LECTURES 4 & 5: POINCARÉ SERIES

Transcription

1 LECTURES 4 & 5: POINCARÉ SERIES ANDREW SNOWDEN These are notes from my two letures on Poinaré series from the 2016 Learning Seminar on Borherds produts. I begin by reviewing lassial Poinaré series, then redo the theory in the vetor-valued setting, and finally disuss a non-holomorphi variant. Referenes: Bruinier, Borherds produts on O(2,l) and Chern lasses of Heegner divisors (Leture Notes in Math 1780). I basially over 1.2 and 1.3 here. Leis, The Poinaré series (notes). Warning: There are many formulas and omputations in these notes. I have not verified them myself, but simply opied them from the soures I used. They re probably mostly right, but there ould be some mistakes. 1. Classial Poinaré series Suppose we want to onstrut modular forms for some group Γ SL 2 (R). One way to try to onstrut a funtion with the right invariane is to start with any funtion and then average it over Γ. Unfortunately, in this partiular situation, suh averages will usually not onverge. However, a modifiation of this idea works: we instead start with a funtion that s already invariant under Γ (assuming is a usp), namely an exponential funtion, and then average over the osets of Γ in Γ. These averages are the Poinaré series, and they span the spae of modular (or usp) forms. While this theory works for quite general Γ, we only present the ase of Γ = SL 2 (Z) here Notation. Before getting into things, we set some notation. We let Γ = SL 2 (Z) and let Γ Γ be the subgroup of upper-triangular matries. These groups at on the upper half-plane h by linear frational transformations: γz = az + b z + d, γ = ( ) a b. d Note that Γ is the full stabilizer in Γ of the usp. We let j γ (z) = z + d (in the above notation) be the usual automorphy fator. For a funtion f on h, we define f k γ to be the funtion on h given by (f k γ)(z) = j γ (z) k f(z). This defines an ation of Γ on funtions, and the transformation property of modular forms of weight k is exatly invariane under this ation. Finally, we put e(z) = e 2πiz. 1

2 2 ANDREW SNOWDEN 1.2. Definition. Let k > 2 and m 0 be integers, with k even. The Poinaré series is defined by (1.1) P m,k (z) = e(mγz) k γ γ Γ \Γ Note that the funtion e(mz) is invariant under the k aiton of Γ, and so the sum is well-defined. We have a bijetion Γ \Γ {(, d) Z 2 0, (, d) = 1} Γ γ ±(0, 1)γ, where the sign is taken to make the first oordinate positive (and (0, 1) denotes a row vetor). Using this, we find (1.2) P m,k (z) = e(mγ,d z) (z + d), k,d Z 2 0,(,d)=1 where γ,d denotes any element of Γ with bottom row (, d). From this we see that P 0,k (z) is the usual Eisenstein series E k (z) of weight k. Moreover, it is lear that the the terms of P m,k (z) are bounded (in absolute value) by those of E k (z), and thus (appealing to the standard onvergene result for E k ) we see that the series (1.2) onverges absolutely. Thus P m,k (z) is a well-defined holomorphi funtion on the upper half-plane invariant under the k ation of Γ. Furthermore, the bound P m,k (z) E k (z) implies that P m,k (z) is holomorphi at the usps, sine the same is true for E k (z), and so P m,k (z) is a modular form of weight k Fourier expansion. Fix m and k. Being a modular form, the Poinaré series admits a Fourier expansion: P m,k (z) = n 0 a n e(nz). We now give a formula for the oeffiients a n. We need to introdue some auxiliary quantities. The Kloosterman sum k(m, n, ) is defined by k(m, n, ) = ( ) nd + md 1. d (Z/Z) e This is a finite sum of roots of unity. The Bessel funtion is defined by J k 1 (x) = 1 z k e (z z 1)x/2 dz. 2πi We let z =1 σ k 1 (n) = d n be the usual funtion, and let ζ be the Riemann zeta funtion. We an now give the formula: =1 d k 1 Theorem 1.3. For m > 0 we have ( n (k 1)/2 ( ) k ( ) 2πi 4π nm a n = k(m, n, )J k 1, m)

3 while for m = 0 we have LECTURES 4 & 5: POINCARÉ SERIES 3 a n = (2πi)k σ k 1 (n) (k 1)!ζ(k). Proof. We just provide a sketh of the proof. Let y > 0 be a real number. We have The = 0 terms are a n = 1+iy 0+iy = d=±1,d Z 2 0,(,d)=1 1+iy 0+iy P m,k (z)e( nz)dz 1+iy 0+iy e(mγ,d z) (z + d) k e( nz)dz e(mz)e( nz)dz = 2δ n,m. Write a n = 2δ n,m + a n, so that a n is the sum over > 0. We break this sum up over osets mod. Preisely, write d = d + l where 0 d < is prime to and l Z. We have Thus a n = z + d = (z + l) + d, =1 0 d<, (,d)=1 l Z 1+iy 0+iy γ,d z = γ,d (z + l). e(mγ,d (z + l)) ((z + l) + d) k e( nz)dz Changing z to z l (and dropping the overline on d), we find Next, we have a n =,d a n =,d l+1+iy l Z l+iy +iy +iy e(mγ,d z) (z + d) k e( nz)dz e(mγ,d z) (z + d) k e( nz)dz (z + d) k = k (z + d )k, γ,d z = a 1 2 (z + d ). We hange z to z d in the integral, to obtain +iy a e( ma m n = =1 d (Z/Z) +iy k z k = k nd )e( =1 e( ma d (Z/Z) ) 2 z ) e( nz + nd )dz +iy +iy z k e( m 2 z nz)dz Note that in the first sum, a is the inverse of d modulo. Thus this sum is simply k(m, n, ). If m > 0 then, after some manipulation, the integral turns into the Bessel funtion and other fators. [What happened to the δ n,m in the formula for a n?] If m = 0 then the integral is

4 4 ANDREW SNOWDEN easy to evaluate using the residue theorem, and one reahes the final formula by applying the identity ( k(0, n, ) = 2πi ) k 1 σ k 1 (n). k n ζ(k) =1 [This doesn t seem quite right...] Corollary 1.4. For m > 0 the series P m,k is a usp form. Proof. It is lear from the formulas that a n = 0 if m > 0. Corollary 1.5. We have P m,k = 0 for all m > 0 if k {4, 6, 8, 10}. Proof. There are no non-zero usp forms for Γ for these weights. Remark 1.6. This vanishing is not at all lear from the definition of P m,k or the formula for its Fourier oeffiients. Apparently, it is a diffiult problem in general to determine when Poinaré series vanish identially Petersson inner produt. Let f and g be weight k modular forms, at least one of whih is uspidal. Reall that their Petersson inner produt is defined by (f, g) = f(z)g(z)y k 2 dxdy. Γ\h The fator y k 2 ensures that this is well-defined, i.e., independent of the hoie of fundamental domain. The Petersson inner produt defines a non-degenerate hermitian inner produt on the spae S k (Γ) of usp forms of weight k. The following result desribes how it interats with Poinaré series Theorem 1.7. Let f be a usp form of weight k. Then (f, P m,k ) = Γ(k 1) (4πm) k 1 a m(f), where Γ is the usual Γ-funtion and a m (f) denotes the mth Fourier oeffiient of f. Proof. We have (f, P m,k ) = We now make use of the identity to obtain Γ\h (f, P m,k ) = y k f(z) γ Γ \Γ Im(γz) = Im(z) j γ (z) 2 Γ\h γ Γ \Γ ( ) k dxdy j γ (z) e(mγz) y 2 Im(γz) k f(γz)e(mγz) dxdy y 2. Note that the measure dxdy is invariant under the ation of Γ. We an therefore unfold y 2 the integral to obtain (f, P m,k ) = y k 2 f(z)e(mz)dxdy. Γ \h

5 LECTURES 4 & 5: POINCARÉ SERIES 5 Now, the ation of Γ on h is generated by the translation z z + 1, and so has for a fundamental domain the region defined by 0 x 1. We thus find (f, P m,k ) = y k 2 f(x + iy)e 2πimx 2πmy dxdy. Write f(z) = n 0 a ne(nz) with a n = a n (f). Then we obtain (f, P m,k ) = n 0 a n y k 2 e 2πi(n m)x e 2π(n+m)y dxdy. The x-integral is δ n,m and the y-integral (for n = m) is Γ(k 1)(4πm) (k 1). This gives the stated formula. Corollary 1.8. The series P m,k with m > 0 span the spae S k (Γ) of usp forms. Proof. Suppose f S k (Γ) is orthogonal to all of the P m,k. Then a m (f) = 0 for all m > 0 by the Theorem, and so f = 0. The non-degeneray of the inner produt thus implies that the P m,k span. Example 1.9 (k = 12). Let (z) be the modular disriminant, the unique normalized usp form of weight 12. We have (z) = q n 1 (1 e(nz)) 24 = n 1 τ(n)e(nz), where τ(n) is the ustomary notation for the nth Fourier oeffiient. (The funtion τ is alled the Ramanujan funtion.) From the Theorem, we find (, P m,12 ) = 10!τ(m) (4πm) 11 Sine S 12 (Γ) is 1-dimensional, P m,12 is a salar multiple of. We find P m,12 (z) = 10! τ(m) (2πm) 11 (, ) (z). In partiular, we find P m,12 = 0 if and only if τ(m) = 0. Lehmer onjetured τ(m) 0 for all m 1. This has been verified for all m up to about 10 24, but has not been proved. 2. Vetor-valued Poinaré series We now give variants of the onstrutions and results from the previous setion in the vetor-valued setting. The proofs are exatly the same, so we omit them Notation. We let Γ and Γ be as before, though we now write M for a typial element of Γ. We let Γ be the metapleti over of Γ. Its elements are pairs (M, φ) where M Γ and φ is a ontinuous square-root of the automorphy fator j γ on h. We let Γ Γ be the subgroup onsisting of elements of the form (M, 1) with M Γ. We put T = (( ) ), 1, Z = (( ) ), i. The element Z generates the enter of Γ and {1, Z 2 } = ker( Γ Γ).

6 6 ANDREW SNOWDEN Fix an even lattie L of signature (b +, b ). We assume (b +, b ) is of the form (2, l), (1, l 1), or (0, l 2). We let q(x) = 1 x, x be the quadrati form. We put κ = 1 + l. This 2 2 will be the weight of the modular forms we produe. Let L be the dual lattie and let ρ be the Weil representation of Γ on the group algebra C[L /L]. For γ L /L we let e γ be the orresponding basis vetor in the group algebra, and put e γ (τ) = e(τ) e γ. For a funtion f : h C[L /L] and (M, φ) Γ, we define f κ(m, φ) to be the funtion defined by (f κ(m, φ)) (τ) = φ(τ) 2k ρ (M, φ) 1 f(mτ). Here ρ (M, φ) is just the omplex onjugate of the matrix ρ(m, φ), sine the Weil representation is unitary and the standard basis is orthonormal. The transformation law for a weight κ vetor-valued modular form is exatly invariane under the the κ ation of Γ. Remark 2.1. (1) The assumption on the signature of L does not seem to be used. (2) The weight κ of the form is derived from the signature of the lattie, but this seems unneessary. That is, it seems that one an fix the lattie L and then work with arbitrary hoies of κ Definition (m positive). Fix β L /L and m Q positive suh that m + q(β) Z. Then e β (mτ) is invariant under κt. Indeed, we have ρ (T ) 1 e β = e(q(β))e β [is this right, or should it be e( q(β))?], and so e β (mτ) κt = ρ (T ) 1 e β e(m(τ + 1)) = e(q(β) + m)e β e(mτ) = e β (τ), as e(q(β) + m) = 1 sine q(β) + m is an integer. We define the Poinaré series by Pβ,m(τ) L = 1 e 2 β (mτ) κ(m, φ). (M,φ) Γ \ Γ The sum is well-defined sine e β (mτ) is invariant under κt. One easily sees that this defines a weight κ vetor-valued modular form. We often drop the L from the notation Fourier expansion. We have a Fourier expansion P β,m (τ) = p β,m (γ, n)e γ (nτ). γ L /L n>0, n+q(γ) Z Define the generalized Kloosterman sum by H sgn()κ/2 e iπ ( ) ( ) ma + nd a b (β, m, γ, n) = ρ β,γ ( γ)e, M =. d M Γ \Γ/Γ Here γ denotes a lift of γ to Γ and ρ β,γ ( M) denotes the entry of the matrix ρ( M) in row β and olumn γ. As previously mentioned, Γ \Γ is in bijetion with relatively prime pairs (, d) with 0. Sine is fixed, summing over M Γ \Γ/Γ is the same as summing over d (Z/Z), and then a is just d 1. Theorem 2.2. We have ( n (κ 1)/2 p β,m (γ, n) = δ m,n (δ β,γ + δ β,γ ) + 2π m) where J k 1 is the Bessel funtion used previously. Z\{0} ( H ) (β, m, γ, n)j 4π mn k 1,

7 LECTURES 4 & 5: POINCARÉ SERIES Petersson inner produt. Let, be the Hermitian inner produt on C[L /L] for whih the basis vetors e γ are orthonormal. We then define the Petersson inner produt of weight κ vetor-valued modular forms f and g (at least one of whih is uspidal) by (f, g) = f(τ), g(τ) y κ 2 dxdy. Γ\h This defines a non-degenerate hermitian form on the spae of usp forms S κ,l. We have: Theorem 2.3. Let f S κ,l be a usp form and let (β, m) denote its Fourier oeffiients. Then Γ(κ 1) (f, P m,β ) = 2 (β, m). (4πm) κ 1 Corollary 2.4. The series P m,β with m > 0 and β L /L span S κ,l Eisenstein series (m = 0). As we have seen, the Poinaré series with m > 0 span the spae of usp forms. We d like to be able ot span all modular forms by inluding the Poinaré series at m = 0, as in the lassial ase. However, there is an issue here: the funtion e β (mτ) is only invariant under κt if q(β) + m Z. Thus we an only hope to define the Poinaré series at m = 0 under the assumption q(β) Z. However, this turns out to be enough. We now give some details. Fix β L /L with q(β) Z. We define the Eisenstein series by Eβ L (τ) = 1 e 2 β κ(m, φ). (M,φ) Γ \ Γ Here e β denotes the onstant funtion h C[L /L] with value e β. This is easily seen to be a modular form. It is a theorem that the E β and P m,β span the spae M κ,l of all modular forms. Consider the Fourier expansion of E β : E β (τ) = q β (γ, n)e γ (ntau). We have and, for n > 0, γ L /L n 0 q(γ)+n Z q β (γ, n) = (2π)κ n κ 1 Γ(κ) q β (γ, 0) = δ β,γ + δ β,γ Z\{0} 1 κ H (β, 0, γ, n). It is a theorem that this quantity is in fat a rational number. Remark 2.5. It is not diffiult to see that E β = E β. It is a theorem that there are no other linear dependenies among the Eisenstein series. Thus the dimension of the spae of Eisenstein series (i.e., the dimension of the omplement of S κ,l in M κ,l ) is equal to #{β L /L 2β = 0, q(β) Z} #{β L /L 2β 0, q(β) Z}. 3. Non-holomorphi vetor-valued Poinaré series We now give one more variant of the Poinaré series: non-holomorphi series of negative weight. We only do the vetor-valued theory, though there is a simpler salar theory (and perhaps it would have been good to do this first for pedagogial reasons).

8 8 ANDREW SNOWDEN 3.1. Notation. We fix an even lattie L as before, with the same assumptions on the signature. We now let k = 1 l, whih will be the weight of our forms. For a funtion 2 f : h C[L /L], we put (f k (M, φ))(τ) = φ(τ) 2k ρ(m, φ) 1 f(mτ) We let ( ) k = y 2 2 x y 2 be the weight k Laplaian. It satisfies ( + iky x + i ) y k (f k (M, φ)) = ( k f) k (M, φ). We let M ν,µ (z) and W ν,µ (z) be the Whittaker funtions. They span the spae of solutions of the Whittaker differential equation ( d 2 w dz ν ) z µ2 1/4 w = 0 z 2 and are distinguished from eah other by different asymptoti behavior at 0 and. For s C and y > 0 we put M s (y) = y k/2 M k/2,s 1/2 (y), and for s C and y R \ {0} we put W s (y) = y k/2 W k sgn(y)/2,s 1/2 ( y ) Definition. Fix β L /L and m Q negative suh that m + q(β) Z. The funtion M s ( 4πmy)e β (mx) is invariant under k T (same alulation as previous setion, the M s fator is irrelevant) and an eigenfuntion of k of eigenvalue s(1 s) + (k 2 2k)/4 (this is where the M s fator is important). We define the Poinaré series by F β,m (τ, s) = 1 2Γ(2s) (M,φ) Γ \ Γ (M s ( 4πmy)e β (mx)) k (M, φ), for Re(s) > 1. This is invariant under the k ation of Γ and an eigenfuntion for k of eigenvalue s(1 s) + (k 2 2k)/4. Sine it is an eigenfuntion of k, it is real-analyti in τ. It is holomorphi in s Fourier expansion. Sine F β,m (τ, s) is invariant under x x + 1, it admits a Fourier expansion F β,m (τ, s) = (γ, n, y)e γ (nx). γ L /L n+q(γ) Z Note that the oeffiients depend on y, and we have e γ (nx) instead of e γ (nτ). Bruinier gives a formula for the Fourier oeffiients. It is quite involved, taking an entire page to state fully. It again uses generalized Kloosterman sums and (different) Bessel funtions, and now also involves the Whittaker funtion W s (y). When s = 1 k/2 the formula simplifies somewhat, but is still more ompliated than I want to try to reprodue here! The important point seems to be that it has the following form: F β,m (τ, s) = e β (mτ) + e β (mτ) + (bounded as τ ).

9 LECTURES 4 & 5: POINCARÉ SERIES 9 (Reall that m < 0, so e(mτ) is not bounded as τ.) Using this, Bruinier obtains the following result: Theorem 3.1. Let f be a vetor-valued modular form of weight k that is holomorphi on h and meromorphi at (or nearly holomorphi in Bruinier s terminology). Then f is a linear ombination of the series F β,m (τ, 1 k/2) with m < 0. Proof. The idea of the proof seems to be the following. By the previous formula for the first terms in the Fourier expansion of F β,m, one an onont a linear ombination g of the F β,m s that has the same prinipal part of f. ( Prinipal part meaning negative terms in Laurent expansion. ) The differene f g is then killed by k (sine s = 1 k/2), invariant under the k ation of Γ, and bounded as τ, and therefore vanishes by some sort of maximum modulus prinipal.