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# ALGEBRA MATH 102

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ABC ESTIMATE7 INTEGRAL POINTS7 AND GEOMETRY OF P MINUS HYPERPLANES JULIE TZU YUEH WANG Institute of Mathematics Academia Sinica Nankang7 Taipei 11529 Taiwan7 RiOiCi January 22 1998 ABS39I RAC39I Let K be a eld and H be a set of hyperplanes in PquotKV When K is a function eld we show that the following are equivalent a H is nondegenerate over KY b The height of the S H7integral points of PquotK 7 H is bounded c P 7 H is an abc variety When K is a number eld and H is nondegenerate ver K we establish an explicit bound on the number of S H7integral points of PquotK 7 HY Finally we discuss the geometric properties of holomorphic maps into P quot1 omitting a set of hyperplanes with moving targets 0 INTRODUCTION Let F be a number eld and H be a set of hyperplanes in P Fi Let S be a nite set of valuations of F including all the archimedean valuations When H is in general position and the number of hyperplanes in H is at least 2n 17 Ru and Wong RW proved that the number of the S Hintegral points is nite later the author WaZ provided an explicit bound on the number Ru then found a necessary and suf cient condition on H such that the number of the S Hintegral points of P F 7 H is nite he also showed that this is a necessary and suf cient condition of Brody hyperbolicityi However7 an explicit bound on the number of the S Hintegral points was not obtained in Ru Let C be an irreducible nonsingular projective algebraic curve of genus 9 de ned over an algebraically closed eld k of characteristic p 2 0 Let K be the function eld of C and H be a set of hyperplanes in Let S be a set consisting of nitely many points of C When p 07 the author showed that if H is in general position and the number of hyperplanes in H is at least 2n 1 then the height of the S Hintegral points is bounded and the bound is a linear function of When p gt 07 the author WaS showed that if H is in general position and Typeset by AM57TEX 2 JULIE TZUiYUEH WANG the number of hyperplanes in H is at least 2n 2 then under certain condition the height of the S Hintegral points is bounded and the bound is a linear function of 539 Recently motivated by the abc theorem for function elds cf Ma BM Vol Wall and No Buium de ned abc varieties and proved that any af ne open subset of an abelian variety over function elds of characteristic 0 with trace zero is an abc varietyi cf Bu The de nition of abc varieties is closely related to the S D integral points of a projective space V deleting a very ample divisor D It turns out that the previous results on function elds done by the author are all theorems about abc varietiesi In the geometric case as mentioned before that Ru gave a necessary and suf cient condition for P C 7 H to be Brody hyperbolici A more general question to consider is when the hyperplanes in H are moving ie the coef cients of the linear forms corresponding to H are holomorphic functions In Wa4 the author applied the method in WaZ and obtained a generalization of the Picard s theorem with moving targets In this paper we will improve the number eld result in Ru by giving an explicit bound on the number of the S Hintegral points In the function eld case of zero characteristic we will show that the condition on H given in Ru is also necessary and suf cient for the height of the S Hintegral points to be bounded and is also a necessary and suf cient condition for PI 7 H to be an abc varietyi Therefore we will prove that PI 7 H is an abc variety if and only if the height of the S H integral points of P K 7 H is bounded Finally in the geometric case we deal with the situation when the coef cients of the linear forms corresponding to the hyperplanes in H are holomorphic functionsi Ackowlegements The author wishes to thank Jing Yu and Min Ru for helpful suggestions 1 ABC VARIETIES AND SD1NTEGRAL POINTS In this section we will restrict ourselves to function eldsi However the de nition of abc varieties and S Dintegral points can be easily adapted to number eldsi Let C be an irreducible nonsingular projective algebraic curve of genus 9 de ned over an algebraically closed eld k of characteristic p 2 0 Let K be the function eld of Cl Given a point P E C we denote by up the normalized valuation associated to Pi For elements f0 M fn of K not all zeros we de ne the height as We m 7 Z 7minvpltfogt m7vpltfngti P60 Pquot MINUS HYPERPLANES 3 For an element f of K we de ne the height as I Z 7min07 vpf PEC We now recall the de nitions of S Dintegral points cf Voj and abc varieties cf Bun Let V be a projective variety de ned over K Let D be a very ample effective divisor on V and let 1 450 4151qu be a basis of the vector space D is a rational function on V such that f 0 0r 2 7D Then 45 450 H 451V de nes a morphism from V to PN and 7 7gt 1510 m NT is an embedding of VK 7 D into KN De nition A point 739 of VK 7 D is said to be an SDintegral point if UP iT 2 012 N7 for every P Si Following Bu we may de ne height and conductor as following h 739 h 739 h4507 A7 NT7 Cond 739 P E C minvpq 1739vpq5NT lt 07 cond 739 lCond Tl De nition We say that V 7 D satis es the abc estimate over K if h 739 ltlt cond 739 017 for every 739 E VK 7 D7 where ltlt 77 means the inequality holds up to multiplication with a positive con stant Remark This de nition does not depend on the choice of the embeddingcf Bub When there is no confusion7 we will omit the subscript De nition VK 7 D is an abc variety if it satis es the abc estimate over every nite extension L of K In this paper7 we only consider the case when V PI and D is a set of hyper planes in Let H be a set of 4 distinct hyperplanes in P K and let Li 1 S i S L be the linear forms corresponding to H Then we can x an embedding from P 7 H to KN in the following form 13 HLixz ue H 4 JULIE TZU7YUEH WANG where each coordinate of the embedding is in the form 13 I H nil Lix with 20 ij qr Let as 113 ung i zH LAX H Suppose that 77 is a point in P and is represented by fo7 m Then from the de nition of height q Mfng f3 S MH Lifowfn7fgv mfgw h 77 i1 On the other hand7 if Li 20 aijzj then up Lifo7 in S rninvp fo7 in up rninvp aw7 m up ami Therefore hq5n M Lif0wfnfgwfgm S qhfom7 fn ihwiopwam 11 i1 Together we have 4hf07fn S WW7 S qhfow7fn 01 Proposition 1 Let H be a set of 4 distinct hyperplanes in P K If P 7 H is an abs vaiiety then the height of the S Hintegml points of P 7H is bounded linearly in 5 Proof Let 739 be an S Hintegra1 point of P K 7 H and be represented by fo7 in Then from the de nitions of Cond and S Hintegra1 points we have cond 739 S P 7 H is an abc variety7 hence h 739 ltlt cond 739 01 Since qhltfo7 m fn S h 7397 we have hfomfn ltlt 5 on Remark This proposition is true for number elds Pquot MINUS HYPERPLANES 5 2 FURTHER RESULTS IN FUNCTION FIELDS Let F be a number eld and let H be a set of hyperplanes in P F Ru gave a necessary and suf cient condition on H such that P F 7 H has only nitely many SHintegral points We will show in this section that for a function eld K this is a necessary and suf cient condition for P 7 H to be an abc variety and also a necessary and suf cient condition such that the height of the S Hintegral points of P K 7 H to be bounded We recall some de nitions and results from Ru Notation Let L be a set of linear forms in n 1 variables which are pairwise linearly independent We denote by LF the vector space generated by the linear forms in L over F De nition Let F be a eld and H be a set of hyperplanes in P F We let L be the set of linear forms corresponding to H We note here that all linear forms in L are pairwise linearly independent over F H is said to be nondegenerate over F if dimLF n I and for each proper nonempty subset L1 of L L1F O L7 L1F L ll Remark If H is in general position and the number of hyperplanes of H is no less than 2n 17 then H is nondegenerate over F De nition Let F be a eld and H be a set of hyperplanes in P F Let V be a subspace of P F V is called Hadmissible if V is not contained in any hyperplane in H PropositionRu Let H be a set of hyperplanes in P F Then H is non degeneate over F and only for every Hadmissible subspace V of P F of projective dimension greater than or equal to one H N V contains at least three distinct hyperplanes which are linearly dependent over F We will need the following version of the abc theorem BrMa for function elds of characteristic 0 Theorem BrownawellMasser Let the characteristic ofK be zero If f07 m fn are Sunits and f0 fn I then either some proper subsum of f0 fn vanishes or hf0mfn g wimp 2972mm 1 We also need the following version of abc theorem Wall for function elds of positive characteristic 6 JULIE TZUiYUEH WANG Theorem Wang Let the characteristic of K be a positive integer p Suppose that fowl7 fn1 are Sunits of K If f0 fn fn1 and f07lll7fn are linearly independent over Kpm for some positive integer m then hf0wfn g wiomilmax4 072g72ls l 2 The main results in this section are the following Theorem 1 Let K be the function eld of a nonsingular projective algebraic curve C which is de ned over an algebraically closed eld h with zero characteristic Let S be a set consisting of nitely many points of C such that there exist nonconstant Sunits Let H be a set of hyperplanes in P Then the following are equivalent a H is nondegenerate over K b P 7 H is an abc variety c The height of the SHintegral points of P K 7 H is bounded linearly in 539 d The height of the SHintegral points of P K 7 H is bounded Theorem 2 Let K be the function eld of a nonsingular projective algebraic curve C which is de ned over an algebraically closed eld h with characteristic p gt 0 Let S be a set consisting of nitely many points of C Let Li Xi 0 S i S n and Ln1i in ainj 0 S i S n where aij are elements of K Let H be the set of 2n2 hypgrplanes de ned by Li 0 S i S 2n 1 Let Sn be the permutation group of 0127H7n If H are in general position ie any n 1 linear forms corresponding to H are linearly independent and the set H0ai0il a 6 Sn is linearly independent over h then P 7 H is an abc variety Proof of Theorem 1 We rst show that a implies Let L L1Hl7Lq be the set of linear forms corresponding to Hl Let 739 be a point of P K 7 H and be represented by fowl7 Denote by li LZfo7 Let SH be a set consisting of nitely many points of C such that every coef cient of each linear form Li has no zero or pole outside SHl Therefore7 vpli 2 rninvp f07 7 UP for P 571 On the other hand7 from the de nition of Cond7397 we have qvp 2 i vpli7 0 S j S n7 for every P Cond739l Therefore i1 vpli rninvpf0wvpfn7 for P Cond739 U SH 1S i S 4 1 Suppose that the set Li17 W Lim is linearly dependent over K and every proper subset of LilquLim is linearly independent over Kl Then we have a linear equation ailLZ1XaimLimXEU7 2 Pquot MINUS HYPERPLANES 7 where aii E Kxi We call equation 2 a minimal relationi Since elements of L are linear forms in n 1 variables and are pairwise linearly independent over K we have 3 S m S n 2 It is clear that up to a nonzero factor in K there are only nitely many such minimal relations for the set L Throughout the proof we will x a nite set of minimal relations representing all minimal relations for L up to a nonzero factor in Ki Without loss of generality we can enlarge the size of 57 Therefore we will assume that every coef cient of the minimal relations in this nite set has no zero or pole outside of 5H1 Let S Cond739 U 5H1 We now consider equation After rearranging the index we may assume that a1L1XamLmXEO 3 where ai 1 S i S m is an STuniti Then we have the following equation alllamlm 01 After rearranging the index we may assume that alll aulu 0 with u S m and no proper subsum of alll aulu vanishes Therefore a2Z2 aulu 7 71 4 1111 1111 7 By 1 if 1 S i S q is an S uniti Hence by the theorem of Brownawell and Masser we have ili 1 hLSwmax02972lSTl 199 5 alll 2 From the de nition of height M5 lt Mi hlt gt lt6 ll 7 alll a1 The coef cients ags in the representing set of minimal relations only depend on L and the number is nite Therefore inequalities 5 and 6 imply li n n1 ME lt w 01 lt7 where 1 S i S u and 01 only depends on L and can be determined effectively From now 0 always represents a constant which only depends on L and can be determined effectively 1f the dimension of the vector space spanned by L1 MLu over K is n 1 then after a linear transformation one can show that cfi Wa2 2 hfowfn le 02 8 8 JULIE TZUiYUEH WANG 1f the dimension of the vector space spanned by L1 W Lu over K is less than n 1 then the set L1 W Lu is a proper subset of 1 Since H is nondegenerate L1MLuK Lu1wLqK 9 0 9 Suppose that Li 6 L1MLuK Lu1 MLqKi 1f1 S i S u then after rear ranging the linear forms we have Li au1Lu1 au2Lu2 awLw where aj 0 and is assumed previously to be an STuniti Similarly after rearranging the index we have an equation 1 au1lu1 au2lu2 39 39 39 aylm V S w where no proper subsum of the equation vanishes Therefore we have lu 1 MT 1571 03 10gt Hence lu lu ll hi h1 he ll lZ ll Snn1lSTlO4i 11 lfi 2 u 1 after rearranging the index we may assume that i u1i Then we have Lu1 ailLi1 ailew where i1 11120 is an index subset of 1111 u and aii is an STuniti Similarly we have lu 1 M7 w 05 Therefore h 1w h 1w h 11 7 S ll 11 1i S nn1lSTl 05 12 Hence we have showed that lt 071571 01 for 1 lt i S u 11 1f L1MLu1 L then we are done Otherwise we can repeat the same ar gumenti Since dim K n 1 after repeating the argument nitely many times we can nd linear forms L1 iiiLw such that dimL1 M LwK n 1 and S OngTlJrOg 1 S i S wi Therefore after alinear transformation cfi Wa2 hf0wfn S 0101571 011 ltlt cond739 01 13 It is clear from the proof that the abc estimate holds for every nite extension of Ki This shows that PI 7 H is an abc varietyi Pquot MINUS HYPERPLANES 9 It follows from Proposition 1 that b implies c implies d triviallyi It remains to show that d implies a We follow the arguments in Ru Assume that H is not nondegenerate over Ki Then there exists an Hadmissible subspace V of P of projective dimension greater than or equal to 1 such that H V does not contain at least three distinct hyperplanes which are linearly dependent over Ki After linear changing of basis we may assume that V PmK m S n Then H V contains exactly 4 distinct hyperplanes which are linearly independent over K and q S m 11 Without loss of generality we may assume that V P and H contains exactly 4 S n1 distinct coordinate hyperplanes Let f be a nonconstant Suniti Then the point in P K 77 represented by 1 flslf M flslf is an S H integral point and h17flSlT7uqflSlT lSlThU 2 lSlT 14 This shows that d implies a The proof in can be easily modi ed to show Theorem 2 For convenience of readers we give an outline of the proof Proof of Theorem 2 Since k 7190sz GV if the set HLOaZUa a 6 Sn is linearly independent over 16 then there exists a positive integer m which depends only on aij such that the set Ugoawa a 6 Sn is linearly independent over Kpmi Let f0wfn represent a point 739 in P K 7 Hi Denote by Z Lif0wfni Let SH be a nite subset of C such that every coef cient of each linear form has no zero or pole outside 5H1 Therefore 1 gives vpli minvpf0 iiivpfn for P Cond739 U SH 1S 239 S qr 15 Since li 0 0 S i S n we may assume that fn 1 Let S Cond739 U 57 15 then implies that li is an STunit for 0 S i S 2n 11 Since L0 ML2n1 are in general position aij 0 Without loss of generality we let am 1 for every n 1 S i S 2n 1 Then we have the following STunit equations aiofo aiifi 39 39 39 aimilfnil 1ln1i7 0 S i S TL 1f a50f0a51f1 ina5n1fn1 and 1 are linearly independent over Kpm for some 0 S B S n then by the theorem of Wang for 0 S j S n 71 Hawk S hao0foyao1f17 u ya zilfnily 1 1 S pm l max07 2g 7 2 1571 10 JULIE TZUiYUEH WANG From the de nition of height we have the following abc estimate 1 l hf07 my fn S h0 i0f07 ailfl7u 7ain71f il7 ainfn ME my l Spm4mmI072972l H Z Mao 05139an lt wpmil max0 29 7 2 cond739 SH Z haiji 7 093an Therefore we only need to consider the case where each set aloft mainfn 0 S i S n is linearly dependent over Kpmi The next lemma shows that this is impossible if ULOaZUa a 6 Sn is linearly independent over Kpmi The theorem is then provedi D Lemma Let and aij 0 S ij S n be nonzero elements of a eld E If each set ai0f0ai1f1mainfn 0 S i S n is linearly dependent over a sub eld F of E then the set HLOaZUa a 6 Sn is linearly dependent over F Proof See WaSl 3 THE EXPLICIT BOUND FOR NUMBER FIELDS The proof of Theorem I can be adapted to the number eld case directlyi How ever the S unit theorem for number elds only provides an explicit bound on the number of S unit solutions Therefore our method can provide explicit bound on the number of S Hintegral points but can not say anything about the abc esti mate Let F be a number eld of degree d Denote by MF as the set of valuations of F and by M00 as the set of archimedean valuations of Fl We rst recall the S unit theorem by Schlickewei Sc Theorem Schlickewei Let a1man be nonzero elements of F Suppose that S is a nite subset of MF of cardinality s containing Moo Then the equation alzl anzn 1 16 has no more than 4sd235 d56 17 solutions in Sunits 11 qr such that no proper subsum ailzil aimzim vanishes When H is nondegenerate to provide an explicit bound on the number of S H integral points we can apply the same method in the proof of Theorem 1 Since it Pquot MINUS HYPERPLANES 11 is completely parallel to the function eld case7 we will only reproduce the parts which need the S unit theoremi Following the rst part of the proof of Theorem 17 we apply the S unit theorem to equation 4 in the number eld casei Then we showed that the number of the S unit solutions 57 in is no more than 4sdl235ud56i Therefore the number of 57 2 S i S u7 which satis es 47 is no more than 48d235ud56 Again if the dimension of the vector space spanned by L17 W Lu over F is n17 then without loss of generality we may assume that L17 m Ln1 are linearly independent over Fl Therefore the number of SHintegral points is equal to the number of 17 73 in lquotl which is bounded by 48dn236nn1d56 18 If the dimension of the vector space spanned by L17 W Lu over F is less than n17 then we can repeat the method in Theorem 1 and establish the same bound 18 for the number of SHintegral pointsi Therefore7 together with Ru s result cfi Ru we have the following Theorem 3 Let F be a number eld of degree d Suppose that S is a nite subset of MF of cardinality s containing Moo Let H be a set of hyperplanes in P H is nondegenerate if and only if the number of S7 Hintegral points of P F 7 H is nite Furthermore the number of SHintegral points of P F 7 H is bounded by mungewm 4 A GENERALIZATION OF THE PICARD S THEOREM A complex space M is called Brody hyperbolic if every holomorphic curve f C 7gt M is constant Ru proved the following cfi Rui Theorem Ru Let H be a set of hyperplanes in P C Then P C 7 H is Brody hyperbolic and only ifH is nondegenerate over C In we extended the classical Picard s theorem to the case where the coef cients of the linear forms corresponding to H are holomorphic functions In this section we will improve the results by adapting the proof of Theorem 1 First7 we explain our notation and terminologiesi Let X E 9172va7 j0 1 S i S 417 2 E C where gij are holomorphic functions and for each i7 giopiqgin 12 JULIE TZUiYUEH WANG has no common zeroes i Denote by zopiqzn E P Cl Li2zo 1 In 0 as the corresponding moving hyperplane of LZ27 1 S i S 417 2 E C and let 712 H12MHq2i Let for wfn be holomorphic functions without common zeroes We say that a holomorphic map f represented by fo7 m is a holomorphic map omitting 712 if fo27M7 f 0 for each 2 E C and i 17 m 41 Denote by HolC as the ring consisting of all holomorphic functions on C and MeroC as the eld consisting of all meromorphic functions on C One can iden tify Hi as a hyperplane in P HolCi Then L Llprq can be identi ed as a set of linear forms with holomorphic functions as coefficients Suppose that the set Li17 WLZm is linearly dependent over MeroC and any proper subset of Li17 WLZm is linearly independent over MeroCi Then we have a minimal relation ailLi1XaimLimX EU 19 where aij is a nonzero holomorphic function and ail 1 i i aim has no common zerosi De nition 71 is said to be unitary related if every holomorphic function aii which appears in any of the minimal relations 19 has no zero We also need the following Unit Theorem which is a consequence of the Borells Lemmai Unit Theorem Let uo7 gum be holomorphic functions without 2eroes and uo um 1 Suppose that no proper subsum uo um 7 1 0 vanishes then uo7 un are all constants The main result in this section is the following n Theorem 4 Let X E gij 2Xj7 1 S i S q 2 E C where gij are 390 holomorphic functions Denote by Hi the corresponding moving hyperplane of Ll27 1 S i S q Let71 H17 in Hg be unitary related Then 71 is nondegenerate over MeroC if and only if there exist nitely many n 1 X n 1 invertible matrices with holomorphic functions as entries such that every holomorphic map omitting 712 multiplied by one of the matrices is constant In addition this set of matrices depends only on the hyperplanes and can be determined e ectively Proof Since the proof is completely parallel to the proof of Theorem 17 we will only reproduce the necessary parts Let for wfn be holomorphic functions on C without common zeroes and fo7 m represents a holomorphic map f into P C omitting Let li Llme7 From 19 we have after rearranging the Pquot MINUS HYPERPLANES 13 index the following unit equation 1ailQmauluiq 20 a111 a111 7 allI a1l1 7 1 S i S u is constant The argument in the proof of Theorem 1 shows that where no proper subsum vanishesi Then by the Unit Theorem we have there exist after rearranging the index L1MLw C L and holomorphic units b1mbw chincw such that dimL1MLwK n 1 and 51 1 lt i S w is a nonzero constant In addition bi and Ci 1 S i S w are coef cients of some minimal relations as 19 Therefore there are only nitely many such holomorphic unitsi After rearranging the index we may assume that L1 MLn1 are linearly independent over MeroCi Hence f0 2 m multiplied by but 9102 152920 27 m Cnign1oz i i a b 91n27 92nz7 gn1nz is constant in P HolCi Since L1 M Ln1 are linearly independent over MeroC and bi Ci are units the matrix in 21 is invertible It is also clear from the proof that the matrices used in 21 can be determined effectively and the number of the matrices is nite Conversely if H is not nondegenerate over MeroC we assume that there exist nitely many n 1 X n 1 invertible matrices with holomorphic functions as entries such that every holomorphic map omitting Hz multiplied by one of the matrices is constant In the following proof we refer to Rub for some basic results and de nitions of Nevanlinna theoryi Let be a holomorphic function such that its characteristic function grows much rapidly than the characteristic function of the entries of the above matrices iiei Thu 0TT7 f for every entry az of the above matricesi Since H is not nondegenerate over MeroC there exists an H admissible subspace V of P MeroC of projective di mension greater than or equal to one such that H V does not contain at least three distinct hyperplanes in P MeroC which are linearly dependent over MeroCi We may assume without loss of generality that V P MeroCi Let H H1 in Hg Then 4 S n1 and H1 in Hq are linearly independent over MeroCi We may assume that H1 M Hq are the rst 4 coordinate planesi Let explnlfz is the function de ned recursively by exp llfz e z and explj1lfz eexpm zh Then the holomorphic map represented by 1 explllfz exp Qlfz M exp nlfz 14 JULIE TZUiYUEH WANG omits If A aij is one of the above matrices such that the product With this holomorphic map is constant then m5 2 aaj2expljlf2 ma 2 amzexpmfz7 22 170 170 Where 0 S a B S n ma and m5 are constant Since Traij 0Tr and Tr expljlf 0Trexpl71l 22 implies that mgaaj maagj 23 for 0 S j S n1 Therefore the determinant of the matrix A is zero Which contradicts the assumption that A is invertible Therefore the proof is completed REFERENCES BM Brownawell D and Masser D Vanishing Sums in Function Fields Math Proc Cambridge Philos Soc 100 1986 4277434 Bu Buiuln A The abc theorem for abelian varieties International Mathematics Research Notices 5 1994 2197233 CV Garcia A and Voloch J F Wronshians and linear independence in elds of prime char acteristic Manuscripta Math 53 1987 4577469 Ma Mason RC Diophantine Equations over Function Fields LMS Lecture Notes 96 Cam bridge Univ Press 1984 No Noguchi J Nevanlinnaecartan Theory and a Diophantine Equation over Function Fields J rein angeW Math 487 1997 61783 Ru Ru M Geometric and arithmetic aspects of Pquot minus hyperplanes Amer J Math 117 1995 3077321 Rub Rubel L A Entire and meromorphic functions Springer 1995 RW Ru M and Wong P7M Integral Points of Pquot 7 2n 1 hyperplanes in general position Invent Math 106 1990 1957216 Sc Schlickewei H P Seunit Equations over Number Fields Invent Math 102 1990 957107 Voj Vojta P Diophantine Approximations and Value Distribution TheorgLect Notes Math Vol 1239 Springer Berlin Heidelberg New York 1987 Vol Voloch J F Diagonal Equations over Function Fields Bol Soc Brazil Math 16 1985 29739 Wal Wang J T7Y The Truncated Second Main Theorem of Function Fields J of Number Theory 58 1996 1397157 WaZ Wang J T7Y Seintegral points of Pquot 7 2n 1hyperplanesingeneralposition over number elds and function elds Trans Amer Math Soc 348 1996 33793389 WaS Wang J T7Y Integral points of projective spaces omitting hyperplanes over function elds of positive characteristic preprint 1 Wa4 Wang J T7Y A generalization of Picard s theorem with moving targets Complex Varie ables and its pplication to appear lNS39l l39l C39l E OF M ATEEMATICS ACADEMIA SINICA NANKANG TAIPEI 11525 TAIWAN ROI Eemail address jwangQ mathsin1cae u tw OUTLINE OF A COURSE ON ELLIPTIC CURVES AND GROSSZAGIER THEOREMS OVER FUNCTION FIELDS ARIZONA WINTER SCHOOL 2000 DOUGLAS ULMER 1 REVIEW OF ELLIPTIC CURVES OVER FUNCTION FIELDS 0 De nitions and examples Constant7 isotriVial7 and non constant curves 0 The Mordell Weil theorem 0 Constant curves The lattices of Elkies7 Shioda7 et al 0 Torsion is uniformly bounded Ranks are unbounded o L functions o Grothendieck7s analysis of L functions gives analytic continuation7 functional equation 0 L functions should be Viewed as functions of characters of the idele class group 0 Zarhin7s theorem X gt gt LEX determines E up to isogeny o The conjecture of Birch and Swinnerton Dyer 0 Work of Tate and Milne ords1LEs 2 Rank with equality if and only if M is nite 0 Outline of the proof 7 The elliptic surface SFq corresponding to LEs det17 q sFrlH for a certain H Q H26 Oz 7 Points on E correspond to curves on S Heights are essentially intersection numbers 7 Cycle classes of curves give rise to zeroes of the L function Finiteness of M gt weak BSD comes from the Kummer se quence on S and M Br 0 Other work Brown7 Riick Tipp7 Longhi7 Pal 0 References Gross7 Zarhin7 Groth7 Milne807 Tate667 Milne757 C Z7 Cross in Storrs Date March 157 2000i DOUGLAS ULMER 2 AUTOMORPHIC FORMS AND ANALYTIC MODULARITY o Additive characters7 multiplicative characters7 conductors and real parts 0 De nition of x4K7 i57 automorphic forms of level K and central character 15 o Analogue with functions on upper half plane The double coset space X where automorphic forms live 0 X is the set of isomorphism classes of rank 2 vector bundles with level structure up to twisting by a line bundle 0 Structure of X Riemann Roch and stability 0 Petersson inner product 0 Cusp forms 0 Hecke operators7 new and old forms 0 Fourier expansions o L functions 0 Functional equations 0 Harmonic forms 0 Constructions of forms7 classically and in terms of vector bundles Eisenstein series Poincare series Theta functions Converse theorems Deligne7s theorem there is a form f such that LEs LU 5 Drinfeld7s geometric Langlands construction 0 Interesting linear functionals on AK are represented as PIP with interesting forms f E x4K7 b 0 Half of the Gross Zagier computation is to nd the Fourier expan sion of the form representing f gt gt L Kf7 1 0 References Weill7 Serre7 Gek7 Del7 Drin83 GROSS ZAGIER OVER FUNCTION FIELDS 3 3 DRINFELD MODULAR CURVES AND GEOMETRIC MODULARITY o The ring A of functions regular outside 00 o For k of characteristic p7 EndkGa is the twisted polynomial ring k7 7 Ta apT 0 De nition of Drinfeld modules Rank7 characteristic7 height 0 Examples 0 Morphisms 0 Division points 0 lsogenies o Endomorphisms 0 Complex multiplication 0 Level structures 0 Modular curves 0 Analytic description of Drinfeld modular curves 0 The adelic version of the analytic description 0 The building map 0 Drinfeld reciprocity relating the cohomology of the modular curve to automorphic forms 0 Geometric modularity via Drinfeld reciprocity7 Deligne converse theorems7 and Zarhin 0 References Drin747 D H7 G R7 AB7 Ohio DOUGLAS ULMER 4 OVERVIEW OF THE GROSS ZAGIER COMPUTATION AND APPLICATION TO ELLIPTIC CURVES o Heegner points on X001 existence7 construction Galois action The Heegner point PK 6 J0n 0 Goal L Kf7 1 c htPKf c a non zero constant for new eigen forms f E AP0nool l 0 Key reduction do it for all f at once i Let hm be the form such that f7 hmp1p L Kf7 1 i Let hazy be the form with Fourier coef cients ltPK7TmPKgtht A formal Hecke algebra argument shows that the goal is equiv alent to the equality hm chm Prove this coef cient by coef cient 0 The analytic computation Rankin7s method shows that LKfs f7hsp1p where hS is the product of a CM form theta series and an Eisenstein series which is a function of s 7 Compute a trace to make the level of hS noo Take the derivative at s 1 h ih 51 Do a harmonic projection nd hm harmonic such that f7 hmp1p f7 hp1p for all harmonic forms f o The algebraic computation lnterpret height as a sum of local intersection numbers 7 At nite places7 intersection number counts the number of isogenies between certain Drinfeld modules L y over nite rings OvWZ Use the moduli interpretation of points 7 Count these isogenies using the ideal theory of the quaternion ring End At 00 there is no convenient moduli interpretation Compute the local height using a Green7s function7 exactly as in the original G Z This is a very analytic way to calculate a ratio nal number7 but it meshes well with analytic aspects of the harmonic projection calculation 0 Application to elliptic curves Show ords1 LEs S 1 i BSD for EF by using G Z formula and non vanishing results for L functions In function eld case7 non vanishing results are used for some useful preliminary reductions7 and to nd a good 0 References G Z AB CZl Del DHl Drin74 Drin83 Gek GR Gross Groth G Z1 Milne75 Milne80 Ohio Serre Storrs Tate66 Tate75 Weill Zarhin GROSS ZAGIER OVER FUNCTION FIELDS 5 REFERENCES Gekeler EiUi et all Eds Drinfeld Modules Modular Schemes and Ap plications Proceedings of a workshop at AldenBiesen Belgium 1996 World Scienti c Singapore 1997 Cox Di and Zucker Si Intersection numbers of sections of elliptic sur faces lnventiones Math 53 1979 1744 Deligne Pi Les constantes des equations fonctionnelles des functions L In Modular functions of one variable 11 Lecture Notes in Math 349 1973 5017597 Deligne Pi and Husemoller Di Survey of Drinfeld modules Contempo rary Math 67 1987 25791 Drinfeld ViGi Elliptic modules Russian Mat Sb NS 94 136 1974 5947627 656 Drinfeld ViGi Twodimensional ladic representations of the fundamen tal group of a curve over a nite eld and automorphic forms on GL2 Amer 1 Math 105 1983 857114 Gekeler EiUi Automorphe Formen uber FqT mit kleinem Fuhrer Abhi Mathi Semi UniVi Hamburg 55 1985 1117146 Gekeler EiUi and Reversat Mi Jacobians of Drinfeld modular curves 1 Reine AngeWi Math 476 1996 27793 Gross Bi Hi Group representations and lattices J Amer Math Soc 3 1990 9297960 Grothendieck Al Formule de Lefschetz et rationalit des fonctions L Seminaire Bourbaki 196566 Expose 279 Gross Bi and Zagier Di Heegner points and derivatives of Lseries lnventi Math 84 1986 2257320 Milne JiSi On a conjecture of Artin and Tate Annals of Math 102 1975 5177533 Milne JiSi Etale Cohomology Princeton University Press Princeton 1980 Goss Di et al Eds The Arithmetic of Function Fields Proceedings of a conference at Columbus OH 1991 de Gruyter Berlin 1992 Serre JiPi Trees Springer Berlin 1980 Cornell Cl and Silverman Ji Eds Arithmetic Geometry Proceedings of a conference at Storrs CT 1985 Springer 1986 Tate 1 On the conjecture of Birch and SwinnertonDyer and a geomet ric analog Seminaire Bourbaki 196566 Expose 306 Tate 1 Algorithm for determining the type of a singular ber in an elliptic pencil 1n Modular Forms of One Variable 1V Lecture Notes in Math 476 1975 33752 Weil A Dirichlet series and automorphic forms Lecture Notes in Math 189 Springer Berlin 1971 Zarhin Jul Gi A niteness theorem for isogenies of abelian varieties over function elds of nite characteristic Russian Funkcionali Anali i PriloZeni 8 1974 31734

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