Linear Algebra and Differential Equations
Linear Algebra and Differential Equations MATH 320
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THE ARITHMETIC OF BORCHERDS EXPONENTS JAN H BRUINIER AND KEN ONO 1 INTRODUCTION AND STATEMENT OF RESULTS Recently Borcherds 13 provided a striking description for the exponents in the naive infinite product expansion of many modular forms For example if Ek denotes the usual normalized weight k Eisenstein series let Cn denote the integer exponents one obtains by expressing E4z as an infinite product 11 E4Z1 240 i Z dag 7 17 IVWG 7 q226760 10 011 7 171 711 din 711 q 2 62quot throughout Although one might not suspect that there is a precise description or formula for the exponents Cn Borcherds provided one He proved that there is a weight 12 meromorphic modular form 02 Z bnq q 3 4 7 240g 26760q4 7 4096240q9 71273 with the property that 001 19012 for every positive integer n It is natural to examine other methods for studying such exponents Here we point out a p adic method which is based on the fact that the logarithmic derivative of a meromorphic modular form is often a weight two p adic modular form To illustrate our result use 11 to define the series Cq 0 12 0q 6 Z Z cddq 71440q 319680q2 i 73733760q3 711 din Both authors thank the Number Theory Foundation and the National Science Foundation for their generous support The rst author acknowledges the support of a Heisenberg Fellowship The second author acknowledges the support of an Alfred P Sloan Foundation Fellowship a David and Lucile Packard Foundation Fellowship an H l Romnes Fellowship and a John S Guggenheim Fellowship Typeset by AMS TEX 2 JAN H BRUlNlER AND KEN ONO lf 712 1124 H1l 7 q denotes Dedekind7s eta function then it turns out that Cq E q 7 9g2 qu3 7 2q4 7 q5 E n2zn2llz mod 11 Therefore if p 37 11 is prime then aEp E 1 6epp mod 11 where aEp is the trace of the pth Frobenius endomorphism on X0ll This example illustrates our general result Let K be a number field and let OU be the completion of its ring of integers at a finite place i with residue characteristic p Moreover let A be a uniformizer for OH Following Serre S2 we say that a formal power series f Zaltngtqi 6 OquH 710 is ap adic modular form of weight k if there is a sequence fi 6 OUHqH of holomorphic modular forms on SL2Z with weights hi for which ordfi 7 f 7 00 and ordk 7 7 00 Theorem 1 Let qh l 7 221 anq E OKHqH be a meromorphic modular form on SL2Z where OK is the ring of integers in a number eld K Moreover let Cn denote the numbers de ned by the formal in nite product FZ qh Hlt17 a c 711 pr is prime and is good at p see 3 for the de nition then the formal power series B h 7 22mm 711 dl71 is a weight two p adic modular form Here we present cases where is good at p As usual let be the modular function 32 q 1 744 196884q 21493760q2 Let H be the upper half of the complex plane We shall refer to any complex number 739 E H of the form 739 717 W with a b c E Z gcda b e l and b2 7 4ac lt 0 as a Heegner point Moreover we let dT 2 b2 7 4ac be its discriminant The values ofj at such points are known as singular moduli and it is well known that these values are algebraic integers A meromorphic modular form on SL2Z has a Heegner diuisor if its zeros and poles are supported at the cusp at infinity and Heegner points Although we shall emphasize those forms which have Heegner divisors we stress that Theorem 1 holds for many forms which do not have a Borcherds product For example Ep1z is good at p for every prime p 2 5 The next result describes some forms with Heegner divisors which are good at a prime p BORCHERDS7 EXPONENTS 3 Theorem 2 Let qh 1 1 221 anq E ZHqH be a meromorphic modular form on SL2Z with a Heegner diuisor whose Heegner points 7172 rs E llllSL2Z have red discriminant d The following are true 1 pr 2 5 is a prime for which 6 071 and Hjmxjm i 1728 0 mod p 21 then is good at p 2 Ifs 1 and 7391 71 x732 resp 7391 i then is good at every prime p E 23 5 11 mod 12 resp p E 23 7 11 mod 12 3 If 2 resp p 3 and ldl E 3 mod 8 resp ldl E 1 mod 3 then is good at p 4 Suppose that p 2 5 is a prime for which E 0 71 and Hj rZ E 0 mod p 2391 IfQd 31 Qx73 or 71 then is good at p 5 Suppose that p 2 5 is a prime for which E 0 71 and s Horn 7 1728 E 0 mod p i1 If 31 or 71 then is good at p Remarks 1 Since 1728 resp j71 x732 0 Theorem 2 2 applies to the modular form 7 1728 resp jz as well as the Eisenstein series E6z resp E4z 2 By the theory of complex multiplication the singular moduli j rl j rs associated to the points in Theorem 2 form a complete set of Galois conjugates over Q and the multiplicities of each 71 is xed in the divisor of 3 For fundamental discriminants d the work of Gross and Zagier G Z provides a simple description of those primes p Which do not satisfy the condition in Theorem 2 4 Theorem 2 admits a generalization to those forms With algebraic integer coe icients and Heegner divisors In particular it can be modified to cover such forms Where the multiplicities of the 72 in the divisor of are not all equal 4 JAN H BRUINIER AND KEN ONO Theorem 2 has interesting consequences regarding class numbers of imaginary quadratic elds If 0 lt D E 03 mod 4 then let H7D be the Hurwitz class number for the discriminant 7D For each such D there is a unique meromorphic modular form of weight 12 on P04 which is holomorphic on the upper half complex plane whose Fourier expansion has the form Lemma 142 B 13 fD2 q D Z 013an 6 leall 1 nEO1 mod 4 Borcherds7 theory implies that 0 14 FD2 q HHD H1QWCDW2 711 is a weight zero modular function on SL2Z whose diVisor is a Heegner diVisor consisting of a pole of order H7D at z 00 and a simple zero at each Heegner point with discriminant 7D At face value to compute this correspondence one needs the coef cients of fD z and the class number H7D Here we obtain in many cases a p adic class number formula for H7D in terms of the coef cients of fD Therefore in these cases the correspondence is uniquely determined by the coef cients of fD Corollary 3 If0 lt D E 03 mod 4 and 7D is fundamental then the following are true 1 IfD E 3 mod 8 then as 2 adz39c numbers we have H7D i i is 4 2 24 710 D 2 IfD E 1 mod 3 then as 3 adz39c numbers we have M D 1 f WW 7 7 c 12 710 D 3 IfD E 023 mod 5 then as 5 adz39c numbers we have M8 H7D cD25 5 l 6 l l o n 4 IfD E 0 l 24 mod 7 then as 7 adz39c numbers we have H7D cD49 7 n l l o BORCHERDS7 EXPONENTS 5 ACKNOWLEDGEMENTS The authors thank Scott Ahlgren Winfried Kohnen Barry Mazur and Tonghai Yang for their helpful comments and suggestions 2 PRELIMINARIES We recall essential facts regarding meromorphic modular forms on SL2 Z and the arith metic of infinite products If Fq 271 anq then let G be the standard differential operator on formal q series defined by 7 21 FQ 2 WWW 712710 Throughout let Fq be a formal power series of the form 22 F01 qh 1 0 WW1 7 711 and let the 001 be the numbers for which 23 F01 qh 10 0117 a c 711 Proposition 21 If Fq and the numbers 001 are as in and then 24 1a 7 Z 000 As formal power series we have logltFltqgtgt mm Z cltngtloglt1i q mm 7 Z cltngt Z 711 711 m1 iogqh Z Hg 6 JAN H BRUINIER AND KEN ONO By logarithmic differentiation with respect to q we obtain quq 9ltFltqgtgt m m W M Wh zHq q hizzcmm D Following Ramanujan let Pz denote the nearly modular Eisenstein series 25 132 1 24iqu 711 din Lemma 22 Let Fq be a weight k meromorphic modular form on SL2Z satis fying If the numbers Cn are as in then there is a weight k 2 meromorphic modular form on SL2Z for which 1 F 2 00 E 11132 h i Z Z Cddq 711 dl71 If is a holomorphic modular resp cusp form then is a holomorphic modular resp cusp form Moreover the poles of are supported at the poles of Proof It is well known p 17 O that the function de ned by 2 129F2 7 is a meromorphic modular form of weight k2 on SL2 Moreover if is a holomorphic modular resp cusp form then is a holomorphic modular resp cusp form The result now follows immediately from Proposition 21 D The remaining results in this section are useful for computing explicit examples of The orem 1 and for proving Theorems 2 and 3 As usual if k 2 4 is an even integer then let Ek denote the Eisenstein series 2k 26 Em 217 37 Z ak1nqi 711 Throughout let to be the cube root of unity 27 to z BORCHERDS7 EXPONENTS 7 Lemma 23 Suppose that k 2 4 is even 1 We have E 1 mod 24 2 If 2 5 is prime and p 71 1k then El mod p 3 Ifk i 0 mod 3 then Ekto 0 4 Ifk E 2 mod 4 then 0 Proof Since 2 i resp z to is fixed by the modular transformation 52 712 resp A2 72 1z the de nition of a modular form implies that 0 whenever k E 2 mod 4 and Ek to 0 whenever k i 0 mod 3 The claimed congruences follows imme diately from 26 and the von Staudt Clausen theorem on the divisibility of denominators of Bernoulli numbers p 233 l R D 3 PROOFS OF THE MAIN RESULTS We begin by de ning what it means for a modular form to be good at p De nition 31 Let qh 1 1 221 anq E OKHqH be a meromorphic modular form on SL2Z whose zeros and poles away from 2 00 are at the points 212225 We say that is good at p if there is a holomorphic modular form 82 with p integral algebraic coe cients for which the following are true 1 We have the congruence 82 E 1 mod p 2 For each 1 S i S s we have 0 Proof of Theorem 1 By Lemma 22 there is a weight k 2 meromorphic modular form on SL2Z whose poles are supported at the poles of Fz for which 1 F 2 00 E 102 h i Z Z Cddq n1 dln Since Serre S2 proved that Pz is a weight two p adic modular form it suf ces to prove that is a weight 2 p adic modular form For every j 2 0 we have 31 my 2 1 mod p71 Since has weight two it follows that SzijzFz for suf ciently large j is a holomorphic modular form of weight pjb 2 where b is the weight of lf SzijzFz does not have algebraic integer coef cients then multiply it by a suitable integer tj1 E 1 mod pl so that the resulting series does Therefore by 31 the se quence 33112 tj18zpj 39zFz defines a sequence of holomorphic modular forms 8 JAN H BRUINIER AND KEN ONO which p adically converges to with weights which converge p adically to 2 In other words is a p adic modular form of weight 2 D Proof of Theorem 2 In view of De nition 31 it suf ces to produce a holomorphic modular form 82 on SL2Z with algebraic p integral coef cients for which 8Ti 0 for each 1 S i S s which satis es the congruence 82 E 1 mod 1 First we prove For each 1 S i S 3 let A be the elliptic curve 32 A z y x3 A 108jltngtltjltngt e 1728gtx e 432jnjn e 1728 Each AZ is defined over the number field Qj7 i with j invariant j rZ A simple calculation reveals that if p is a prime ideal above a prime p 2 5 in the integer ring of Qj for which 33 jnjn1728 0 mod P then A has good reduction at p Suppose that p 2 5 is a prime which is inert or rami fied in satisfying 33 for every prime ideal p above p By the theory of complex multiplication p 182 L it follows that j rZ is a supersingular j invariant in Np A famous observation of Deligne see for example 81 Th 1 K Z implies that every supersingular j invariant in characteristic 1 is the reduction of jQ modulo 1 for some point Q which is a zero of Ep12 Therefore there are points Q1 Q2 Q5 in the fundamental domain of the action of SL2Z not necessarily distinct for which Ep1QZ 0 for all 1 S i S s with the additional property that 34 HltX 7 M2 2 HltX A M mod p in FAX Now define 82 by 7 s j2jn 3 5 82 EP M M 7 M2 By Lemma 23 2 34 and 35 it follows that 8Ti 0 for each 1 S i S s and also satisfies the congruence 82 E 1 mod 1 Moreover 82 is clearly a holomorphic modular form and so is good at p This proves Since 1728 resp jw 0 Lemma 23 shows that is good at every prime p E 23711 mod 12 resp p E 23511 mod 12 This proves BORCHERDS7 EXPONENTS 9 To prove 3 one argues as in the proof of 1 and 2 using Lemma 23 1 3 4 and the Gross and Zagier congruences Cor 25 G Z ldl E 3 mod 8 gt j rZ E 0 mod 215 ldl 21 mod 3 mi 2 1728 mod 36 In View Of 27 to PrOVe 4 and 5 we may assume that H loWU i 1728 7g 0 i1 We use a classical theorem of Deuring on the reduction of differences of singular moduli mod ulo prime ideals p In particular if Z QiQx73 then since j rl j rs forms a complete set of Galois conjugates over Q Deuring7s result implies that see Th 1321011DD 36 j7 i E 0 mod p gt p E 2 mod 3 2391 37 7 1728 E 0 mod p gt p E 3 mod 4 2391 The same conclusion in 36 resp 37 holds if Qx73 resp provided that 191 d A straightforward modification of the proof of 1 using Lemma 23 2 3 4 now proves that is good at p D Proof of Corollary 5 Since 7D is fundamental Theorem 2 shows that FD z is good at the primes p as dicated by the statement of Corollary 3 Define integers AD by 38 iADnq iZcDd2dq 711 711 din Therefore by the conclusion of Theorem 1 it follows that 39 7Hlt7Dgt e 2 wow 711 is a weight two p adic modular form for the relevant primes p S 7 10 JAN H BRUINIER AND KEN ONO Serre proved Th 7 S2 for certain p adic modular forms that the constant term of the Fourier expansion is essentially the p adic limit of its Fourier coef cients at exponents which are pth powers In these cases we obtain i iimnnloo AD2 11 D E 3 mod 8 H D 5111mm AD3 11 D E 1 mod 3 7 hmnnloo 14135 11 D E 0 23 mod 5 iiimnnloo ADm 11 D E 0124 mod 7 4 SOME EXAMPLES Example 41 Let f7z 2277 07nq be the weight 12 modular form on F04 de ned in 13 lts q expansion begins with the terms g f7 4119q 8288256114 7 52756480115 By the Borcherds isomorphism Th 141 B there is a modular form of weight 0 on SL2Z with a simple pole at 00 and a simple zero at z 1 1772 with the Fourier expansion 1 F7 z f1 l l 17 q c7 2 7 4119 196884q 21493760q2 864299970q3 q 711 Since j1 1772 7153 E 0 mod 5 by the proof of Theorem 2 4 there is a weight 6 holomorphic modular form on SL2Z which is congruent to 717 Z 207d2dq E 4 4q 2g2 q3 mod 5 711 dl71 This is 4E6z mod 5 and so 07012 E 1 mod 5 if n i 0 mod 5 Example 42 Here we illustrate the class number formulas stated in Corollary 3 If D 59 then we have that f59 z 1 59 Z 059nq 1 59 7 30197680312q 455950044005404355712q4 711 By Corollary 3 1 we have 1 0 H759 37 4712 lt gt 24 gem gt One easily checks that the first two terms satisfy 1 H59 3 E 730197680312 455950044005404355712 2 mod 29 BORCHERDS7 EXPONENTS 1 1 REFERENCES RI EI Borcherds Automorphic forms on 052 2R and in nite products lnventiones Mathe maticae 120 1995 161213 DI Cox Primes of the form m2ny2 Fermat Class Field Theory and Complex Multiplication Wiley Publi New York 1989 MI Deuring Teilbarkeitseigenschaften der singularen moduln der elliptischen funktionen und die diskriminante der klassengleichung Comml Mathi HelVI 19 1946 7482 B Cross and DI Zagier On singular moduli JI reine angewl mathl 355 1985 191220 KI Ireland and MI Rosen A classical introduction to modern number theory SpringerVerlag New York 1990 MI Kaneko and DI Zagier Supersingular jinvariants hypergeometric series and Atkin s or thogonalpolynomials Comp perspectives on Number Theory Chicago Illinois 1995 AMSIP Studl AdVI Mathl Amer Math Soc 7 1998 97126 SI Lang Elliptic functions 2nd edition SpringerVerlag 1987 AI Ogg Sur39oey of modular functions of one variable Springer Lect Notes in Math 320 1973 136 JIPi Serre Congruences et formes modulaires d apres H P F SwinnertonDyer Seml Bour baki 416 19711972 7488 JIPi Serre Formes modulaires et fonctions zeta padiques Springer Lect Notes in Math 350 1973 191268 GI Shimura Introduction to the arithmetic of automorphic forms lwanawi Shoten and Prince ton UniVI Press 1971 DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON WISCONSIN 53706 E mail address bruinieerathwiscedu DEPARTMENT OF MATHEMATICS UNIVERSITY OF WISCONSIN MADISON WISCONSIN 53706 E mail address ononathwiscedu FW Math 320 TakeHome Exam Thur Sept 26 2002 This is a TAKE HOME EXAM You must work on this alone by yourself With no one s help besides your own You can use the book Edwards and Penney your class and homework notes but no other reference material So any information obtained from the Internet in any way is also forbidden except of course copies of this exam You may use your calculator You can work on this for as long as you want but must state how long it took you and turn it in by the due date and time 120 pm Monday Sept 30 2002 YOUR NAME TIME for completion 1 Solve ww1ysinwiwy y0 1 Your answer may contain integrals but all constants and limits of integration must be speci ed Estimate y7r 2 Make a beautiful hand sketch no computer plot of solution curves of y y 1 What can you say about limzn0yw about limgHiFOO yw about the solution with initial condition y0 yo a 0 Extra credit same questions but for wzy y w 3 An initially empty cylindrical tank of constant cross section A is lled with water at the constant rate k The water drains from a hole of cross section a at the bottom of the tank under the action of gravity Assume Torricelli7s law for the water velocity at the hole a Draw a sketch of this problem identifying all relevant variables b Derive the di erential equation governing the evolution of the water depth h as a function of time t c Determine the water depth as t a 00 explaining brie y your answer assume the tank is deep enough that it does not over ow d Find a de nite integral expression relating h and t lntegrate 3 Edwards and Penney 2120 4 Edwards and Penney 2320 5 The mass m8 of an electron is much less than the mass Mp of a proton so as a rst approximation we can assume that the proton does not move during the interaction of an electron and a proton just like we neglect the e ect of a space shuttle on the earth Assume that the interaction force between the proton and the electron is Fr E ar2 7 br4 where r is the distance from the center of the proton to the center of the electron a and b are positive constants and the force is attractive if Fr gt 0 repulsive if Fr lt 0 a Derive the equation that governs the radial velocity v E drdt of the electron where t is time b What is the equilibrium position of the electron c ls that equilibrium stable or unstable d What is the escape velocity for an electron located at the equilibrium position 6 Use elimination to answer Edwards and Penney 3228 Make sure you clearly identify what operations you are performing on the equations to eliminate variables for instance by writing E1 4 E2 E etc
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