Rational Points over Finite Fields on a Family of Higher Genus Curves and Hypergeometric Functions. Yih Sung

Transcription

1 TAIWANESE JOURNAL OF MATHEMATICS Vo. xx, No. x, pp. 1 25, xx 20xx DOI: /tjm.xx.20xx.7724 This paper is avaiabe oie at Ratioa Poits over Fiite Fieds o a Famiy of Higher Geus Curves ad Hypergeometric Fuctios Yih Sug Abstract. I this paper we ivestigate the reatio betwee the umber of ratioa poits over a fiite fied F p o a famiy of higher geus curves ad their periods i terms of hypergeometric fuctios. For the case y = xx 1x λ we fid a cosed form i terms of hypergeometric fuctios associated with the periods of the curve. For the geera situatio y = x a1 x 1 a2 x λ a we show that the umber of ratioa poits is a iear combiatio of hypergeometric series, ad we provide a agorithm to determie the coefficiets ivoved. 1. Itroductio 1.1. Backgroud The Legedre famiy of eiptic curves is defied expicity by X λ = { y 2 = xx 1x λ } o C 2, where λ C {0, 1} = PC 1 {0, 1, }. It is geeray uderstood that the umber of ratioa poits of X λ over a fiite fied F p with the prime p is reated to a period itegra o X λ, which i tur is reated to the Gauss hypergeometric series 2 F 1 1 2, 1 2, 1; λ moduo p. This hypergeometric series is a soutio of a hypergeometric differetia equatio i which the derivatives are give by the Gauss-Mai coectio of the famiy. The first goa of this paper is to uderstad correspodig situatios for more geera famiies of Riema surfaces {X λ } of higher geus. We wat to give a expicit formua of the umber of ratioa poits o X λ over a fiite fied F p with the prime p i terms of period itegras or hypergeometric series, as i the case of the Legedre famiy. We are particuary iterested i famiies associated with triage groups, i which the Legedre famiy is a specia case. It is importat to ote that a fibre curve X λ i this famiy may have siguarities, which makes the situatio more compicated ad iterestig. We aso ivestigate F p Received Jue 22, 2016; Accepted August 16, Commuicated by Sai-Kee Yeug Mathematics Subject Cassificatio. 14G05, 0F0, C05. Key words ad phrases. fuctios. because Riema surfaces, Ratioa poits, Hoomorphic differetias, Hypergeometric 1

2 2 Yih Sug the case of > 1 is more subte tha the case of = 1. We wi fiay cosider the couts moduo p ad moduo p. The former situatio is expaied competey i this paper. For the atter situatio, we wi provide exampes to demostrate that the probem at had is more subte so that the geera probem remais ope. The cassica correspodece betwee the period of X λ ad the umber of ratioa poits o X λ over F p ca be proved through brute force, as show i [1]. cacuatio, the umber of ratioa poits o X λ is 1.1 p 1/2 X λ 1 p 1/2 r=0 = 1 p 1/2 2F 1,p 1/2 1 p 1/2 2 λ r mod p r 2, 1 p 2, 1; λ mod p. By direct To carify the subidex of F, p 1/2 refers to the trucatio i the summatio. Note that the Gauss hypergeometric fuctio 2 F 1 a, b, c; λ satisfies a secod-order differetia equatio 1.2 xx 1 d2 u + a + b + 1x cdu + ab u = 0. dx2 dx It is surprisig that the umber of ratioa poits o X λ is reated to a soutio of a differetia equatio defied o the base of the famiy. I papers [] ad [4] Mai expaied this pheomeo by appyig the Lefschetz Fixed Poit Formua. Sice h 0 X λ, K = 1 the hoomorphic differetia ω λ = dx/y geerates H 0 X λ, K. Mai observed that by takig the oca coordiate x of X λ ad fixig a base poit q, ω λ ca be expressed as ω λ = dx + r 1 a r x xq r dx. The by the Lefschetz fixed-poit theorem Mai showed that a p 1 satisfies the Picard- Fuch equatio 1.2 moduo p. Therefore periods of X λ are reated to the umber of ratioa poits o X λ moduo F p ad satisfy the hypergeometric equatio Statemet of resuts We cosider the famiy of curves defied by y = xx 1x λ ad y = x a 1 x 1 a 2 x λ a with assumptios that a 1, a 2, a Z >0,, a 1, a 2, a = 1 ad α = a 1 + a 2 + a. For the first famiy, we offer a formua i a cosed form: Theorem 1.1. Let m 4, be itegers. Let Xλ m be the famiy of agebraic curves defied by y m = xx 1x λ over the fiite fied F q, q = p, with the parameter λ Q. Let = m, q 1, so that satisfies q 1. If = 1, the umber of ratioa poits o X λ is X m λ,p 0 mod p.

3 Ratioa Poits ad Hypergeometric Fuctios If 2, et S reg ad S irr be sets such that { S reg = 0, s, 1, s ad deote a = 2 s+1 X m λ,q where δ =, 1, ad M r,s = [ 2 1 s { S irr = 0, s ] 2, 0 s 0 s < } 2 1, r, b = 1 s+1, c = 2 r,s S reg k r,s 2F 1,Nr,s a, b, c; λ + N r,s = 1 1 s+1 [ ] } 2 2, r = 2b r. The r,s S irr k r,s λ Mr,s 2 F 1,N r,s a c + 1, b c + 1, c + 2; λ δ mod p, 2 r 2s + 1 s + 1 q 1, N r,s = s + 1 s 1q 1 q 1, k r,s = 1 Nr,s N r,s I a more geera case, we derive the foowig resut. s 1q 1 q 1, k r,s = 1 Nr,s. M r,s Theorem 1.2. Let m 4, be itegers, ad Yλ be the famiy of agebraic curves defied by y = x a 1 x 1 a 2 x λ a over the fiite fied F q where q = p. Assume, a 1, a 2, a Z >0,, a 1, a 2, a = 1, α = a 1 + a 2 + a, ad q 1. The Y λ,q k=1 α B c α F Nα a α, b α, c α ; λ m δ k,m λ m mod p, where B is a basis of hoomorphic oe forms o Y λ ad F N α a α, b α, c α ; λ are the associated hypergeometric fuctios for some a α, b α, c α Q, ad δ k,m are ratioa umbers refectig the siguarities of the curves. A expicit agorithm to fid the costats ivoved i the above theorem is preseted i the appedix. We ote that to combie the cassica coutig techique ad the Lefschetz fixed-poit theorem is ecessary. If we appy oy cassica coutig methods, it is difficut to see how coutig is reated to the periods of hoomorphic differetias; o the other had, if we appy oy the Lefschetz fixed-poit theorem we caot determie precise costats for each hypergeometric fuctio.

4 4 Yih Sug Compared to the Legedre famiy of eiptic curves, there are a few sigificat differeces that we eed to address. First, the agebraic curves we are iterested have siguarities. Therefore, we eed to appy ormaizatio or to use a desiguarisatio mode of the curves i order to appy the Lefschetz Fixed Poit Formua. The differece i coutig o the umber of ratioa poits i the affie part of the ormaizatio ad the curve itsef gives rise to a expressio that we ca the correctio term i this artice, represeted by δ ad δ k,m i the above theorems. Secody, there are more tha oe choice of basis of the space of hoomorphic differetias. Thus, we eed to cosider a appropriate iear combiatio of period itegras or appropriate hypergeometric fuctios i order to compute the expicit coefficiets i Theorem 1.2. Fiay, we cosider the fiite fied F q where q = p ad > 1. For most of our resuts, we cosider the umber of ratioa poits i F p moduo p. The situatio of ratioa poits i F p moduo p wi be expaied by expicit exampes i the ast sectio. 1.. Cotets Throughout this artice, we assume that λ C {0, 1}. We derive the cosed formua for the case y = xx 1x λ i Sectio 2. I Sectio, we give a agorithm to hade the case y = x a 1 x 1 a 2 x λ a. I Sectio 4, we remark o a extesio of resuts from the fiite fied F p to F p for eiptic curves ad make some observatios about the trucatio eves of the hypergeometric fuctios ivoved. I the ast sectio, we ist severa exampes to iustrate subte poits i the formuatios ad computatios of our theorems. 2. Case of X defied by y = xx 1x λ 2.1. Geus formua ad Abeia differetias Let us cosider a famiy of curves X λ defied by y = wx = xx 1x λ over the fiite fied F q where q = p. Let C λ be a smooth mode of the projectivizatio of X λ. Let X λ,q be the curve defied over F q. For brevity, we drop the depedece o λ ad q ad simpy deote a curve i the famiy by X. By the defiig equatio, we kow H 0 X, K X is geerated by the Abeia differetias 2.1 ω r,s = x r y s dx y 1 = xr [xx 1x λ] [s 1]/ dx ad we wi fid the appropriate rage of r ad s ater. For 4, by checkig the smoothess at after chage of coordiates we may simpy appy the geus formua. For

5 Ratioa Poits ad Hypergeometric Fuctios 5 > 4, after compactificatio i P 2, the curve is defied by W 1 = W 0 W 0 W 2 W 0 λw 2 W 2. Speciaize to the affie ope set U W0 =1, ad the curve is defied by y = 1 z1 λzz which has a siguarity at 0, 0. Let C be a smooth mode of X. To fid the geus of C, the stadard method is to appy the Hurwitz Formua. Lemma 2.1. [2, Theorem ] Let > 4. a The geus of C is give by 2 if, gc = 1 if. b Deoted by [a] the itegra part of a. A basis of hoomorphic oe forms o C is give by dx y i ad x dx, where [ 2 y j ] + 1 i 1 ad [ ] + 1 j 1. After resovig the siguarities of X, we get a smooth mode C i P 2 P 1 P 1. The coordiates are x, y, z; z 1, t 1 ; y 1, w 1 ;... ; y i, w i ;... ad C is defied by y = xx 1x λ ad the associated equatios of bowup. Oce x ad y are determied, the rest of the vaues are determied accordigy. Thus, away from the siguarities ad their preimage o the bowup there is a oe-oe correspodece of ratioa poits betwee X ad C. This impies that we ca cout the umber of ratioa poits o C. Sice the Lefschetz Fixed Poit Formua requires that the curve is smooth, we must cosider the smooth mode C rather tha X. The we cosider the Frobeius map F b x, y, z; z 1, t 1 ; y 1, w 1 ;... ; y i, w i ;... = x q, y q, z q ; z q 1, tq 1 ; yq 1, wq 1 ;... ; yq i, wq i ;..., ad the cassica argumet appies. For the computatio of the trace map, we ocaize the computatio to a affie ope set U of C by choosig U = C = X. The we take the oca parameter x to cotiue o the computatio Hypergeometric fuctios ad periods By Lemma 2.1, we kow that the basis of hoomorphic 1-forms ca be chose as 2.2 ω 0,s, 0 s [ ] 2, ad ω 1,s, 0 s [ ] 2 2.

6 6 Yih Sug Recaig the formua of the period 2F 1 a, b, c; x = Γc ΓbΓc b 1 0 t b 1 1 t c b 1 1 xt a dt, ad comparig ω r,s with the differetia i the itegra, we have 2. a = s 1, b = r + s + 1, c = r + 2s + 1. Hece a chage of coordiate λ = 1/x is eeded. We have a techica observatio: Propositio 2.2. Lettig λ = 1/x, the aaytic cotiuatio of x a 2F 1 a, b, c; x at is 2F 1 a c + 1, a, a b + 1; λ. Proof. The chage of variabe x = 1/λ meas that we study the behavior of the hypergeometric series at after aaytic cotiuatio. Note that here does ot mea the of X. It simpy meas the chage of variabe x = 1/λ. By takig a appropriate brach cut i the domai to take roots of 1 we ca cosider the period itegra 2.4 λ a 2F 1 a, b, c; 1/λ = Γc 1 a b+c 1 ΓbΓc b with c > b > 0. Mutipyig λ α to 1.2 we have 1 0 t b 1 t 1 c b 1 t λ a dt 2.5 λ 1λ α+2 d2 u du + 2 cλ + a + b 1λα+1 dλ2 dλ abλα u = 0. Our pa is to repace u with λ α u ad fid a appropriate α such that uλ α satisfies a ew hypergeometric differetia equatio. By direct cacuatio, the above equatio ca be rewritte as λλ 1 d2 dλ 2 λα u + [2 c + 2aλ + a + b 1] d dλ λα u + aa c + 1λ α u = 0 by takig α = a ad dividig two sides by λ. Comparig with 1.2 we have a + b + 1 = 2a c + 2, a = a c + 1, c = a + b 1, = b = a, a b = aa c + 1, c = a b + 1. Hece the proof is compete. Let us ed the subsectio by itroducig a otatio. Notatio 2.. The trucated hypergeometric series is defied by 2F 1,N a, b, c; λ := N k=0 a k b k c k k! λk. Simiary, we deote F N a, b, c; x the trucated hypergeometric series of F a, b, c; x, which is a soutio to the hypergeometric equatio 1.2.

7 Ratioa Poits ad Hypergeometric Fuctios Coutig ratioa poits Let us cosider the Frobeius map F b x = x q o the ormaizatio C λ of X λ. Appyig the Lefschetz fixed-poit theorem to F b, we have [1, 2.4] 1 TrFb H 1 C λ,o = umber of fixed poits of F b. Reca that the umber of ratioa poits o X λ is deoted by X λ. Therefore, we get 2.6 X λ = TrFb H 1 C λ,o poits at o C λ 1 k r,s F Nr,s 2 r s+1, s 1, 2 r 2s+1 ; λ δ mod p r,s S for some costats k r,s N r,s, δ, where S is the set of subscripts defied i 2.2, ad δ = poits at o C λ 1. Note that i the secod cogruece idetity, We take {ω r,s } r,s S as the basis of H 1 C λ, O, appyig the simiar techique i the cacuatio of the cassica case of eiptic curves, ad the each eemet ω r,s i the basis wi cotribute k r,s F Nr,s a, b, c ; λ to the trace of Fb. The parameters of the hypergeometric fuctios a = 2 r s+1, b = 1 s+1 = s 1, c = 2 r 2s+1 are determied by 2. ad Propositio 2.2. I additio, the term δ i 2.6 deotes the differece betwee the umber of ratioa poits at o C λ ad X λ, ad we ca δ the correctio term at. Our goa is to determie these costats by coutig ratioa poits over some primes. First, et us expai how to compute δ. I Lemma 2.1 we saw if, =, the poit at ifiity of the smooth mode C λ spits ito three poits, ad if, = 1 the smooth mode has oy oe poit at ifiity of C λ. O the other had Xλ m oy couts the ratioa poits i U W2 =1, so for the case, = we have to make a correctio 1. These correspods to δ = 0 or 2 respectivey. Cosider q such that 2.7 q 1. This impies, p = 1 ad 0 mod p, so the fractios i 2.6 are we defied. I the foowig coutig process, we eed a criterio of the existece of -roots: Lemma 2.4. Let q 1 ad a F q. The 2.8 a q 1/ 1 iff a = y for some y, other vaues there does ot exist y such that a = y. Proof. The proof foows the ies of the proof of the cassica case, amey the case of = 2. If there exists y such that a = y, the a q 1/ y q 1 1 i F q

8 8 Yih Sug by the itte Fermat theorem. Coversey, we assume a q 1/ 1. Cosider the agebraic cosure Ω of F q such that Ω cotais a roots of the agebraic equatios y b for b F q. Thus a = y is sovabe i Ω. The the assumptio a q 1/ 1 impies y q 1 1 i Ω. However, the equatio y q is sovabe i F q. Let F q = α, the y q 1 1 y αy α 2 y α q 1. Therefore, y F q. Now we wat to cout the umber of ratioa poits o X λ,q. Give a pair x, y X λ,q we appy 2.8 to xx 1x λ to see if x, y is a ratioa poit o X λ,q. Let t = [xx 1x λ] q 1/. We ited to fid a poyomia ft satisfyig f0 = 1, f1 =, fζ i = 0 for 0 i 1. This meas that if xx 1x λ = 0, there is oy oe poit x, 0 o X λ,q ; if xx 1x λ 1/ exists i F q, there are poits o X λ,q ; if xx 1x λ 1/ does ot exist i F q, x, y is ot a ratioa poit o X λ,q. Observe that the simpest fuctio f satisfyig fζ i = 0 for 0 i 1 is ft = t ζt ζ 2 t ζ 1 = t 1 t 1 = 1 + t + + t 1, ad ft aso satisfies f0 = 1 ad f1 =. Hece f is a coutig fuctio ad we have X λ x F q t + + t 1 mod p. Let us compute each term x tk. The highest power of x i x tk is k q 1. By the observatios 2.9 x k 1 mod p if q 1 k, x F q 0 mod p if q 1 k, we eed the power to be a mutipe of q 1, which impies k. By the rage 0 k 1, we kow k 1. For simpicity, et us first cosider the case δ = 0 ad we wi come back to the case δ 0 ater. Assume that δ = 0

9 Ratioa Poits ad Hypergeometric Fuctios 9 or, = 1 ad reca that q 1. By the power series expasio, we get t k kq 1 kq 1 = x 1 m x kq 1 kq 1 m λ x kq 1 m x F q x F q m = kq 1 kq 1 x kq 1 1 m+ λ x 2kq 1 m m x F q m, = 1 N x F q N m+=n kq 1 m kq 1 λ x 2kq 1 m+ x kq 1. Defiitio 2.5. For fixed k the coutig i 2.10 is caed a weight-k coutig of X λ,q. Let us examie the umerica coditios cosey. m, i 2.10 come from the biomia expasio, so they are required to be itegers. Pus 2.9 we ca write N = k r 1 q 1 for some r Z 0 satisfyig 2.11 kq 1 N = r + 1 2k q 1 0 = r + 1 2k 0. If r = aways hods, because k 1 <. If r = becomes 1 2k 0 = 2 k. Thus, we divide the situatio ito two cases. Reguar parameters. r = 1, k 1; ad r = 0, k 2. Irreguar parameters. r = 0, 2 < k Cotributios from reguar parameters By 2.9 we observe that the o-zero terms i 2.10 correspodig to the power of x satisfy 2kq 1 m + + for some r Z 0. Hece k 2.12 N = m + = for some r Z 0, ad the coefficiets are N N =0 kq 1 1 N N kq 1 N kq 1 kq 1 λ = r + 1q 1 r q 1 2F 1,N k r 1, k, 2k r; λ mod p.

10 10 Yih Sug Compare 2.1 with 2.6 ad we have 2.14 k = s 1. Therefore, N r,s = 2 r s + 1 q 1, s 1q 1 k r,s = 1 Nr,s N r,s By costructio, N r,s Z >0. This impies r s + 1 > 0 = r = 0, 1. For r = 0, the iequaity becomes 2 1 > s = 2k + 2γ = 2k + γ 1 γ > s, 1 > s, by writig = k + γ which matches 2.2, because i 2.2 the equatio reads s [ ] 2 = s k + γ k 2 = s 2k + γ 1 1 < 2k + γ 1 γ. For r = 1, 2.15 becomes 1 > s = k + γ 1 > s = k + γ 1 2γ > s, which aso matches 2.2, because i 2.2 the equatio reads s [ ] [ ] 2 2γ 2 = s k + γ 2k + 2 [ ] 2γ = s k + γ < k + γ 1 2γ. This cocudes the computatio of the reguar parameters.

11 Ratioa Poits ad Hypergeometric Fuctios 11 We redo the computatio: 2.5. Cotributios from irreguar parameters N N =0 kq 1 N kq 1/ = 1 N = 2k 1q 1 kq 1 = 1 N 2k q 1 1 k q 1 kq 1 kq 1 N λ λ 2k q 1 kq 1 2k q 1 kq 1 λ 2F 1 k q 1, k 2kq 1 q 2 1 q, ; λ λ 2k q 1 2F 1 k, k, 2 k ; λ mod p. Referrig to 2.1, ettig a = k, b = k, c = 2k because r = 0, oe ca recogize that the parameters i 2.16 are a c + 1, b c + 1, c + 2. Therefore, for the irreguar parameters, F N is chose to be x 1 c 2F 1,N a c + 1, b c + 1, c + 2; x. This ca be justified i the partia summatio because the ower boud 2k q 1 > 0. I the above cacuatio, we assumed δ = 0 or equivaety, = 1, so N = m+ 0. Cosider the case, = = ad set N = 0 which impies k = k = r + 1, ad we kow r = 0, 1 which eads to k =, 2 1 = 1. This meas that there are two situatios i which N = 0 ad the summatio [xx 1x λ] kq 1/ cotais x q 1 or x 2q 1. Cosequety, these two terms cotribute 2 after summig over F q. The above discussio, together with 2.1 ad 2.16, cocudes the proof of Theorem 1.1 i the case of m =, with p Cocusio of proof of Theorem 1.1 Lemma 2.6. Let d =, q 1 = gcd, q 1 ad S be the set of F q. Deote S = {a a S}. a If q 1, the the umber of ratioa poits of y = xx 1x λ over F q is the same as the umber of ratioa poits of y d = xx 1x λ over F q.

12 12 Yih Sug b If d = 1, the S = S, amey for every a F q the equatio y = a has a uique soutio i F q. Proof. For a, this is a basic property of the uits F p of a fiite fied F p. For b, by the same property of F q, oe ca show S = S. By this emma, we ca aways assume q 1 ad if d =, p 1 = 1, the equatio y = xx 1x λ has a uique soutio i F q for every x F q. Thus the coutig poyomia is ft = 1 ad we have X λ,q x F q 1 0 mod p. Now we ca compete the proof of Theorem 1.1 for geera m as stated. Observe the foowig reatio betwee H 0 Cλ m, K ad H0 Cλ, K, where C λ is the smooth mode of the curve defied i P 2 with the defiig equatio W 1 = W 0 W 0 W 2 W 0 λw 2 W 2. I the affie piece U W2 =1 P 2, by 2.2 we wat to show [ m ] 2.17 m [ ] ad Write m = k, ad = q + r, 0 r 2. The [ m ] m [ ] [ ] 2m m 2qk 1 r + rk sice 2qk 1 r 2[qk 1 1] 0. For the other iequaity, [ ] 2m m sice r [ 2r the oca expressio [ ] 2 ] 1 if r = 2, [ 2r x s 1 r+ qk 1 r [ ] 2. [ ] 2r + rk [ ] rk [ ] 2rk 0 ] [ = 1; if r = 1, 2r ] = 0. These two iequaities aow x 1 s 1 x λ s 1 dx of the differetia w r,s to have the same format i H 0 C m, K ad H 0 C, K, ad aow the idices to match. Therefore, we ca safey proceed the reductio ad compete the proof.

13 Ratioa Poits ad Hypergeometric Fuctios 1. Case of X defied by y = x a 1 x 1 a 2 x λ a.1. Basic facts We ca appy the method deveoped i the precedig sectio to a more geera situatio. Let p be a prime ad q = p. Let X λ be the curve defied by y = x a 1 x 1 a 2 x λ a where, a 1, a 2, a Z >0,, a 1, a 2, a = 1, α = a 1 + a 2 + a, q 1 ad C λ be a smooth mode of X λ. The the geus of C λ is cacuated by the techique appied i Lemma 2.1. Lemma.1. [2, 4] Let C be a smooth mode of the curve X defied by y = m j=1 x q j a j. Deote α = m i=1 a i. Assume, a 1,..., a m = 1 ad α. The gc = 1 2 m 1 1 m, a 2 j +, α + 1, where a, b = gcda, b. Proof. If a j 2, X has a siguarity at q j. The by bowig up there are, a j poits above q j with brach idex /, a j. Hece by the Hurwitz formua, m 1 gc = m, a j +, α 2 2. I our case the geus of C λ is cacuated by j=1 j=1 gc λ = , a 1 +, a 2 +, a +, α. O the ope set U W2 =1 {0, 1, λ} of C λ, cosider the differetias defied by ω k1,k 2,k,k = xk 1 x 1 k 2 x λ k dx y k ad ω r,k = xr dx y k which has the same form of ω r,s itroduced i the precedig sectio. Reca 2.14: k = 1 s. The we have the foowig techica emma for ater use. Lemma.2. There exists a basis B = {ω k1,k 2,k,k} such that the umber α a2.1 Mk 1, k, k = k k 1 + k 1 q 1 is uiquey determied by k 1, k ad k. Deote B k the subset of B cotaiig the hoomorphic differetias of the type,,, k. Proof. We wi give a expicit agorithm to costruct B i the appedix.

14 14 Yih Sug.2. Proof of Theorem 1.2 We break the argumet ito three steps. The first step is to use the Lefschetz fixed-poit theorem to get a rough idea of the shape of the summatio. The secod step is to examie the oca behavior of each siguar poit ad make ecessary correctios, which we amed correctio fuctios. The ast step is to cacuate the umber of ratioa poits by the cassica techique which we deveoped i the precedig sectios. We ca the determie the precise vaues of the costats accordig to the formuas derived i the first step. Step 1: Cout ratioa poits by the Lefschetz fixed-poit theorem. Let us reca 2.6 ad compute the umber of ratioa poits. Sice a 1, a 2, a might be greater tha 1, there might be siguarities at 0, 1, λ,. Thus, we must take correctios o those poits. Hece 1 TrF b H 1 C λ,o = umber of fixed poits of F b o C λ, which impies the umber of ratioa poits o X λ is X λ = TrF b H 1 C λ,o δ + δ 0 + δ 1 + δ λ, where δ 0, δ 1, δ λ are correctios at 0, 1, λ respectivey ad δ = poits at 1. Let δ = δ 0 + δ 1 + δ λ + δ. The.2 X λ ω k1,k 2,k,k B for some costats c k1,k 2,k,k ad N k1,k 2,k,k. c k1,k 2,k,k F Nk1,k 2,k,k δ mod p Now we wat to determie the correspodig parameters a, b, c for every differetia ω k1,k 2,k,k B. Reca the expicit expressio of the differetia ω k1,k 2,k,k B: ω k1,k 2,k,k = x k 1 a 1 k x 1 k 2 a 2 k x λ By comparig with the differetia t b 1 t 1 c b 1 t λ a dt, ad a = a k a = j=1 k, k j + αk 1, b = k 1 a 1k b = a k k a k + 1, c = k 1 + k 2 a 1k k, dx. a 2k c = k 1 k + a 1k a k. For simpicity, et r = j=1 k j, which pays a simiar roe as r defied i the previous case y = xx 1x λ. The we kow the correspodig trucated hypergeometric serious is. c k1,k 2,k,kF Nk1,k 2,k,k αk r 1, a k k, k 1 k + a 1k + a k ; λ,

15 Ratioa Poits ad Hypergeometric Fuctios 15 where c k1,k 2,k,k ad N k1,k 2,k,k are two costats to be determied ad F ca be either 2F 1,N a, b, c ; λ or λ c 1q 1 2F 1,N a c +1, b c +1, c +2. Note that the expoet of λ i frot of the hypergeometric series satisfies c 1q 1 1 c mod p, which is cosistet with the soutio to the hypergeometric differetia equatios, ad this is exacty the umber M we itroduced i.1. Lemma.. Let N k1,k 2,k,k = αk r 1q 1 ad N k 1 = k a 1k q 1. Let M = Mk 1, k, k be defied as i Lemma.2. If M < 0, k 1, k 2, k, k beogs to the reguar parameters ad the soutio to the hypergeometric differetia equatio i F q with parameters a, b, c is 2F 1,Nk1,k 2,k,k a, b, c ; λ. If M 0, k 1, k 2, k, k beogs to the irreguar parameters ad the soutio to the differetia equatio i F q is λ M 2F 1,N k1 a c + 1, b c + 1, c + 2; λ. Proof. The oy issue is the egth of trucatio. Sice we are workig o F q, either a = 0 i F q or b = 0 i F q ca ed the series. Note that for the first case, a +N k1,k 2,k,k = αk r 1q 0, so we ca take it as the egth of trucatio. Simiary, for the secod case, sice b c + 1 = k a 1k the egth of trucatio ad it is precisey the vaue of N k 1. Step 2: Compute the correctio fuctios. we ca take k a 1k q 1 as I this step we geeraize the correctio terms defied i the ast sectio, which is used to reate the umber of ratioa poits o X to the umber of ratioa poits o the ormaizatio C. I geera, the correctio terms might deped o λ, which is differet from the case i the ast sectio. Thus istead of requirig a correctio costat, we eed a correctio fuctio δλ. By the defiig equatio of X λ, there might be siguar poits aog 0, 1, λ. Aroud x = 0, the defiig equatio of X λ reads y = x a 1 1 a 2 λ a + higher order terms. Let d 1 =, a 1, the there are ratioa poits o C λ over x = 0 if ad oy if a d 1 -th root of 1 a 2 λ a exists i F q. By 2.8, we wat to desig a fuctio such that δ1 = d 1 1, δζ i = 1 for 1 i d 1 1. This meas if a d 1 -th root of 1 a 2 λ a exists, there are d 1 ratioa poits o C λ ad we must make a correctio d 1 1. If a d 1 -th root of 1 a 2 λ a does ot exist, we must make a correctio 1 because 0, 0 is a ratioa poit o X λ.

16 16 Yih Sug Ceary, the fuctio δr = 1+r+ +r d satisfies the properties we discussed above. Hece the correctio fuctio at x = 0 ca be defied by.4 δ 0 r = where d 1 =, a 1 ad r = 1 a 2 λ a q 1/d 1. By the same idea, we fid that the d 1 1 j=1 oca defiig equatios of X λ aroud x = 1, x = λ are y = x 1 a 2 1 λ a ad y = λ a 1 λ 1 a 2 x λ a, ad their associated correctio fuctios are.5 δ 1 s = d 2 1 j=1 r j, s j, δ λ t = where d 2 =, a 2, a =, a ad s = 1 λ a q 1/d 2, t = λ a 1 λ 1 a 2 q 1/d. d 1 The siguarity at ifiity is simpy δ =, α, so the tota correctio fuctio is.6 δ = δ 0 + δ 1 + δ λ + δ. Lemma.4. r κ 1, s κ 2, t κ i.4 ad.5 such that cotribute to the weight-k coutig of X λ,q. j=1 κ i q 1 d i = k, 1 i Proof. The proof is directy from costructio. The we ca decompose δ with respect to the weight-k structure ad write.7 δ = δ k,m λ m. k m Note that the costat δ is at weight 0. Step : Determie the costats. Let us appy the cassica techique agai cf t j, X λ x F q t + + t 1 mod p. The.8 x F q t k = x F q x = N a 1 kq 1 m a kq 1 1 N m+=n a2 kq 1 1 m x a 2 kq 1 m λ x a kq 1 a2 kq 1 m m a kq 1 λ αkq 1 x N. x F q

17 Ratioa Poits ad Hypergeometric Fuctios 17 For brevity, we deote.9 N 2 k = a 2kq 1 By 2.9 the power of x is αkq 1.10 N = Nk = N, N k = a kq 1. N = r + 1q 1 for some r Z 0. Hece αk r 1 q 1 for some r Z 0, ad.8 ca be simpy writte as 1 N N2 m m+=n N λ. By usig the property of fiite fieds, we kow that there are two reatios i F q : λ q 1 = 1 ad 1 + λ + + λ q 2 = 1 for λ 0, 1. Fixig k, the weight-k coutig of X λ,q is.11 X λ,q k 1 N Nk m+=n N2 m N λ ω k1,k 2,k,k B k c k1,k 2,k,kF Nk1,k 2,k,k a, b, c; λ δ k,m λ m i F q, where B k is defied as i Lemma.2. Accordig to the reatios betwee N, N 2, N, there are four differet types of summatio. Fix k, the N 2, N are fixed cf..9. Lemma.5. Assume 0 < N N 2 + N, the a for N N 2, N N, N =0 N2 N b for N 2 < N N, N =0 N2 N N c for N < N N 2, N =0 N2 N N λ = N λ = N2 2F 1,N N, N, 1 N + N 2 ; λ; N N2 λ N N 2 2 F 1,N2 N 2, N N 2 + N, N N 2 + 1; λ; N λ = N2 2F 1,N N, N, 1 N + N 2 ; λ; N m

18 18 Yih Sug d for N 2 < N, N < N, = N =0 N2 N N2 N λ N λ N N 2 2 F 1,N2 +N N N 2, N N 2 + N, N N 2 + 1; λ. Proof. These idetities ca be justified directy. We eed to eumerate possibe k, rs such that αkq 1 Z >0. This correspods to δ. If d > 1, we fid the umber of k satisfyig αk r 1 Z as we did i defiig the correctio i the case of y = xx 1x λ. Let = d ad α = α d. Cosider the coditio αk r 1 = α k r 1 Z 0 = k = k, ad k has to satisfy α k 1. This impies 1 k d 1 because if k = d α r 1 0 which vioates r α 2. Therefore, the correctios is d 1 which is exacty δ, the correctio at ifiity. Now we are ready to determie the costats c k1,k 2,k,k. The assumptio 0 N N 2 + N impies mi {[ ] αk 1, [ a2 k ] } + k 1 + k r max { a1 k } 1, 0, ad this gives the rage of r for fixed k. Uike the case of y = xx 1x λ, the differetia ω r,k associated with the summatio N2 N m+=n m λ might ot be hoomorphic o X λ. However, by Lemmas.2,. ad.11, for each weight k, we kow that there exist {c k1,k 2,k,k} such that.12 X λ,q k + m δ k,m λ m i F q, where a = αk r 1, b = a k proof of Theorem 1.2. ω k1,k 2,k,k B k c k1,k 2,k,kF Nk1,k 2,k,k k, c = k 1 k + a 1k a, b, c; λ + a k. This cocudes the.. Agorithm to fid the coefficiets Let G k λ = X λ,q k + m δ k,mλ m. The by the cassica techique of takig derivatives we ca geerate eough equatios to sove for {c k1,k 2,k,k}. Assumig B k = m, we take

19 Ratioa Poits ad Hypergeometric Fuctios 19 derivatives with respect to λ ad get a system of equatios G k λ = c k1,k 2,k,kF Nk1,k 2,k a, b, c; λ,,k.1 G k λ = c k1,k 2,k,kF N k1 a, b, c; λ,,k 2,k,k. G m 1 k λ = c k1,k 2,k,kF m 1 N k1 a, b, c; λ.,k 2,k,k By Lemma.5, we have G k λ = f f where f i = C i 2 F 1,Ni a, b, c; λ or f i = C i λ c 1 2F 1,Ni a, b, c; λ for every i. Let g 1, g 2,..., g m be the vectors of fuctios o the right-had side of.1. Each etry of g j has the form 2 F 1,Nj a, b, c ; λ or λ c 1q 1 2F 1,Nj a c + 1, b c + 1, c + 2; λ. Let c 1,..., c m be the ukows ad G k be the coum vector G k λ, G k λ,..., Gm 1 k λ T. The, by Cramer s rue oe has c j = W [g 1,..., G k,..., g m ] W [g 1,..., g m ] = costat, where W represets the Wroskia. Thus we ca itroduce ay umber ito λ. For simpicity, we ca set λ = 1. Notice that f i ad g j have the form λ M F N a, b, c; λ ad the derivative of a hypergeometric series with respect to λ is This impies F Na, b, c; λ = ab c F N 1a + 1, b + 1, c + 1; λ. λ M F N a, b, c; λ s λ=1 = Csλ s M s + Cs 1λ s M s 1 F Na, b, c; λ + = = s j=0 s j=0 Appy this formua to every f s i of coefficiets i.12. λ=1 C s s j M M s + j + 1 a jb j c j F N j a + j, b + j, c + j; 1 s s j M s j a j b j F N j a + j, b + j, c + j; 1. c j j! ad g s j, 1 s m 1 ad we get a expicit expressio 4. Remarks o the Legedre famiy of eiptic curves over F q The resuts of Sectio 2 i the case of = 2 ad q = p give rise to the cassicay kow resuts for the Legedre famiy of eiptic curves. I the case of q = p with > 1, apart from the method preseted i Sectio 2, oe ca aso obtai a simiar expressio by a

20 20 Yih Sug cassica approach of cosiderig a trucated hypergeometric series reated to p istead of p. The goa of this sectio is to show that after appyig Wei s resuts o the umber of ratioa poits over a fiite fied the two coutigs with differet trucated hypergeometric series are actuay the same Arithmetic geometry Let us first geeraize the cassica argumets i coutig. There are two steps i this argumet. The first step is to appy Fermat s itte theorem to cout the umbers of ratioa poits directy: 4.1 E λ 1 + xx 1x λ p 1/2 mod p. x F p Here we foow the same guid ie. O the fiite fied F q, the idetity a q 1 1 for a F q hods. By the costructio of F q, we have a criterio of quadratic roots: a q 1/2 1 if there exists y such that a = y 2, 1 otherwise, which is a specia case of 2.8. Hece we have E λ xx 1x λ q 1/2 i F q. x F q By 2.9 we ca cocude q 1/2 4.2 E λ 1 1 q 1/2 which impies r=0 1/2 2 λ r i F q, r 4. q 1/2 E λ,q 1 q 1/2 r=0 1/2 2 λ r mod p 1 q 1/2 2F 1,q 1/2 1 2, 1 2, 1; λ. r For the iterpretatio, reca the Picard-Fuch equatio λ 1 + λλ 1 4 λ λ 2 a p 1 λx xq p d 2 dx c pλx xq p 0

21 Ratioa Poits ad Hypergeometric Fuctios 21 over F q, where q = p. O F q, the Frobeius map is F x = x q. Therefore, we oy have to repace p by q ad get 4.5 E λ a q 1 λ k q 1/2 r=0 1/2 2 λ r i F q = E λ k 2F 1,q 1/2 1 2, 1 2, 1; λ mod p. Let us expai the first idetity. Sice a q 1 satisfies 4.4, a q 1 has a series expressio q 1 a q 1 = c k λ k. k=0 This is because F q has q eemets which impies that the upper boud of the summatio is q 1. Aother expaatio is by Fermat s itte theorem, which says λ q = λ for every λ. Hece the highest meaigfu power of λ is q 1. The, through the expicit soutio of the hypergeometric differetia equatio, we have 4.6 a q 1 k q 1/2 r=0 1/2 2 λ r i F q. r By comparig 4.2, 4.5 ad 4.6, we have k 1 q 1/2 i F q, which impies that k ca be take as a iteger ad k 1 q 1/2 mod p. Therefore by 4.2, 4. ad 4.6 we ca cocude Propositio E λ,q a q 1 λ 1 q 1/2 2F 1,q 1/2 1 2, 1 2, 1; λ for every q = p. r 4.2. Computatio after Wei Let us approach the same probem by Wei s cojecture/theorem o smooth agebraic curves. By usig Tate s modue or Étae cohomoogy oe ca derive 4.8 #EF q = 1 α β p 1 α + β mod p, where β = α ad α = β = q cf. [5, p. 16]. Let a = α + β Z. Note that the otatio #EF q icudes the ifiity poit. Thus by our otatio we have #E λ F q = 1 + E λ,q.

22 22 Yih Sug We ca cacuate a by ettig = 1 ad comparig with 1.1, which impies a 1 p 1/2 2F 1,p 1/2 1 2, 1 2, 1; λ mod p. To abbreviate, we deote F = 2 F 1,p/2 1 2, 1 p 2, 1; λ. The we ca derive E λ,q for 1 by a simpe observatio α + β α + β a mod p. Proof. The key is to show α + β Z. We wi prove this by iductio o. The case of = 1 is obvious. For the geera case, accordig to the biomia expasio 4.9 α + β = Ck α k β k k=0 = α + β + C 1 αβα 2 + β 2 +, ad the by the iductio hypothesis C 1 αβα 2 + β 2 + Z, we ca cocude that α + β Z. Agai by 4.9, sice αβ = α 2 = q, Therefore we ca cacuate α + β = α + β + q C 1 α 2 + β 2 + α + β mod p. #E λ F q 1 α + β 1 a mod p 1 1 p 1/2 F mod p = E λ,q 1 p 1/2 F mod p. It is easy to verify 1 p 1/2 = 1 p 1/2, so we get a equatio 4.10 q 1/2 r=0 1/2 2 λ r r p 1/2 r=0 1/2 2 λ r r mod p. Remark 4.2. Icidetay, this equatio eads to the foowig o-obvious idetity: 2F 1,q 1/2 1 2, 1 2, 1; λ 2F 1,p 1/2 1 2, 1 2, 1; λ mod p.

23 Ratioa Poits ad Hypergeometric Fuctios 2 5. Exampes 5.1. I this subsectio, we wi use exampes to iustrate subteties betwee takig moduo p ad moduo q = p for > 1, which expais why Theorem 1.1 was stated i terms of mod p istead of mod q. Exampe 5.1. Let X 2 λ be defied by y2 = xx 1x λ ad p be a prime. Let q = p. Reca the formua 4.7 Let λ =, the Fλ,q 2 = 1q 1/2 2F 1,q 1/2 1 2, 1 2, 1; λ. q = 5, the X,5 2 = ad F,5 2 mod 5. q = 5 2, the X,5 2 = 1 ad F 2 2,5 2 1 mod 5, F,5 2 6 mod q = 5, the X 2,5 = 147 ad F 2,5 2 mod 5, F 2,5 97 mod 5. These three resuts shows that takig moduo p is ecessary. The idetities wi be faied if oe takes moduo q = p. Exampe 5.2. Let X 4 λ be defied by y4 = xx 1x λ ad p be a prime. Let q = p. By usig Theorem 1.1, we have where Fλ,q 4 = k 1λ q 1/2 F N1 1 4, 4, 1 2 ; λ k 2F N2 1 4, 4, 1 2 ; λ k F N 1 2, 1 2, 1; λ, q 1 k 1 = 1 5q 1/4 4 2q 1 4 Let λ =, the q 1, k 2 = 1 q 1/4 4 q 1, k = 1 q 1/2, 4 N 1 = q 1 4, N 2 = q 1 4, N = q 1 2. q = 5, the X,5 4 = ad F,5 4 mod 5. q = 5 2, the X,5 4 = 19 ad F 4 2,5 2 4 mod 5, F, mod q = 5, the X 4,5 = 147 ad F 4,5 2 mod 5, F 4,5 17 mod 5. q = 17, the X,17 4 = 2 ad F, mod 17.

24 24 Yih Sug q = 7, the X,7 4 = ad F,7 2 mod 7. q = 11, the X,11 4 = 15 ad F, mod 11. The first three resuts shows that takig moduo p is ecessary. The idetities wi be faied if oe takes moduo q = p. For q = 5, 17, they satisfy the assumptio m, q 1 = 4 where m = = 4. For the ast two idetities, they satisfy the assumptio m, q 1 = 2 where m = 4, = The foowig exampe shows that the assumptio of = m, q 1 i Theorem 1.1 is ecessary. Exampe 5.. Let Xλ 6 be defied by y6 = xx 1x λ. As before, we deote Fλ,q 6 the formua provided i Theorem 1.1. Let F λ,q 6 = F λ,q 6 δ, i.e., without cosiderig the correctio terms. Let λ =, the q = 7, the X,7 6 = ad F,7 6 5 mod 7. q = 1, the X,1 6 = 27 ad F,1 6 mod 1. q = 17, the X,17 6 = 2 ad F, mod 17, F, mod 17. The first two resuts shows that the correctio terms are ecessary. For the ast resut, the formua Fλ,q 6 does ot provide the correct resut i the case Istead, we must cosider the right power ad do the reductio: Fλ,q 2 where 2 = 6, Appedix We wi preset a expicit agorithm metioed i the proof of Lemma.2. First et us reca: Lemma 6.1. [2, Theorem ] ω k1,k 2,k,k is hoomorphic if ad oy if k 1, k 2, k, k satisfies Test1: k j + 1 ka j +, a j for j = 1, 2,, ad Test2: kα k 1 + k 2 + k + 1 +, α. Now we decare variabes m =, a = α, k1 = k 1, k2 = k 2, k = k. The we have the foowig agorithm.

25 Ratioa Poits ad Hypergeometric Fuctios 25 Listig 1: Fid a basis for k=i t g e r P a r t m/a +1; k<=m 1; k++{ = 1; // c o t r o f a g. for k =0; k<=a 2; k++{ for k2 =0; k2<=a 2; k2++{ for k1 =0; k1<=a 2 && k1+k2+k> ; k1++{ Test1 ; Test2 ; =k1+k2+k ; // r e s e t t h e f a g. }}} } Note that for each fixed k, we move k first, the k2, the k1. I this maer we ca make k1 + k2 + k keep growig. Oe ca easiy justify that this feature makes B satisfy the requiremet i Lemma.2. Ackowedgmets We wat to speciay thak Professor Sai-Kee Yeug for usefu discussio ad geerous advice o this paper. Refereces [1] C. H. Cemes, A Scrapbook of Compex Curve Theory, Secod editio, Graduate Studies i Mathematics 55, America Mathematica Society, Providece, RI, [2] J. K. Koo, O hoomorphic differetias of some agebraic fuctio fied of oe variabe over C, Bu. Austra. Math. Soc , o., [] Ju. I. Mai, Agebraic curves over fieds with differetiatio, Amer. Math. Soc. Tras. Ser , [4], The Hasse-Witt matrix of a agebraic curve, Amer. Math. Soc. Tras. Ser , [5] J. H. Siverma, The Arithmetic of Eiptic Curves, Secod editio, Graduate Texts i Mathematics 106, Sprigger, Dordrecht, Yih Sug Departmet of Mathematics, Uiversity of Wiscosi-Madiso, Va Veck Ha, 480 Lico Drive, Madiso, WI 5706, USA E-mai address: ysug26@wisc.edu