU E XU as U schemes We now verify that A and X satisfy the conditions of Theorem 11 Conditions 1 and 2 are clear from the construction The following lemma applied to the smooth bers Ct shows that WxXkv D Uhv and condition 3 then follows from compactness of WXXkv Lemma 23 Let h be a local eld ie a nite extension of R Qp or Fpt and leth denote a separable closure of k Let X be a smooth projective geometrically integral curve over h of genus g and let 7 X gtltk h Then there epists an element of degree g 7 1 in Pic7 which is stable under Galhk Proof This is a consequence of Tate local duality See Theorem 7 on p 133 of Lic for the case where h is a nite extension of Qp or Section 4 of PS for the general case The real case is related to W D Geyer7s modern version of work of Weichold and Witt see Sc Thm 20151 p 221 We now prove condition Suppose condition 4 fails so that X has a closed point of odd degree lts image under 7139 is a a closed point of P1 of odd degree Since U contains all points of P1 of odd degree this closed point corresponds to some t E Uk with k Q odd such that the ber X has a closed point of odd degree over k Since Ct admits a 2 to 1 map to Pi it possesses a point in a quadratic extension of k hence Xt Picak possesses also such a point Thus the principal homogeneous space X Picak of At Picak is annihilated by coprime integers hence X is trivial On the other hand we will show that X cannot be trivial because of the following which is Corollary 12 in PS the assertion c E MA is Lemma 23 above Lemma 24 Let h be a global eld ie a nite eptension on or Fpt Let C be a smooth projective geometrically integral curve over k Let A be the canonically polarized jacobian of C and let c be the class of Pica in H1kA Then c E MA and the Cassels Tate pairing satis es cc N2 E QZ where N is the number of places v of h for which 0 has no Irv rational divisor of degree g 7 1 6 JEANLOUIS COLLIOT THELENE AND BJORN POONEN Let us apply this to the ber 0 Ct Then 9 27 c is the class of 26 and N equals the number of real places of k by Lemma 22 Since k Q is odd7 N is odd Thus ltc7cgt 31 07 so c 31 0 ie7 X is non trivial in mAt This contradiction proves condition We next prove condition The curve CU is projective and smooth over U There is an associated Q morphism f from U to the coarse moduli space M2 of smooth curves of genus 27 and the jacobian functor yields a Q morphism j M2 a A2 to the coarse moduli space of principally polarized abelian surfaces over Q By Torelli7s theorem7 the map j is injective over algebraically closed elds MFK p 143 Thus to prove 57 it is enough to show that f is not constant Suppose f is constant Let R be the completion of the local ring of P1 at the closed point where CP l has bad reduction7 and let K be the fraction eld of R Since f is constant7 there exists a nite eld extension LK such that C gtltp1 L is L isomorphic to a constant curve The latter has good reduction over the ring of integers of L7 but for a stable curve the property of having good reduction does not depend on the eld extension7 so the stable curve C gtltp1 R must be smooth This is a contradiction Finally we must show that the generic ber of A a P1 is absolutely simple Let 539 denote the good mod 3 reduction of the ber C4 and let J be the jacobian of Calculating C F3 1 and C F9 157 we nd that the characteristic polynomial of Frobenius for J is x4 7 3x3 7x2 7 9x 9 A root 04 generates a quartic eld L having a unique non trivial proper sub eld F of degree 2 The LF conjugate of 04 is 304quot 7 and 304404 is not a root of unity7 so a Z F and a is of degree 4 for all n 2 1 Hence J is absolutely simple To deduce from this that the generic ber of A a P1 is absolutely simple7 it suf ces to apply the following well known result twice Lemma 25 Let A be an abeliari scheme over a discrete valuation ririg R If the special ber is absolutely simple theri s0 is the gerieric ber Proof If the generic ber is not absolutely simple7 then there exists a nite extension K of the fraction eld K of R and an isogeny A XRK a B gtltKz C where B and C are non trivial abelian varieties over K Let R be a discrete valuation ring in L containing R7 and let k h be the corresponding extension of residue elds By ST7 the Neron models 8 and C of B and C are abelian schemes over R The universal property of the Neron model extends the isogeny to the Neron models A gtltR R a B gtltRz C Taking special bers shows that A gtltR h is not simple7 so A gtltR h is not absolutely simple III This completes the proof of Theorem 11 25 Review of the Brauer Manin obstruction Manin Ma1 in 1970 introduced what is now called the Brauer Manin obstruction Now suppose X is a smooth proper irreducible variety over a number eld h Let Ak denote the adele ring of k If i is a place of k then we may de ne a local evaluation pairing77 evv BrX gtlt Xkv a QZ7 by letting evvax E QZ be the invariant of i204 E Brkv where im Spec kw a X corresponds to z E Xkv Given 04 the properness of X implies that the map z gt gt evvax is zero for almost all i so by summing over all i we obtain a global pairing ev BrX gtlt XAk a QZ7 FAMILIES OF SHAFAREVICHTATE GROUP ELEMENTS 7 continuous in the second argument such that evaz 0 if 04 comes from Brk or if z E One says that there is a Brauer Manin obstruction for X if for every x E XAk there exists 04 E BrX with evaz 74 0 It is conjectured that for any smooth proper geometrically rational surface X over a number eld k the Brauer Manin obstruction is the only obstruction to the Hasse principle ie that if Xk 0 it is because x E XAk evaz 0 for all 04 E BrX 0 This appeared as a question in CTS and as a conjecture in CTKS once many examples were available Any rational surface over It is k birational either to a conic bundle over a conic or to a Del Pezzo surface of degree d 1 S d S 9 Del Pezzo surfaces of degree at least 5 satisfy the Hasse principle The Del Pezzo surfaces of degree 4 are the smooth intersections of two quadrics in P4 and those of degree 3 are the smooth cubic surfaces in P3 There exists theoretical evidence for the conjecture in the case of conic bundles and Del Pezzo surfaces of degree 4 and numerical evidence for diagonal cubic surfaces In another direction Skorobogatov Sk recently gave an example of a bielliptic surface over Q for which the Brauer Manin obstruction is not the only one He and Harari Ha2 HS have discovered that the new obstruction in this and some other examples can be explained by non abelian unrami ed covers 26 The evaluation pairing and the Cassels Tate pairing It is possible to de ne the Cassels Tate pairing in terms of the pairing ev de ned above Suppose X and Y are torsors of an abelian variety A over k representing elements of We may identify A with Picqu and then Y corresponds to an element of H1k Pic0 X which may be mapped to an element 6 E H1k PicX The Leray spectral sequence Mi1 lll118a for X a Speck gives rise to an exact sequence kerBrX a BrX a H1k P107 a H3k Gm 0 the last equality holding because It is a number eld We pick z E XAk and pick 04 E BrX mapping to B in the sequence above Then the Cassels Tate pairing ltXYgt equals evoz which does not depend on the choices made See Theorem 4124 in Ma2 for the case of genus 1 curves or the homogeneous space de nition77 of the pairing in 138 27 The Brauer Manin obstruction for X It is conjectured in CT Conj 2 that the Brauer Manin obstruction to the existence of zero cycles of degree 1 the de nition is an obvious extension of the one given for rational points on smooth projective irreducible varieties over number elds is the only one In this section we show that the non existence of zero cycles of degree 1 on our variety X can indeed be explained by this obstruction moreover it can be explained by the obstruction attached to a single element of BrX Lemma 26 Let k be a riumber eld arid let G be a smooth projective geometrically integral curve over k Theri H3kC Gm 0 Proof If F is any eld of characteristic zero 00 denotes the group of roots of unity in F and Q Fliioo then Q is uniquely divisible so HiF Q 0 for i gt 0 and we deduce that H3FGm H3Floo H3F QZ1 For real places i of the number eld k let k kw HE C E denote the corresponding real closure of k Theorem 1 of Ja states that H3 CQZr vreal H3kCQZr for r at 2 Taking r 1 we nd that H3kO Gm e gamed H3k0 Gm 8 JEANLOUIS COLLlOTTHELENE AND BJORN POONEN For 1 real let G GalhCh0 Galhk Z2 The spectral sequence Egg HPG HqZO Gm HPqkO Gm together with the fact Theoreme 11 in Gr2 that HqhC Gm 0 for q gt 0 implies that H3k07Gm H3G 730W By periodicity the latter is H1Gh0 which is zero by Hilbert7s Theorem 90 III Corollary 27 Let h be a number eld and let G be a smooth projective geometrically integral curve over h Then li mHg JV Gm 0 where the direct limit is over the dense open subschemes V of C with respect to the restriction maps Proof The direct limit equals H3kC Gm by a result of Grothendieck See Corollaire 59 in SGA 4 VII Expose Vll Site et Topos etale d7un schema par A Grothendieck SGA4 or Lemma 116 and Remark 117a in Mi1 III Remark With more work one can show that if U C C is an af ne open subset then H2tUGm is nite but not necessarily zero Moreover there exists a dense open sub set U C C such that H2tU Gm 0 for all open subsets U Q U On the other hand H2tC Gm is always in nite We now return to our situation with C and U as in Section 22 and X as in Section 24 Let V Q U be a dense open subset Recall that XV Pic VV Let i CV a XV be the natural inclusion of V schemes We obtain a commutative diagram of exact sequences of etale sheaves over V 0 PICXVV a PICXVV q q 0 a PichV a PiCCVV a Z a 0 and an argument similar to the one of Proposition 69 on p 118 of shows that the rst vertical map is an isomorphism Taking cohomology we obtain PicQYVV HtV7PiCXvV Z HMV PichV HVPicCVV The image of 1 E Z in H MV PichV is the class V of the torsor XV of PichV The analogous result for the generic bers with V replaced by its generic point 77 Spec Qt follows from a cochain calculation in Galois cohomology The claim for V follows using functoriality and applying the injectivity of HtVB a H177B7 for abelian V schemes B to the case B PichV Let B be the image of i 1 y under HMV PicQYVV a HtVPicXVV Then 6 maps to 0 in HMV PicCVV The Leray spectral sequences for WXV XV a V and 7TCV CV a V FAMILIES OF SHAFAREVICHTATE GROUP ELEMENTS 9 yield a commutative diagram of exact sequences 23 BrV ker 13er a H tVR27rXV1Gm Hamrim H tVGm BrV a ker BrCV a HtV7 R2 7TCV1Gm a HMV7 PicCVV By Corollary 277 after shrinking V and restricting if necessary7 we may assume that B E H MV7 PicXVV maps to zero in Hg tV7 Gm7 so that B lifts leftwards to an element 04 6 Br XV Since 6 maps to 0 in H MV7 PicCVV7 we may adjust Oz by an element in the image of BrV if necessary7 to assume that 04 restricts to 0 in Br CV Suppose that t E VQp for some nite or in nite prime p7 and consider the ber Xi which is a torsor of an abelian variety over Qp The restriction of y to an element 7 of H1QpPicYtQp represents the torsor PicgEtQp7 which is trivial by Lemma 23 Then the corresponding restriction t 6 H1QpPicXtQp is 07 so the analogue of the top row of 23 for X a Spec Qp shows that the restriction at comes from Br Qp ln particular7 evpoztx is a constant function of z E 24Qp For any nite p7 Lemma 22 gives us a point z E CQp7 but 04 restricts to 0 6 Br Ct so the constant function must be identically zero Thus for all nite p7 evpoz 0 for all z E XVQP By Ha7 Theoreme 2117 this implies that 04 extends uniquely to an element 07 6 Br X By continuity7 evo7 a 0 for all z E 2Qp7 for any nite p The evaluation of 07 on an adelic point z xv E XVAQ depends only on zoo The union of the points x00 6 XR lying above rational points t E VQ is dense in 2R7 and by Section 267 evozz for any such z equals the value of the Cassels Tate pairing lt24 24gt which is 12 Hence by continuity evozz 12 for all z E XLAQ7 and we deduce a posteriori that evoooquotzz 12 for all z E XR The same argument which showed that evpol 0 for all z E 2Qp shows that if kw is any nonarchimedean local eld containing Q and z E 2kw7 then evwo7 a 0 From the above two paragraphs7 we see that for any zero cycle 2v on X gtltQ Qw7 the element evvolzv equals 0 E QZ ifi is a nite place7 and equals dv2 E QZ if 1 is the real place7 where dv is the degree of the zero cycle 2v over R If the 2v for all 1 arise from a zero cycle 2 on X of odd degree7 then summing over 1 shows that evoz7 z 127 which is impossible By de nition7 this means that 07 gives a Brauer Manin obstruction to the existence of such a zero cycle z 3 A FAMILY OF GENUS 1 CURVES We now prove Theorem 12 31 Cubic surfaces Violating the Hasse principle Swinnerton Dyer SD1 disproved a conjecture of Mordell Mo by constructing a smooth cubic surface V in P3 over Q violating the Hasse principle ie7 such that V has points over each completion of Q7 but not over Q Soon afterwards7 Cassels and Guy CG gave the diagonal cubic surface 5x3 9y3 1023 1213 0 violating the Hasse principle Manin Ma1 used the Brauer Manin obstruction to explain Swinnerton Dyer7s counterex ample See also Ma2 Much later CTKS explained the Cassels Guy example from this 10 JEANLOUIS COLLIOT THELENE AND BJORN POONEN point of view7 and gave a very explicit algorithm for computing the Brauer Manin obstruction for diagonal cubic surfaces We x a number eld h and a smooth cubic surface V C Pi for which the Hasse principle fails 32 Lefschetz pencils By 252 in Ka7 the embedding V lt gt Pi is a Lefschetz embed ding77 This implies that the statements in the rest of this paragraph hold for a suf ciently generic choice of a k rational line L in the dual projective space P2 Suf ciently generic77 here means for L corresponding to points outside a certain Zariski closed subset of the Grass mannian Let L denote the aais of L7 that is7 the line in P2 obtained by intersecting two hyperplanes in the family given by L Blowing up V at the scheme theoretic intersection V L results in a smooth projective variety V over h isomorphic to the reduced subvariety of V gtlt L whose geometric points are the pairs o7H where i E V is on the hyperplane H C P2 corresponding to a point of L Each ber of V a L g Pi is a proper geometrically integral curve of arithmetic genus 17 and if it is singular7 there is only one singularity and it is a node The generic ber is smooth 33 Local points in the pencil Let kw denote the completion of h at a non trivial place i Let OH be the ring of integers7 let my be its maximal ideal7 and let FE Ovmv Variants of the following lemma have appeared in various places in the literature eg CTSSD7 CT7 21 The key point is that all the geometric bers of V a P1 are integral Lemma 31 There ewists a nite set S ofplaces ofh such that for any i Z S and any nite extension K of kw the map V K a P1K is surjectioe Proof The morphism f V a P is proper and at Combining Theorem 1111 and Theorem 1224viii of EGA IV7 we see that there exists a a nite set S of places of k with associated ring of S integers 97 such that f extends to a proper at morphism f V a P1 7 all bers of which are geometrically integral We may assume that S contains all the real places and none of the complex places The desired surjectivity is automatic at complex places7 so it remains to prove the surjectivity for nite i Z S Let i be a place not in S7 let Khv be a nite extension with associated valuation w7 let Ow be the ring of integers of K and let Fw be the residue eld Let q Fw If t E P1Ow P1K then the special ber of the ber V is a geometrically integral7 proper curve of arithmetic genus 1 The number of smooth Fw points on this special ber is at least 7 12 gt 0 Hasse if the special ber is smooth7 and is equal to q 7 17 q7 or q 1 if not in any case there is at least one Hensel7s Lemma then shows that the generic ber of V has a K point7 as desired Let U be the largest open subscheme of Pi over which V is smooth Lemma 32 For each completion kw ofk the image ofV kv a P1kv contains anonempty open subset Wu in the o adic topology Proof Recall that Vkv 31 Q by choice of V Since V and V are birational smooth projective varieties7 V kv 31 Q too7 and in fact V kv is Zariski dense in V In particular we can nd P E V kv mapping into U The image of V kv a P106 will then contain a neighborhood of the image of P III FAMILIES OF SHAFAREVICHTATE GROUP ELEMENTS 11 We may choose S as in Lemma 31 to contain no complex places For 1 E S choose Wu as in Lemma 32 We may assume that there exists 1 E S for which Wu C U hv Then the bers W for t 6 WE are smooth 34 Base change of the pencil Lemma 33 Let S be a nlte set of non complem places of k let T be a nlte set of closed points of Pi and for each 1 E S let Wu be a nonempty open subset of P1hv Then there emsts a non constant h morphlsm f Pi a Pi such that f is smooth le unraml ed above the points of T and fP1kv Q Wu for all 1 E S Proof First we show that for each 1 6 S7 we can nd a non constant fv de ned over kw such that fvP1kv Q WE Choose an af ne point A 6 Wu lf 1 is real7 we may take fvz A 2 n lL for n gt 0 lf 1 is nite7 then we may choose 9 E kvx such that 9 maps 0100 to A and then let fvz 9x for n qkq 7 1 where q Fv and k gt 0 Preceding each fv with an arbitrary non constant rational function of appropriate degree7 we may assume that deg fv is the same for all 1 E S Let Bd denote the open af ne subset of a07 7ad7b07 7bd 6 AM for which the homogeneous polynomials ZaiXin i and Z biXle l have no non trivial common factor We have a morphism Bd gtlt P1 a P1 which constructs the rational function of degree d which is the quotient of the two polynomials7 and then evaluates it at the point in P1 Choose by E Bdkv representing fv By compactness of P1kv in the o adic topology7 any point of Bdhv suf ciently close to by represents a rational function still mapping P1hv into WE Weak approximation gives us a point b E Bdh close enough o adically to by for each 1 E S so that the corresponding rational function f over h maps P1hv into Wu for o E S Moreover f will be smooth above the points of T provided that b avoids a certain closed subset of Bd of positive codimension7 so this is easily arranged III Apply Lemma 33 to obtain f for the S and the WW at the end of Section 337 and with T Pi U We let my X a P be the base extension of V a P by f7 so that the following is a cartesian square X a V l l P L P Each factor is smooth over k and above each point of the lower right P at least one of the two factors is smooth by choice of T7 so X is smooth over h The generic ber of V a Pi is geometrically integral7 so X is geometrically integral Since V is projective over k so is X Let U f 1U By choice of f7 fP1h C U Hence U is an open subscheme of Pi containing 131k and XU a U is a proper smooth family of geometrically integral curves of genus 1 We may construct S as the minimal proper regular model of the jacobian of the generic ber of X a Pi Then 8U E PicQYUU is an abelian U scheme lf 1 is a place of k and t E P lhw7 then X E Vf t has a hv point by choice of f The property of having a rational point is a birational invariant of smooth projective varieties7 and Vk 0 so V k 0 Since X maps to V 7 Xh Q too 12 JEANLOUIS COLLIOTTHELENE AND BJORN POONEN It remains to show that the j invariant of my X 7 Pi is non constant7 or equivalently that the j invariant of 7139 V 7 Pi is non constant Euler Poincare characteristic compu tations BPV7 Prop 1147 p 97 show that the bration V 7 P1 has exactly 12 nodal geometric bers Thus the j invariant of the generic ber has poles on Pl7 so it cannot be constant 35 Generic CasselsTate pairing Let h be a number eld7 let U C Pi be a dense open subscheme of Pi and let A be an abelian U scheme Assume that A 7 U is projective7 so that the dual abelian scheme A 7 U exists Gr1 Now let XU be an A torsor over U such that each ber X for t E Uk represents a nonzero element of mAt ls it then automatic that there exists a A torsor yU over U or at least over some dense open subscheme such that for all t E Uh7 the class of y is in and the Cassels Tate pairing Xi7 34 gives a nonzero value in QZ which does not depend on t The special case where the families A and X are split is the well known conjecture that the Cassels Tate pairing is nondegenerate For our rst example7 the XU in Section 2 built out of the PicL of the relative curve of genus 27 we can identify A with A and take yU XU because we showed Xi7 Xi 12 E QZ for all t E UQ For the second example7 the one from the Lefschetz pencil in a cubic surface V over k we can construct yU if we assume as is conjectured that the failure of the Hasse principle for V is due to a Brauer Manin obstruction7 as we now explain The cokernel of Brk 7 BrV for a cubic surface V is nite7 but need not be cyclic see SD2 for the possibilities Nevertheless one has the following Lemma 34 Let V be a smooth cubic surface in P3 over a number eld k Assume that the Hasse principle for V fails and fails because of a Brauer Manin obstruction for V Then there epists 04 E BrV such that evozz is a nonzero constant independent ofz E VAk Proof Let G be the cokernel of Brk 7 l3rV7 and let 6 HomGQZ These are nite7 and are given the discrete topology Let S be the image of the continuous map b VAk 7 6 induced by ev We claim that 31 z7y726Sgtyz7z S Because b is continuous and V is smooth7 given zyz 6 S7 we may choose P PH 6 VAk with P z and similarly Q and R giving y and 2 so that for each i PW7 Qv and RU are not collinear7 and the plane through them intersects V in a nonsingular genus 1 curve Cy Applying Riemann Roch to CE yields a point TE 6 Ckv linearly equivalent to the divisor Qv Rm 7 PH on Cy As in Lic7 the pairing BrCv gtlt Cvkv 7 Brkm Q QZ extends to a pairing BrCv gtlt DivCv 7 QZ which induces a pairing BrCv gtlt PicOv 7 QZ For B E l3rV7 ev T can be obtained by restricting B to Cu pairing with Ty and summing over i It follows that T Q R 7 15137 so y z 7 z E S It follows from 31 and S 31 Q that S is a coset of a subgroup H of But S cannot be a subgroup7 because 0 E S would contradict the existence of a Brauer Manin obstruction Hence we may pick 9 E G g HomC397 QZ annihilating H but not S For any lift 04 E BrV of g evozz is a nonzero constant for z E VAk III Remarks 1 D 00 4 Cf One can prove a similar result7 namely Luz E S gt FAMILIES OF SHAFAREVICHTATE GROUP ELEMENTS 13 Let Y denote the Grassmannian of lines in P3 over k Using correspondences and using the triviality of BrY Brk7 one can prove LyES gt ixiyES which improves 31 Thus 04 may be taken so that its image in BrVBrk has order 3 This follows also from the calculation of possibilities for BrV Brk in SD27 together with Corollary 1 of SD27 which forces BrVBrk to have odd order Class eld theory shows that 3 Brk Brk7 and it follows formally that the order 3 element of BrVBrk can be lifted to an order 3 element of BrV isiyiz E S for del Pezzo surfaces V of degree 4 smooth intersections of two quadrics in P4 In this case BrVBrk is killed by 2 See p 178 in Ma2 or the proof of Proposition 318 in CTSSD If there is a Brauer Manin obstruction7 then it can be explained by a single element 04 E BrV of order 2 If k has real places7 2Brk 74 Brk7 so the argument in the previous remark needs to be modi ed in order to prove this one must make use of the assumption that Vkv 74 Q for all real places 1 Suppose that V is a del Pezzo surface of degree 4 over k as above7 with a Brauer Manin obstruction given by 04 E BrV of order 2 A result of Amer Am and Brumer Br then implies that VL Q for any nite extension L of odd degree over k According to the conjecture in Section 257 there should be a Brauer Manin obstruction for V gtltkL Does the same 04 yield an obstruction over L7 If one replaces V by a smooth cubic surface in the previous question7 and the Amer Brumer result by the conjecture that the existence of a point of degree prime to 3 on V implies the existence of a k point7 then one is led to ask the analogous question for V7 for extensions L with gcdL M3 1 There is a geometric application of the Amer Brumer result that leads to a question about Brauer Manin obstructions for Weil restriction of scalars Let L be a nite separable eld extension of k and W be a quasi projective variety over L If R ResLk W is the Weil restriction of scalars7 then there is a natural L morphism RL R gtltk L a W Now let V be a del Pezzo surface of degree 4 over k and apply the above to W VL V gtltk L We obtain a morphism RL a VL lf L k is odd7 then application of the Amer Brumer result for V to the extension of function elds LRkR7 yields a k rational map R quota V A Brauer Manin obstruction to the existence of a k point on V can be pulled back to obtain a Brauer Manin obstruction for R over k Hence we are led to the following question7 to which a positive answer would imply a positive answer to the question in remark 3 above Let LK be a nite extension of number elds Let V be a smoooth projective geometrically integral variety over L7 and let R ResLkV It is clear that V has points over all completions of L if and only if R has points over all completions of k Suppose that this is the case Then is it true that there exists a Brauer Manin obstruction for VL if and only if there exists a Brauer Manin obstruction for Rk If it is true that the Brauer Manin obstruction is the only one for geometrically rational varieties7 then the answer must be yes for such varieties For arbitrary V7 one direction can be proved without too much work if there is a Brauer Manin obstruction for VL7 then because of the natural map RL a V described above7 there is a Brauer Manin obstruction for RLL7 and compatibility of the corestriction map BrRL a BrR with pullback shows that there is a Brauer Manin obstruction for Rk 14 JEANLOUIS COLLIOT THELENE AND BJORN POONEN We return to the notation of earlier subsections of Section 3 In particular V is a smooth cubic surface violating the Hasse principle By our assumption the conjecture there is a Brauer Manin obstruction Recall that we are trying to show that the lack of rational points on V can be explained by the Cassels Tate pairing Choose 04 as in Lemma 34 Let g be the composite morphism X a V a V and de ne a fa E BrX which gives a Brauer Manin obstruction for X We take U to be the largest open subscheme of Pi over which my X a P is smooth and let XU 7T1U as usual We have an exact sequence of etale sheaves of groups 0 a PicgUU a PicXUU a Z a 0 and HMU Z 0 by Proposition 36ii in Ar This explains the bottom row of BrXU 32 HU PicgUU HUPicXUU a 0 The vertical map is the homomorphism ker Him Gm a HgUR27rGm a HUR1 mom from the Leray spectral sequence for 7139 XU a U since the sheaf R1 iriGm can be identi ed with PicXUU by BLR p 203 and R2 iriGm 0 by Corollaire 32 in Gr2 We restrict a to an element of BrXU map it downwards in 32 to obtain 6 E HUPicXUU and lift leftwards to obtain 7 e HUPic3UU By MiL 11147 7 is the class of some torsor M for PicQYUU 8U over U Proposition 35 With notation as above assuming that there is a Brauer Manin obstruc tion for V the class of y is in for allt E Uh and the Cassels Tate pairing 2634 6 QZ is a nonzero constant independent oft Proof Suppose t E lf 6 is a cohomology class of an etale sheaf on U or XU let 6 denote the restriction to t resp We have a diagram BrXt 33 H1kPic07t H1kPicZ 0 analogous to 32 in which the restrictions 04 on the top and 7 on the left map to t The class of the ber 3 equals 7 Then evoz evozg is a nonzero constant independent of z E 24Ak and also independent of t The former is a sum of local functions evvozv of independent argu ments so each of these must be constant Let E X gtltk kw which is an elliptic curve over kw since 240 31 0 Let F E gtltkU E A point in Pic0E kw is represented by a difference of two points in Ehv and the pairing BrE gtlt PicE a Brkv induced by evaluation is compatible with the pairing H1kv Pic gtlt Pic0E a Brkv lt gt QZ Lic so the image BE of t in H1kvPicF is in the kernel of the latter pairing It follows from Tate local duality or a result of Witt if i is real that BE 0 FAMILIES OF SHAFAREVICHTATE GROUP ELEMENTS 15 In the diagram analogous to 33 but over kw instead of k the bottom surjection is now an isomorphism since the rst map in the exact sequence H0kv PicE a z a H1k P100E a H1k P10E a 0 is surjective Hence the image of 7 in H1kvPic0E which maps to BE 0 must itself be zero Thus 7 E By Section 26 lt24 eva p for any x E 24Ak and the right hand side is independent of t El ACKNOWLEDGEMENTS We thank Sir Peter Swinnerton Dyer for some discussions about cubic surfaces in par ticular for a suggestion which led to the proof of Lemma 34 We also thank Qing Liu for directing us to Liu REFERENCES Am Mi Amer Quadratische Farmen uber Funktianenkb39rpern Dissertation Mainz 1976i Ar Mi Artin Faisceaux constructibles Cohamalogie d une caurbe alg brique Expose IX pp 1742 in Theorie des tapas et cohamalagie tale des sch mas SGA 4 Tame 5 Lecture Notes in Math 305 Springer Berlin 1973 BPV W1 Barth Ci Peters and A Van de Ven Compact complex surfaces Ergebnisse der Mathematik und ihrer Grenzgebiete 3i Folge Band 4 SpringerVerlag 1984 BLR Si Bosch W1 Liitkebohmert and Mr Raynaud N ran models SpringerVerlag Berlin 1990 Br Al Brumer Remarques sur les couples de farmes guadratiques Ci Rt Acad Sci Paris 286 s rie A 1978 6797681i CG Ji W1 Si Cassels and Mr l T Guy On the Hasse principle for cubic surfaces Mathematika 13 1966 1117120 CT JiLi Colliot Th lene The Hasse principle in a pencil of algebraic varieties Contemporary Math 210 1998 197391 CTS JiLi Colliot Th lene et JiJi Sansuc La descente sur les vari t s ratiannelles in Jaurn es de Geometrie Alg brique d Angers Juillet 1979Algebraic Geometry Angers 1979 2237237 Sijthoff amp Noordhoff Alphen aan den Rijn 1980i CTKS JiLi Colliot Th l ne Di Kanevsky et JiJi Sansuc Arithm tigue des surfaces cubiques diagonales pp 17108 in Iquot L quot r 39 quot and t J theory Bonn 1985 Lecture Notes in Math 1290 Springer Berlin 1987 CTSSD JiLi Colliot Th l ne JiJi Sansuc and Sir Peter SWinnerton Dyer Intersections of two quadrics and Chatelet surfaces I l Reine AngeWi Math 373 1987 3771071 DM Pi Deligne and Di Mumford The irreducibility 0f the space of curves of given genus Instr Hautes Etudes Scii Publ Mathi Not 36 1969 757109 EGA IV A Grothendieck Elements de g om trie alg brique r dig s avec la collaboration de J Dieudonn IV Etude locale des sch mas et des morphismes de sch mas Traisieme Partie Instr Hautes Etudes Scii Publi Mathi Not 28 1966 Grl Al Grothendieck Technique de descente et theoremes d existence en g om trie alg brique V Les sch mas de Picard theoremes d ezistence in S minaire Bourbaki Vol 7 Exp 232 1437161 Soci Mathi France Paris Gr2 Al Grothendieck Le graupe de Brauer U exemples et complements pp 887188 in Diet exposes sur la cohamalagie des sch mas AdVi Studi Pure Math 3 NorthHolland Amsterdam 1968 Ha Di Harari Methade des bratians et obstruction de Manin Duke Math J 75 1994 2217260 Ha2 Di Harari Weak r 39 quot and 1 739 I J 39graups preprint October 1998 l l HS Di Harari and Al Ni Skorobogatov Nanabelian cohamalagy and rational points preprint May 1999 Ja Ui Jannsen Principe de Hasse cahamalagique S minaire de Th orie des Nombres Paris 19891990 Sinnou David edi Progress in Mathematics 102 1992 1217140 16 JEANLOUIS COLLIOTTHELENE AND BJORN POONEN Ka Ni Katz Pinceaux de Lefschetz theoreme d existence Expose XVH ppi 2127253 in Groupes de mon odromie en g om trie alg brique SGA 7 U Lecture Notes in Math 340 Springer Berlin 1973 Lic Si Lichtenbaum Duality theorems for curves over padic elds lnventi math 7 1969 1207136 Lin CiEi Lind Untersuchungen uber die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins Thesis University of Uppsala 1940 Liu Liu Modeles entiers d une courbe hyperelliptique sur un corps de valuation discrete Trans Amer Math Soc 348 1996 no 11 457774610 Mal Yu 1 Manin Le groupe de Brauer Grothendieck en g om trie diophantienne in Actes du Congres International des Math maticiens Nice 1970 Tome 1 4017411 GauthierVillars Paris 1971 Ma2 Yu 1 Manin Cubic forms Translated from the Russian by Mr HazeWinkel Second edition North Holland Amsterdam 1986 Mil Ji Milne Etale cohomology Princeton Univ Press Princeton Nil 1980i Mi2 Ji Milne Arithmetic duality theorems Perspective in Mathematics 1 Academic Press 1986 Mo Li J Mordell Rational points on cubic surfaces Publi Mathi Debrecen 1 1949 176i MFK Di Mumford Ji Fogarty and F Kirwan Geometric invariant theory Third edition Ergebnisse der Mathematik und ihrer Grenzgebiete 2 34 SpringerVerlag Berlin 1994 PS Bi Poonen and Mr Stoll The Cassels Tate pairing on polarized abelian varieties to appear in Ann of Math Re Hi Reichardt Einige im Kleinen uberall lo39sbare im Grossen unlo39sbare diophantische Gleichungen Ji Reine AngeWi Math 184 1942 12718 Sc Ci Scheiderer Real and tale Cohomology Lecture Notes in Math 1588 Springer Berlin 1994 ST JiPi Serre and l Tate Good reduction of abelian varieties Ann of Math 2 88 1968 4927517 SGA4 SGA 4 Theorie des topos et cohomologie tale des sch mas Tome 1 Theorie des topos Seminaire de Geometrie Alg brique du Bois Marie 1963719641 Dirige par Mi Artin Al Grothendieck et J Li Verdieri Avec la collaboration de Ni Bourbaki Pi Deligne et Bi SaintDonati Lecture Notes in Mathematics Voli 2691 SpringerVerlag BerlinNew York 1972 Sk Al Ni Skorobogatov Beyond the Manin obstruction lnventi Math 135 1999 nor 2 3997424 SDl H B F SWinnertonDyer Two special cubic surfaces Mathematika 9 1962 54756 SD2 Sir Peter SWinnerton Dyer The Brauer group of cubic surfaces Mathi Proci Cambridge Philosi Soc 113 1993 no 3 4497460 vGY J van Geel and Vi Yanchevskii Indices of hyperelliptic curves over padic elds Manuscripta Math 96 1998 317333

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