CONGRUENCES FOR RAMANUJAN S f AND ω FUNCTIONS VIA GENERALIZED BORCHERDS PRODUCTS. April 10, 2013

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1 CONGRUENCES FOR RAMANUJAN S f AND ω FUNCTIONS VIA GENERALIZED BORCHERDS PRODUCTS JEN BERG, ABEL CASTILLO, ROBERT GRIZZARD, VÍTĚZSLAV KALA, RICHARD MOY, AND CHONGLI WANG Aril 0, 0 Abtract. Bruinier and Ono recently develoed the theory of generalized Borcherd roduct, which ue coefficient of certain Maa form a exonent in infinite roduct exanion of meromorhic modular form. Uing thi, one can ue claical reult on congruence of modular form to obtain congruence for Maa form. In thi note we work out the examle of Ramanujan mock theta function f and ω in detail.. Introduction The goal of thi note i to rove ome exlicit congruence for the following two function: ωq = a ω nq n q nn+ := q q q n+ n=0 n=0 = + q + q + 4q + 6q 4 + 8q 5 + 0q 6 + 4q 7 + 8q 8 + q 9 + 9q 0 + 6q + ; fq = a f nq n q n := + q + q + q n n=0 n=0 = + q q + q q 6 + q 7 + q 9 q 0 + q q + q q 4 + q 5 q 6 +, where q = e πiτ for τ in the comlex uer half-lane H. The function fq and ωq are two of Ramanujan famou mock theta function, which he decribed in hi famou lat letter to Hardy. We now undertand the mock theta function a the holomorhic art of half-integral weight weak harmonic Maa form ee for examle [4] for background on thee function. In [] Bruinier and Ono decribed generalized Borcherd roduct for weak harmonic Maa form and roved that they are weight 0 meromorhic modular form. They ecifically decribe a family of uch Borcherd roduct which i defined uing a vector-valued form whoe coefficient are defined in term of the coefficient of fq and ωq. Thi work indicate that one can rove a multitude of congruence for thee mock theta function by uing the claical theory of congruence for modular form; however, uing thi technique to obtain exlicit congruence take ome effort. Bruinier and Ono howed one examle of thi only for ω in [], and in the reent aer we roduce additional examle. In order to tate our reult, we mut introduce ome auxiliary object which will be defined in Section. We will define a vector-valued form with coefficient c +, which will be given in term of the coefficient a f and a ω in Lemma.4. For a fixed negative fundamental dicriminant and an integer r uch that r mod 4, the aforementioned 000 Mathematic Subject Claification. F0, F. Key word and hrae. mock theta function, Borcherd roduct, harmonic Maa form.

2 BERG, CASTILLO, GRIZZARD, KALA, MOY, AND WANG Borcherd roduct tudied by Bruinier and Ono ha a logarithmic derivative which will have a q-exanion a follow: Φ,r z, H d = c + 4, rd n. d e q n n= d n mod d where i the quadratic reidue ymbol. For any rime l inert or ramified in Q and any exonent R, the normalization Φ,r = n= bnqn of thi Fourier exanion will be the ame modulo l R a that of a modular form on Γ 0 6. Therefore it make ene to dicu the action of Hecke oerator T for any rime l on Φ,r, modulo lr. Our mot general reult i the following. Theorem.. Let be a negative fundamental dicriminant, r an integer uch that r mod 4, and let Φ,r be the normalization of the form defined in.. Fix a rime l inert or ramified in Q and a oitive integer R. Let be a rime uch that. Φ,r T,k bφ,r mod l R, where k i the correonding weight a in Lemma.8 and T,k i the aroriate Hecke oerator. Write cn = c + n 4, rn. We have. cn c n n mod b vn b vn l R. Moreover, for a given, r, l, and R, the et of rime for which the above tatement hold ha oitive arithmetic denity. For examle, we have the following. Corollary.. Let = 8, r = 4. Let, ; l, R, and k be a in Theorem., uch that Φ,r T,k 0 mod l R. Then for all m we have 4m+ a ω 8 m+ k m mod l R, and 4m a ω m k m mod l R. Exlicitly, 8 = when, mod 8 and when 5, 7 mod 8; = when mod and when mod. Moreover, for a fixed rime ower l R, the et of rime uch that the above tatement hold ha oitive arithmetic denity. Corollary.. Let =, r =. Let,, ; l, R, and k be a in Theorem., uch that Φ,r T,k 0 modl R. Then for all m we have 4m+ + a f 4 m+ k m modl R, and 6 4m + a f 4 m k m modl R.

3 CONGRUENCES FOR f AND ω Moreover, for a fixed rime ower l R, the et of rime uch that the above tatement hold ha oitive arithmetic denity. We will rove thee and related reult in Section. We will decribe the tatement on the denity of rime in Section.. In Section 4, we will decribe how to rove exlicit congruence that follow from our theorem. acknowledgment Thi aer aroe from a roject led by Ken Ono at the Arizona Winter School 0. The author wih to thank Profeor Ono and the organizer of the AWS for roviding thi oortunity and for the huge amount of uort and hel with the roject. The author alo thank William Stein and Sage for hel with our comutation.. Nut and Bolt.. Modular form modulo rime. Let A be the Hae invariant; we recall that A i a modified Eientein erie that vanihe reciely at the ueringular locu; and that, for l 5, A can be lifted to the Eientein erie E l. Moreover, we know that E l mod l []. We combine thee fact in the following lemma. Lemma.. For any rational rime l 5 and any oitive integer R, there exit a holomorhic modular form F, of weight Rl uch that F mod l R and F mod l vanihe reciely at je mod l for all ellitic curve E ueringular over F l... Almot all tatement for congruence for modular form. Following Ono [], we make the following obervation which reult from the exitence of Galoi rereentation of modular form and the Chebotarev Denity Theorem. Lemma.. Let gz be an integral weight modular form in M k Γ 0 6 whoe Fourier coefficient are in O K, the ring of algebraic integer of a number field K. Let l be an ideal of O K. Then a oitive roortion of rime have the roerty that gz T,k 0 mod l. Lemma.. Auming the ame notation a above, a oitive roortion of rime have the roerty that gz T,k gz mod l. In articular, for ufficient rime l, we will have that the logarithmic derivative Φ,r i an eigenfunction for the Hecke oerator T,k modulo l with eigenvalue 0 or for a oitive roortion of rime. Additionally [, Theorem.65], tell u that almot all of the coefficient of Φ,r will be zero modulo l. Thee reult jutify the tatement on the denity of rime atifying the hyothee of Theorem. and Corollarie. and.... Generalized Borcherd roduct. We follow Bruinier and Ono [], and write a weight / harmonic Maa for whoe coefficient are related to thoe of f and ω. Define

4 4 BERG, CASTILLO, GRIZZARD, KALA, MOY, AND WANG the following cuidal weight / theta function n n + e πin+ τ, g τ := n Z g τ := n + e πin+ 6 τ, and 6 n Z g τ := n Z n + e πin+ τ. Let F τ := q 4 fq, q ω q, q ω q T and i g z, g z, g z T Gτ := dz; τ iτ z Zweger howed that Hτ := F τ Gτ i a vector-valued weight / harmonic Maa form. Bruinier and Ono [] how that H give rie to H H /,ρl ; the lemma below give an exlicit relation between the coefficient c + m, h of the holomorhic art of H and thoe of f and ω. Lemma.4. Notation a above. a c + m, Z = c + m, + Z = c + m, 6 + Z = c + m, 9 + Z = 0 b c + m, + Z = c + m, 5 + Z = c + m, 7 + Z = c + m, + Z = a f m + 4 c c + m, + Z = c + m, + Z = d c + m, 4 + Z = c + m, 8 + Z = m a ω m m + a ω m One alo oberve that H atifie the condition for the generalized twited Borcherd lift decribed by Bruinier and Ono [] to be a meromorhic modular function of weight 0 for Γ 0 6, whoe divior i uorted on CM oint. We then ue the following well-known lemma tated below without roof. Lemma.5. Let g be a meromorhic modular function of weight 0 for a congruence ubgrou Γ of SL Z. Then, the logarithmic derivative of g, i.e. d πi dz g g i a weight meromorhic modular form for Γ of weight, with imle ole on the uort of the divior of g, and no other ole. We take the logarithmic derivative of Ψ,r z, H, that i, [ Φ,r z, H q d dq Ψ,r z, H ] := Ψ,r z, H. By the lemma above, thi i a weight meromorhic modular function for Γ 0 6, with at mot imle ole at CM oint and no other ole. There exit an uer bound for the number of uch ole which can be given in term of.

5 The q-exanion of Φ,r z, H i given by. Φ,r z, H = n= d n CONGRUENCES FOR f AND ω 5 d c + 4, rd d mod mod n e q n, d giving u an exlicit relationhi between the Fourier coefficient of Φ,r z, H and the coefficient of the mock theta function fq and ωq. For imlicity of notation, let u denote cn := c + n 4, rn and c := c, ureing the deendence on r and wherever it will not caue confuion. Alo, let Φ = Φ,rz, H := Φ,r z, H/ n c d be the normalized logarithmic derivative i.e., o that it q -coefficient i. To obtain a imler exreion for the coefficient of Φ we evaluate the Gau um : Lemma.6. Let a, Z, a, 0. Let Ga, = e a. mod a If gcda, =, then Ga, = a G,. b In articular, for = 8, we have Ga, 8 = e 8 if a, mod 8 8 if a 5, 7 mod 8 0 if a 0 mod. Proof. a Let a be the invere of a mod. We have Ga, = a e mod = a = mod a mod e e by a imle ubtitution in the ummation and the multilicativity of the Kronecker ymbol. Therfore, a follow ince a = a. The roof of b i an exlicit calculation of the value. Prooition.7. Take n corime to. Then the q n coefficient of the q-exanion of Φ,r z, H i bn := cdd. c n/d Moreover, cn = c n where µd i the Möbiu function. d n d n bdµn/d, n/d

6 6 BERG, CASTILLO, GRIZZARD, KALA, MOY, AND WANG Proof. The q -coefficient of Φ i c G,. Thu the formula for bn follow immediately from Lemma.6. Then note that we have the convolution formula b = b b, where b n = n c cn and b n = n. Therefore, b n i multilicative, and o it Dirichlet convolution invere i µb. Therefore b = b µb, which give exactly the formula for cn. In order to rove congruence in the next ection, we need that Φ i congruent to modular form modulo ower of rime: Lemma.8. Suoe Φ,r z, H and thu alo Φ ha B imle ole. Let l be a rational rime that remain inert or ramifie in Q. Then there exit a holomorhic modular form Φ o of weight + l BR uch that Φ Φ o mod l R Proof. Let α be a ole of Φ,r z, H. Then, α mod l correond to a zero of the holomorhic modular form F defined in Lemma.. We know that α will be a CM oint defined over Q, and that l doe not lit in thi field. Suoe β i a zero of F with α β mod l; letting E α and E β be ellitic curve with j-invariant α and β reectively, we can multily Φ by F jz jα R jz jβ to obtain a meromorhic modular form congruent to Φ modulo l R with one le imle ole. Oberve that thi increae the weight of the form by l R.. Proof of Congruence Throughout thi ection, will denote a fundamental dicriminant and r an integer uch that r mod 4. Lemma.. Let gq = n=0 anqn be a normalized i.e., a = modular form on Γ 0 6 of weight k. a If, i a rime uch that g T = 0, then a m+ = 0 and a m = k m. b Let S be a et of rime, uch that for all S we have g T = ag. Then aman = amn for all corime m, n diviible only by rime lying in S. Alo, for S we have j= a j j = a + k. Proof. Thi i very tandard, o we jut rovide a ketch: The q n -coefficient of g T i an + k an/. Part a then follow by eay induction uing thi formula. Similarly, art b follow by uing the formula for coefficient of g T. Prooition.7 allow u to obtain congruence for the coefficient c + auming we know the value of bn modulo l R for a uitable rime ower l R. For examle, uch information can come from our logarithmic derivative being an eigenfunction for a Hecke oerator T modulo l R, or even an eigenfunction for all the Hecke oerator i.e. congruent modulo l R to a Hecke eigenform.

7 CONGRUENCES FOR f AND ω 7 Theorem.. Fix a rime l which i inert or ramified in Q and R. Let k be the correonding weight a in Lemma.8 and T,k the aroriate Hecke oerator. Let,, l be a rime uch that Φ,r T,k 0 mod l R. If alo doe not divide, then: c m+ c m+ k m mod l R c m c m k m mod l R Moreover, fixing, r, l, R, the et of rime uch that Φ,r T,k 0 mod l R ha oitive denity. Proof. By Prooition.7 we have c M = c M N M b N µ M N M N = c M b M b M, becaue when M N 0,, M N i not quare-free, and o µ M N = 0. The tatement now immadiately follow from Lemma.. Proof of Corollarie. and.. Note that in thi cae c = c + = 4aω 0 = 4. By Lemma.4 we have c m+ = c + 4m+, 4m+ = ±4a 4m+ ω, where the ign ± i exactly. The tatement now follow immediately from.. The econd art and Corollary. follow in the ame way. Similarly we can obtain congruence for coefficient at q n, where n i diviible only by rime a in Theorem.. Proof of Theorem.. All the congruence in thi roof are mod l R, and o we omit writing thi. Throughout the roof, let n be a oitive integer diviible only by rime in S. Note that by Lemma. we have bn n bvn. Alo the quadratic ymbol i multilicative and µd i 0 if d i not quare-free. Thu by Prooition.7 we have n c cn = bn/dµd = bn/dµd. d d n d d n d quare-free Writing d = n ε get, 4 with ε = 0,, and uing the multilicativity of b, n c cn ε n the um being over all oible tule ε. But the lat um i jut the roduct n to how. b vn ε ε,, and µ, we b vn b vn, a we wanted

8 8 BERG, CASTILLO, GRIZZARD, KALA, MOY, AND WANG 4. Exlicit comutation Our theorem, combined with the almot all theorem mentioned in Section., tell u that we can obtain infinitely many exlicit congruence for the coefficient a f n and a ω n. Writing any exlicit congruence down, however, i a different matter. If we want to rove that Φ,r z, H T 0 mod l for certain rime and l, then we mut comute enough coefficient of the q exanion of Φ,r z, H to be able to aly Sturm Theorem ee [6] and [, ection.9]; thi ueful theorem give an uer bound on the number of coefficient of a modular form that can be zero modulo l R without all of them being zero. There are two method for comuting thee coefficient. The firt method ue. directly. Thee coefficient are given in term of the coefficient of fq and ωq. In ractice, thi i quite difficult, a we now decribe. Suoe we wanted to how that Φ,r q T 0 mod l, by verifying thi congruence for the firt N coefficient, where N i determined uing Sturm Theorem. Thi mean we need to have at leat N coefficient comuted for Φ,r q. Uing Lemma.4, we can verify that thi will require comuting N + 4 coefficient of fq and N 8 coefficient of ωq. For examle, if = 8, N = 000, and = 5, thi would mean comuting over 8 million coefficient for ωq and over 6 million for fq. The econd method i to find an exlicit exreion for Φ,r, a in [, Section 8.], in term of well-known erie. The author of [] wrote an exreion for Φ 8,4 a a rational function in Eientein erie and Dedekind eta function ee formula on age 75 of []. Uing thi, one can comute a many coefficient for Φ 8,4 q a one can comute for the Eientein and eta roduct. For the above examle, thi i fater by everal order of magnitude. Uing thi econd method, we were able to comute enough coefficient to rove ome exlicit examle of congruence for ωq uing Φ 8,4. We roved the following comutationally. Examle 4.. Let Φ 8,4 q be the normalized q-erie defined in Section.. We have a Φ 8,4q T 5 0 mod. b Let g be the normalized newform in M 0 Γ 0 6. We have that Φ 8,4q g mod 5. We will only exlicitly dicu art a. Part b follow in the ame manner. We know Φ 8,4 q ha at mot two ole from Bruinier and Ono exlicit formula [, ]. We multily Φ 8,4 q by two twited Hae invariant mod 5 to obtain a holomorhic modular form Φ o Φ 8,4 q mod of weight 46 on Γ 06 by Lemma.8. Uing Sage [5], we calculated that Φ q T 5 0 mod to reciion q 50. Sturm Theorem give u that if h M k Γ, l i a rime, and the coefficient of q n in the q-exanion of h vanih modulo l for all n k[sl Z: Γ] 4, then h 0 mod l. In thi cae, the Sturm bound i 46 [SL Z: Γ 0 6] 4 = a coincidence, and were able to verify that 4. Φ 8,4q T 5 Φ o q T 5 0 mod comutationally, a decribed above.

9 CONGRUENCES FOR f AND ω 9 Equation 4., along with Corollary. give u the exlicit congruence 5 M a ω 5 m ε mod, where M = m + ε, ε {0, }. We can alo roduce an infinite family of uch congruence modulo 5, becaue we verified that Φ 8,4 i congruent modulo 5 to a Hecke eigenform. Reference [] Jan Bruinier and Ken Ono. Heegner divior, L-function and harmonic weak Maa form. Ann. of Math., 7:5 8, 00. [] Jan H. Bruinier and Ken Ono. Identitie and congruence for Ramanujan ωq. Ramanujan J., - :5 57, 00. [] Ken Ono. The web of modularity: arithmetic of the coefficient of modular form and q-erie, volume 0 of CBMS Regional Conference Serie in Mathematic. Publihed for the Conference Board of the Mathematical Science, Wahington, DC, 004. [4] Ken Ono. Unearthing the viion of a mater: harmonic Maa form and number theory. In Current develoment in mathematic, 008, age Int. Pre, Somerville, MA, 009. [5] W. A. Stein et al. Sage Mathematic Software Verion 5.6. The Sage Develoment Team, 0. htt:// [6] Jacob Sturm. On the congruence of modular form. In Number theory New York, , volume 40 of Lecture Note in Math., age Sringer, Berlin, 987. The Univerity of Texa at Autin, Deartment of Mathematic, Autin, TX jberg@math.utexa.edu The Univerity of Illinoi at Chicago, Deartment of Mathematic, Statitic, and Comuter Science, Chicago, IL acati8@uic.edu The Univerity of Texa at Autin, Deartment of Mathematic, Autin, TX rgrizzard@math.utexa.edu Purdue Univerity, Deartment of Mathematic, Wet Lafayette, IN vita.kala@gmail.com Northwetern Univerity, Deartment of Mathematic, Evanton, IL ramoy88@math.northwetern.edu Columbia Univerity, Deartment of Mathematic, New York, NY chongli@math.columbia.edu