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by: Addison Beer


Addison Beer
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This 15 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 136 at University of Washington taught by Staff in Fall. Since its upload, it has received 18 views. For similar materials see /class/192085/math-136-university-of-washington in Mathematics (M) at University of Washington.




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Date Created: 09/09/15
Differential algebraic geometry and abc Alexandru Buium Lecture 1 Motivation abc on a ine varieties We start by explaining a generalization of abc that makes sense for any af ne variety over a global eld The usual abc is simply the case of P10 1 00 We then state our main result that says that this general ized abc holds under certain trace conditions for af ne open sets of abelian varieties over function elds Let K be a eld equipped with a family of absolute values v K a 000 1 E M all of which except nitely many are non archimedian Set Mm flogmv for z E KX Assume my is a collection of positive integers such that the product formula77 Emux 0 z E Kgtlt 1 holds Set yv infvKX 0oo Also for any 77 77177N E KN and 1 E M set 7177 Injinvm De ne the af ne logarithmic height hez ghtAN ANK KN a 000 by the formula heightANM 7 Z Vivi77 Emu max logJr njlv v a vltngtS0 where logx maxlogx0 z 6 000 Note that heightANM heightpw 771 771V where heightp xo zN Emu rnaxlog lzjlv 7 39U is the usual height in projective space On the other hand de ne the loga rithrnic conductor condAN ANK KN a 0oo by the formula condAN 77 Z 777qu vltngtS0 Clearly7 by the very de nition of mu we have condAN 77 S heightAN 777 77 E KN We need the following piece of notation Let g S a 000 be two real functions on a set S we write f S g 01 if there exists a positive real constant C such that fP S gP 0 P E S and we write f g 01 if f S g 01 and g S f 01 We write f ltlt 9 01 if there exist two real positive constants 0102 such that fP 3019P027 P65 and we write f E g 01 if f ltlt 9 01 and 9 ltlt f 01 Corning back to height and 00nd one easily checks that if P ANK a A K a map given by an nituple of polynomials in N variables with Kicoef cients then we have height 0 P ltlt heightAN 01 condAn o P S condAN 01 This permits to de ne the height and the conductor for any af ne variety as follows Let U be an af ne variety over K Let 239 U a AN be a closed immersion and de ne hetghtU UK a 0 oo candy UK a 0 00 by the formulae hetghtUP hetghtANz P P E UK condUP condANz39P P E UK By the above discussion if hetghtU and candy are de ned by a closed im mersion 239 and heightU and BondU are de ned by a closed immersion 2quot then hetghtU E heightU 01 candy BondU 01 In particular we have candy ltlt hetghtU 1 01 Here is our basic de nition De nition We say that the abc estimate holds on U if hetghtU ltlt candy 01 In this lecture we would like to understand what are conjecturally the af ne varieties U on which the abc estimate holds To tackle this question and establish the link with the usual abc we need to be more speci c about our eld K Assume in what follows that we are in one of the following situations 1 Number eld case K is a number eld equipped with its standard family of absolute values l normalized in such a way that they extend the standard absolute values of Q in particular ifi divides a rational prime 3 p then lplv p l We take my Ky Qv So for any non archimedian 77 dividing an unrami ed rational prime p we have yv 771 logp so 777qu log N77 where N77 is the norm of 77 cardinality of the residue eld of 77 Since there are only nitely many rami ed primes we have condAN77 Z logN77 77 E KN vlt gtlt0 2 Function eld case K is a function eld of one variable over an algebraically closed eld k of characteristic zero We equip K with the abso lute values l lv arrising from the kerational points 77 of the smooth projective model V of Kk we normalise them via the condition yv 1 for all 77 and we take my 1 for all 77 So in this case condAN 77 for 77 E KN is simply the number of points 77 E V which are poles for at least one of the rational functions 777 It is a trivial exercise to show that if say K Q or K kt and if there exists a non constant morphism of Kevarieties P10 00 a U into an af ne Kevariety U then the abc estimate fails on U On the other hand the optimist would be tempted to believe that the presence of such mor phisms is the only obstruction to the abc estimate we make optimistically the following Conjecture Assume U is a smooth af ne variety over K and assume that any morphisms of Kevarieties P10 00 a U is constant Then the abc estimate holds on U Our Conjecture should be viewed as an a ne analogue of Lang7s con jecture77 saying that if X is a smooth projective variety over K and if any morphism from an algebraic group G to X is constant then the points of XK have bounded height hence in the number eld case XK is nite ln Lang7s conjecture one allows morphisms G a X de ned over the alge braic closure K of K it might be reasonable to allow this in our Conjecture as well In case K Q and U P10 1 00 the Conjecture above simply says that the abc estimate holds on this particular U note that this is equivalent to the following Variant of abc conjecture For any relatively prime integers abc with a b 0 there exist real numbers 0 gt 1 and M gt 1 such that max al lbl S C radabc As usual radn where n is an integer is de ned as the product of all primes dividing n this Variant is weaker than the abc of Masser and Oesterle which predicts that M can be made as close to 1 as we want However this Variant still implies say the assymptotic Fermat To see the equivalence between the Conjecture and the Variant of abc for K Q and U P10 1 oo embed U into A2 via the map Then for coprime integers 11 0 with a b c we have that height1 equals 2 1337 logmax abct W lagl 2 logmaxmai W 161 01 On the other hand h6 ghtp2lt a2 Cb log radabc c c d 17 d 7 con U a con A2a Note that by a result due independently to Mason and Silverman the abc estimate holds for the projective line minus 3 points On the other hand results of Voloch V Brownawell Masser EM and Wang can suitably be interpreted as abc estimates outside some exceptional loci77 for projective spaces minus unions of hyperplanes Theorem abc for abelian varieties with trace zero B947 In the function eld case the abc estimate holds for any al ne open set of an abelian variety with trace zero Theorem abc for isotrivial abelian varieties B98 Let K be a function eld over k let Ak be an abelian variety over k and Dk a divisor in A which does not contain any translate ofa non zero abelian subvariety in particular Dk is ample so Uk AkDk is af ne Let U Uk 8 K Then the abc estimate holds on U The above Theorems are immediate consequences ofthe following stronger results Theorem Bounded Multiplicity Theorem trace zero case B947 Assume AK is an abelian variety with trace zero and let f E KA be a rational function Then there exists a constant 0 depending only on K A f with the following property For any point P E AK Where f is de ned and does not vanish all zeroes and poles of fP E Kgtlt have multiplicity at most 0 Theorem Bounded Multiplicity Theorem isotrivial case B98 Let X be a smooth projective curve over k A an abelian variety over k and D an effective divisor on A Assume that D contains no translate of a non zero abelian subvariety Then there exists a real constant C gt 0 depending only on X A and D with the property that for any morphism f X a A with fX D all points of the divisor fD have multiplicity at most 0 The proofs of the results above are based on differential algebraic ge ometry A characteristic p version of these results was proved by Scanlon Sc Lecture 2 Differential algebraic geometry Let f be a eld of characteristic zero equipped with a derivation 6 We let C be the constant eld If K is a function eld we will always assume we have xed a non zero kiderivation on it and Kd is embedded into f 6 Following the classical work of Ritt and Kolchin one de nes the 67polynomial ring fT where T is an n7tuple of variables as the usual polynomial ring over f in the variables T 239 2 0 The order ofA E fT is the highest 239 such that a variable Ty is present in A f is called 67closed if for any AB E fT T a variable such that 0rd B lt 0rd A there exists a E f such that Aa 0 and Ba 31 0 We need more de nitions A D7scheme is simply an f7scheme V with a given derivation on OV that lifts 6 D7schemes form a category morphisms are required to commute with the derivations A D7group scheme is a D7scheme which is also an f7group scheme such that the multiplication inverse and unit are morphisms of D7schemes A D7variety is a D7scheme which is also a variety over f An algebraic D7group is a D7group scheme which is also an algebraic group over 7 One has a forgetful functor D 7 schemes 7 f 7 schemes V gt gt V It has a right adjoint f 7 schemes 7 D 7 schemesX gt gt X Homel X HomDVX de ned as follows If X Spec flTlI set X Spec fTI I 161621 In the non af ne case one glues the af ne pieces Note that X is the inverse limit of a system X of varieties obtained by truncat ing everything to order n One has a natural map V XT 7 X f which in coordinates sends a E f into a6a62a Pull backs via this map of Zariski closed sets are called 67closed sets Kolchin Topology If f is 67closed there is a bijection between delta closed sets and reduced closed D7subschemes of X If E C XT is 67closed corresponding to a D7subscheme H C X then one de nes the absolute dimension 12 as the maximum of the transcendence degrees over f of the function elds of the irreducible components of H Example aXf 00 if dz39m X gt 0 If X descends to C ie comes from a variety XC over C then aXC dz39m X For G an algebraic group over 7 the above bijection induces a bijection between the 67closed subgroups of nite absolute dimension of CK and algebraic D subgroups of G By a ifunction on Xf where X is a variety we mean a map XT a f obtained by composing V with a regular map on X A Ficharacter on an algebraic group will mean a ifunction which is also an additive homo morphism Theorem 67density Theorem B93 If X is a smooth projective unira tional f7 variety then any Sifunction on XT is constant Moreover ifX de ned over an intermediate eld L between C and f L 31 C then XL is Fidense in Xf Theorem Finiteness of absolute dimension B92 Let A be an abelian fivariety of dimension g Then the intersection Ait of the kernels of all Ficharacters ofA has nite absolute dimension 9 S aAil 3 2g Conse quently the Seclosure of any nite rank subgroup ofAf has nite abso lute dimension 3 2g r Where r is the rank In particular in the trace zero case the iclosure of the group of division points of AK has nite absolute dimension The 67characters are the incarnation in differential algebraic geometry of the Manin homomorphisms but our niteness result aAil lt 00 is quite different in nature from the Manin Chai Theorem ofthe kernel What the latter says in the trace zero case in our terminology is the following Theorem of the Kernel Manin Chai Let A be an abelian fivariety with fCitrace zero Then for any intermediate Si eld C C L C f of de nition for A which is nitely generated over C we have All AL ALt039rs39 Theorem imaps on curves B94 Let X be a smooth projective curve over f of genus g 2 2 that does not descend to C Then there exists an injective imap b X a A In a certain precise sense one can actually prove more namely that pro jective curves of genus at least 2 are af ne in this geometry Half way towards our abc for abelian varieties we have the following differential algebraic generalisation of the geometric Lang conjecture on sub varieties of abelian varieties Theorem Differential Algebraic Lang B92 Let A be an abelian variety over f with trace zero over C Let E C A be a Seclosed subgroup of nite absolute dimension and X C A a closed subvariety Then there exist in X nitely many translates of abelian subvarieties Whose union contains Xf 2 The above Theorem is indeed a generalisation of the geometric Lang conjecture because together with the Theorem on nite absolute dimension it formally implies the Theorem Geometric Lang Conjecture B92 Let A be an abelian variety over K with trace zero over k Let P C AU be a subgropp of nite rank and X C A a closed subvariety Then there exist in X K nitely many translates of abelian subvarieties Whose union contains XK P Lecture 3 Description of proofs a Descent results An fivariety resp an algebraic group over f is said to descend to constants if it comes via base change from a variety resp an algebraic group over C The next four theorems are proved via complex analytic arguments for the rst theorem Gillet showed me an argument based on formal schemes which should be also considered in some sense analytic this kind of argument in formal geometry does not seem to apply to the other results Theorem Descent of projective varieties B87 Any projective Divariety descends to constants Theorem Descent of linear algebraic groups B92 Any linear algebraic D group descends to constants Remark the above fails for non linear groups A VERY interesting ques tion Let G be an algebraic D group and H be its maximum connected linear algebraic subgroup does H descend to constants 7 We won7t need the previous two Theorems for our diophantine purposes For the abc Theorem for abelian varieties with trace zero we will need Theorem Descent via D groups B92 Let G be an algebraic D group V C G a closed Disubvariety and V a W a dominant morphism to a projective variety W of general type Then the Albanese variety AlbW descends to constants For the abc Theorem for isotrivial abelian varieties we need Theorem Descent via D groups split case B98 Let W be a projective variety of general type over K Assume W is a closed subvariety of AK Where A is an abelian kivariety Let G be any algebraic Digroup V C G a Disubvariety and u V a W be a dominant morphism Then after replacing K by a nite extension of it one may nd a closed kisubvariety Z C A and a point Q E AK such that W ZKQ in AK Moreover if we view W as a Discheme by trivially lifting 6 from K to W 2 ZK Z k K then u V a W is necessarily a morphism of Dischemes Let us sketch the proof of the Theorem on descent via D groups We may assume C is the eld of complex numbers We may further reduce ourselves to the case when V7 W7 G7 Spec 7 6 comes7 via base change7 from a situation in complex algebraic geometry77 VW7 Gr7 S763 where S is an af ne complex curve with a non vanishing vector eld 63 W is projective over S with integral geometric bres of general type G is a group scheme of nite type over S equiped with a vector eld 63 lifting 63 such that the inverse and the multiplication on G are 6G7equivariant call such a 63 group compatible 10 V is a closed subvariety of G horizontal with respect to 6g with a dominant map to W A lemma of Hamm says that an analytic 1 parameter family of complex Lie groups77 whose total space is equipped with a group compatible77 analytic vector eld lifting a non vanishing vector eld on the base is locally analyt ically trivial So locally analitically G G0 gtlt S 63 063 where G0 is some Lie group So we have an analytic splitting note that there is no algebraic splitting in general It follows that V a S is locally analytically trivial Fix 50 E S For any 5 E S consider the analytic map ngonSaWS lts image contains a Zariski open set The Big Picard Theorem Kobayashi Ochiai 75 says that any analytic map from an algebraic variety to a projective variety of general type whose image contains a complex open set is in fact algebraic So our maps 155 are algebraic So they induce maps from the Albanese variety of a smooth projective model of V5O into AlbW5 So all AlbWS are isomorphic to each other and we are done b Sketch of proof of the Finite Absolute Dimension Theorem Set 0 Spec OA One easily checks that U KerA a A is unipotent Then one proves that whenever one has an extension 0gtUgtGgtA with A an abelian variety and U unipotent in nite dimensional then one also has an exact sequence 0 a A a G a U a 0 with A nite dimensional and U unipotent in nite dimensional This is an exercise in the theory of proalgebraic groups The one is done by noting that A can be chosen to be a Disubgroup and that Ait is the 67closed subset correspoding to A c Proof of Differential Algebraic Lang 11 Assume we are in the hypothesis of Differential Algebraic Lang Then 2 corresponds to an algebraic Disubgroup G of A Let V be an irreducible component of X G and let Y C X be the Zariski closure of the image of V in X We claim Y is a translate of an abelian subvariety7 and we shall be done Assume its not By a result of Ueno there is an abelian subvariety A1 C A such that the image W of Y in AA1 is positive dimensional7 of general type By the Theorem on descent via D groups77 it follows that AlbW descends to constants But A was assumed to have trace zero7 a contradiction 1 Proof of abc for abelian varieties Preparation Let A be an abelian variety over K7 with Kkitrace zero and let U be an af ne open set The closed set AU is the support of a very ample effective divisor D on A Note that D is neither irreducible nor reduced apriori Embed A into Pg such that D is given by 0 07 where zo N are a basis of H0P7 01 Let Uj C A be the open sets de ned by Q 31 0 So U0 U We need to show that there is a constant C such that for any point 77 7717 77N E UK and for any place 1 we have 077 2 G This actually easily implies the stronger Bounded Multiplicity Theorem as well For simplicity we shall assume A is simple Constructing the nite set Y Let A 7 D be attached to A and D respectively the whole construction being made over K7 rather than over 7 Then D appears as a closed subscheme of A Exactly as in the proof of Differential Algebraic Lang7 there exists an algebraic Disubgroup G C A such that for all P E AK we have VP E Set V D G scheme theoretic intersection and let Y C A be the Zariski closure of the image of V via the map A a A we view Y with its structure of reduced subscheme of A As in the proof of Differential Algebraic Lang Y is isomorphic7 over the algebraic closure of K7 with a nite union of translates of abelian subvarieties Since A was assumed simple7 Y is nite For simplicity we shall assume the nitely many points of Y are rational over K Uniform discreteness bl7 and 7 It is an easy consequence of the Manin Chai Theorem of the Kernel that one can nd7 for each j a nite 12 family wig aw bl7 E jY U and there exists a real number 7 2 0 such that for any 2 and any P E UjKY there exists an index 2 E 7 such that MaxiPD S 7 Here j of a subscherne denotes the ideal de ning that subscherne ldentify bl 6 0Uj with their pull backs in 0Uf0 Then for a suitable q we have 3 e JltV m U JltD n U JG n U Clearly x jD Uj 39 0Uj 7 So we get E 00 00 0 2 0 72 OD U 6767OU lt 7 7 7 7 gt ltgt We may write for all j and 2 E 17 n 960 961v 960 1 Q imi t 7 397 120 739 95139 3995 97 with gij E jG C OU and Figt differential polynomials with Kicoef cients De ning 6 Clearly we may nd a real number 6 2 0 such that for any 77 E KN and any 2 with 077 2 0 we have 21Fijt77 2 76 for all 239jt Fixing P and 2 Now we x a point P E UK AKD with coordinates g E KNJr1 and x a place 2 Estimate equation 1 at VP E A K Since VP E GK we have gVP 0 So by further taking 2 we get for each j and each 2 E 7 Choosing j Now for our xed P and 2 there exists j such that j 31 0 hence P E UjK and 2 j 2 0 for all 2 ie 2 j 2 Fix such a j Choosing 2 For our xed P2j there exists by uniforrn discreteness an index 2 E 7 such that 3 v ijP S Y 13 Fix such an index 239 Conclusion Now by the choice of B we have for all t 4 Fm Z 5 5739 5739 It is trivial to see that there exists a real number 6 2 0 such that for all z E K and all 1 we have 116z 2 v 7 0 Then putting together 2 3 and 4 we get q 7 Z q WagPD 2 6 M50 7 7157 7119 H 77 1507 N o then we get 77 U 0 20 7 qvn6 and we are done References BM DBrownawell DMasser Vanishing sums in function elds Math Proc Cambridge Philos Soc 100 1986 427 434 B87 A Buium Fields of de nition of algebraic varieties in characteristic zero Compositio Math 611987339 352 B92 A Buium lntersections in jet spaces and a conjecture of SLang Annals of Math 136 1992 583 593 B93 A Buium Geometry ofdifferential polynomial functions 1 algebraic groups Amer J Math 115 6 1993 1385 1444 B94 A Buium Geometry of differential polynomial functions ll alge braic curves Amer J Math 116 4 1994 785 819 B95 A Buium Geometry of differential polynomial functions lll mod uli spaces Amer J Math 117 1995 1 73 B947 A Buium The abc theorem for abelian varieties lntern Math Research Notices 5 1994 219 233 B98 A Buium lntersection multiplicities on abelian varieties Math Ann 1998 K E Kolchin Differential Algebra and Algebraic Groups Academic Press New York 1973 L SLang Number Theory 111 Springer Verlag Mas RCMason Diophantine equations over function elds London Math Soc Lecture Notes Series 96 Cambridge Univ Press 1984 Man YulManin Rational points on an algebraic curve over function elds Transl AMS ll Ser 50 1966 189 234 and Letter to the Editor lzvAkadNauk USSR 34 1990 465 466 Si JHSilverman The S unit equation over function elds Math Proc Camb Phil Soc 95 3 1984 3 4 Sc T Scanlon The abc theorem for commutative algebraic groups in characteristic p lMRN 18 1997 881 898 V FVoloch Diagonal equations over function elds Bol Soc Brasil Mat 161985 29 39 W JTYWang Siintegral points of projective space omitting hyper planes over function elds of positive characteristic preprint Dept Math and Stat University of New Mexico Albuquerque


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