A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages S (XX A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES DRAGOS OPREA Abstract We study the splitting properties of the Verlinde bundles over elliptic curves Our methods rely on the explicit description of the moduli space of semistable vector bundles on elliptic curves, and on the analysis of the symmetric powers of the Schrödinger representation of the Theta group 1 Introduction Recently, Popa defined and studied a class of vector bundles on the Jacobians of curves, which he termed the Verlinde bundles [15] The fibers of these vector bundles are the spaces of nonabelian theta functions on the moduli spaces of bundles with fixed determinant over the curve, as the determinant varies in the Jacobian Popa investigated the splitting properties of these bundles under certain étale pullbacks He further used these results to prove the Strange Duality conjecture at level 1, and to study the basepointfreeness of the pluri-theta series In this note, we study the Verlinde vector bundles in genus 1, correcting some of the statements of [15] The techniques and results of our work are also useful to the understanding of the higher genus case, as we will demonstrate in [13] To set the stage, consider a smooth complex projective curve X of genus g 1, and write U X (r, r(g 1 for the moduli space of rank r, degree r(g 1 semistable bundles on X This moduli space comes equipped with a canonical Theta divisor supported on the locus (11 Θ r = { V U X (r, r(g 1, such that h 0 (V = h 1 (V 0 } Following Popa [15], we define the level k Verlinde bundles on the Jacobian as the pushforwards (12 E r,k = det ( Θ k r under the determinant morphism det : U X (r, r(g 1 Jac r(g 1 (X = Jac(X Among the results Popa proved, we mention: (i the pullback of E r,k under the multiplication morphism r : Jac(X Jac(X splits as a sum of line bundles; (ii E r,k is globally generated if k r+1, and is normally generated if k 2r+1; 2000 Mathematics Subject Classification Primary 14H60, Secondary 14H40 Supported by NSF grant DMS c XXXX American Mathematical Society

2 2 DRAGOS OPREA (iii E r,k is ample, polystable with respect to any polarization on the Jacobian, and satisfies IT 0 In addition, it is known that the Verlinde bundles enjoy the following level-rank symmetry: (iv there is an isomorphism SD : E r,k = Êk,r The hat decorating the bundle on the right hand side denotes the Fourier- Mukai transform with kernel the normalized Poincaré bundle on the Jacobian The morphism (iv, sometimes termed Strange Duality, was constructed in this form by Popa Proofs that SD is an isomorphism can be found in [10] [3] The case of elliptic curves, which will be relevant for us, is simpler; a discussion is contained in [7] To explain the results of this note, assume from now on that X is a smooth complex projective curve of genus 1 For reasons which will become clear only later, let us temporarily write h for the rank of the bundles making up the moduli space We will first show: Theorem 11 Let k, h and q be positive integers The Verlinde bundle E h,k splits as a sum of line bundles iff the level k is divisible by the rank h When k = h(q 1, we have ( (13 E h,h(q 1 = Θ q 1 m L Here, Θ is the canonical Theta bundle on the Jacobian, and the L s are the h- torsion line bundles Each line bundle L of order ω occurs with multiplicity (14 m = ( { } 1 qδ h/ω qδ 2, δ h/δ δ h provided that either h or q is odd If both h and q are even, then (15 m = ( 1 δ ( { } qδ h/ω qδ 2 δ h/δ δ h The symbol {} appearing in the above statement is defined as follows For any integer h 2, we decompose h = p a1 1 pan n into powers of primes We set (16 Here, { } λ h { 0 if p a pn an 1 does not divide λ, = ( n i=1 ɛ i 1 otherwise p 2 i ɛ i = { 1 if p ai i λ, 0 otherwise If h = 1, the symbol is always defined to be 1

3 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 3 Note that it was expected that the splitting of the Verlinde bundles should involve only h-torsion line bundles In fact, Popa proved the isomorphism ( hq h E h,h(q 1 = h Θ N, where N = 1 q h However, the multiplicities m of the nontrivial bundles L were incorrectly claimed to be 0 in Propositon 27 of [15] This led to an erroneous statement in Proposition 53 Our note corrects this oversight As an example, when h is an odd prime, all nontrivial h-torsion line bundles appear in the decomposition (13 with the same (nonzero multiplicity This follows for instance by the arguments of [2], upon analyzing the action of a symplectic group on the h-torsion points This is consistent with the Theorem above, which specializes to Here, n = 1 h 2 ( 1 q Our proof will show that E h,h(q 1 = Θ q 1 0 L n O m ( qh 1, and m = 1 ( ( 1 qh h h q h ( m = dim Sym h(q 1 Hh S h, with S h being the Schrödinger representation of the Heisenberg group H h If h is not prime, the ensuing formulas for multiplicities are more complicated, and their integrality is not immediately clear Theorem 11 is stated for the moduli spaces of bundles of degree zero The case of arbitrary rank and degree, and of arbitrary Theta divisors will be the subject of Theorem 31 in Section 31 The case when the level is not divisible by the rank is slightly more involved, and requires additional ideas We will consider this most general situation separately, in Section 32 To explain the final result, let us first change the notation, writing hr for the rank of the bundles making up the moduli space, and letting hk be the level If gcd(r, k = 1, then, for any h-torsion line bunde, there is a unique stable bundle W r,k, on the Jacobian, having rank r and determinant Θ k We will show: Theorem 12 Assume that gcd(r, k = 1 The Verlinde bundle of level hk splits as (17 E hr,hk = W m r,k, For each h-torsion line bundle on the Jacobian, having order ω, the multiplicity of the bundle W r,k, in the above decomposition equals (18 m = ( 1 (h+1krδ ( { } (r + kδ h/ω (r + kδ 2 rδ h/δ δ h

4 4 DRAGOS OPREA Finally, the methods of this work make use of the characteristic zero hypothesis In positive characteristic, it is likely that the answer is different, and that it depends on the Hasse invariant of the curve After this paper was written, Fabrizio Catanese informed us that computations of similar flavor appear in [5], in connection with certain moduli spaces of complex surfaces; we would like to thank him for pointing out the reference In this note however, we cover most general numerics, making use of stronger representation theoretic results for Heisenberg groups Furthermore, we handle the combinatorics involved in the calculation differently, expressing the final answer in closed form 2 The proof of Theorem 11 The title of this section is self-explanatory The proof of Theorem 11 to be given below relies on two essential ingredients: (i first, the geometric input is provided by the explicit description of the moduli space of bundles over elliptic curves, as found in [1][16]; (ii second, it will be crucial to understand the symmetric powers of the Schrödinger representation of the Heisenberg group An algebraic computation will determine their characters, which are related to the decomposition (13 We will discuss these two items at some length in the next sections, attempting to keep the exposition reasonably self-contained Our arguments are quite elementary, so it is plausible that some of the results below may already exist in the literature; we tried to provide references, whenever possible 21 Geometry Fix an elliptic curve (X, o Throughout the paper we identify X = Jac 0 (X in the usual way, p O X (p o In [1][16], Atiyah and Tu showed that the moduli space U X (h, 0 of rank h, degree 0 semistable vector bundles on X is isomorphic to the symmetric product U X (h, 0 = Sym h X Up to S-equivalence, the isomorphism can be realized explicitly as (21 Sym h X (p 1,, p h O X (p 1 o O X (p h o U X (h, 0 Under these identifications, the morphism taking bundles to their determinants det : U X (h, 0 Jac(X is the Abel-Jacobi map, which in this case becomes the addition a : Sym h X X, (p 1,, p h p p h Note that the fiber of the morphism a over the point p X is the linear series (22 [p] + (h 1[o] = [p] [o] + h[o] In fact, as an Abel-Jacobi map, the morphism a has the structure of a projective bundle P(V h X, where V h is a rank h vector bundle on X To describe V h, we let P be the Poincaré bundle over X X, normalized in the usual way P = O X X ( {o} X X {o},

5 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 5 with X X being the diagonal The Fourier-Mukai transform with kernel P, denoted RS : D(X D(X, is given by for α D(X Here RS(α = Rp 2! (p 1α P, p 1, p 2 : X X X are the projections on the two factors Using (22, we conclude that (23 V h = RS(O X (h[o] A standard computation shows that V h has rank h and determinant [o]; the reference [9] spells out the details Furthermore, we note that V h is simple since it is the Fourier-Mukai transform of a simple bundle [11] In fact, by Atiyah s classic study [1], there is a unique bundle on X with all the above properties, defined inductively as the (unique nontrivial extension (24 0 V h 1 V h O X 0, starting with V 1 = O X ( [o] Alternatively, this exact sequence is obtained as the Fourier-Mukai transform of 0 O X ((h 1[o] O X (h[o] O {o} 0 Note that the line bundle (21 has a section precisely when p i = o for some 1 i h It follows from (11 that the canonical theta divisor Θ h on U X (h, 0 is the image of the symmetric sum (25 [o] + Sym h 1 X Sym h X Thus, the Theta line bundle Θ h agrees, at least fiberwise, with O P(Vh (1 In fact, one can show the isomorphism Θ h = OP(Vh (1 Moreover, the canonical section vanishing along the Theta divisor (25 is the composition O P(Vh O P(Vh (1 a V h O P(Vh (1, with the second arrow given by (24 These observations allow us to compute the level k Verlinde bundle (26 E h,k = a ( Θ k h = a ( OP(Vh (k = Sym k V h For convenience, we will write W h = Vh for the unique stable bundle on X of rank h and determinant O X ([o] More generally, if gcd(h, d = 1, we let W h,d be the unique stable bundle of rank h and determinant O X (d[o] The bundles W h,d can be constructed inductively as successive extensions cf [14], page 180 Indeed, consider two consecutive terms 0 d1 h 1 < d2 h 2 < 1 in the Farey sequence, ie assume that h 1 d 2 h 2 d 1 = 1 Set h = h 1 + h 2, d = d 1 + d 2 Then W h,d is the unique nontrivial extension 0 W h1,d 1 W h,d W h2,d 2 0

6 6 DRAGOS OPREA With these preliminaries out of the way, we proceed to investigate the splitting behavior of the Verlinde bundles E h,k Our analysis relies on the multiplicative structure of the Atiyah bundles [1], which may not be immediately obvious Lemma 21 The Verlinde bundle E h,k splits as a sum of line bundles if and only if h divides k Proof This result will be reproved later in the paper A more direct argument is given below First, observe that E h,k is a direct summand of W k h It suffices to show that these tensor powers split as sums of lines bundles iff h divides k In fact, something more general is true: Claim 22 Assuming gcd(h, d = 1, the tensor powers W k h,d split as sums of rank h bundles of the form W h,dk M where M are various degree 0 line bundles Here, we set h h = gcd(h, k, k k = gcd(h, k To prove the Claim, we first decompose h = h 1 h s into powers of primes, and pick integers d 1,, d s such that Then, d 1 h d s h s = d h W h,d = W h1,d 1 W hs,d s This could be argued as follows: both sides have the same (coprime rank and determinant, and are moreover semistable, in fact stable Therefore, they should coincide by Atiyah s classification With this understood, one checks that it is enough to take h to be a power of a prime p For the latter case, we will need the following rephrasing of Theorems 13 and 14 in [1] Assume e 1, e 2, e are integers not divisible by p, and that e 1 p + e 2 a1 p = e a2 p a Then, Atiyah showed that for certain degree 0 line bundles M, we have (27 W p a 1,e1 W p a 2,e2 = M W pa,e M Thus, when h is a power of a prime, the Claim follows from (27, by a straightforward induction on k Remark 23 Using a sharper version of Atiyah s results, one can prove that when h is odd, the M s appearing in the Claim above are representatatives for the cosets of h-torsion line bundles on X modulo the twisting action of the group of h -torsion line bundles The same statement should hold true for h even, but Atiyah s results only show that the orders of the M s divide 2h In particular, for h odd and gcd(h, k = 1, we immediately conclude that (28 E h,k m = W h,k, with m = 1 h+k ( h+k h i=1 Equation (28 is a particular case of Theorem 12

7 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 7 We will identify the splitting of E h,k = Sym k W h when the level k is divisible by the rank h We set q = 1 + k h Let X h be the group of h-torsion points on the elliptic curve Let G h be the Theta group of the line bundle O X (h[o], which is a central extension 1 C G h X h 1 The assignment η η 2h defines an endomorphism of G h, whose image lies in the center of G h This is easily checked for instance using (29 below Let H h be the kernel of this endomorphism It corresponds to an extension 1 µ 2h H h X h 1, where µ 2h C is the group of 2h-roots of 1 Finally, let S h denote the h- dimensional Schrödinger representation of G h, ie the unique representation such that the center of G h acts by its natural character We refer the reader to Mumford s paper [12] or to Chapter 6 of [4] for more details regarding Heisenberg groups and their Schrödinger representations Picking theta structures, we identify G h with the Heisenberg group (29 G h = C Z/hZ Z/hZ The multiplication on the right hand side is defined as (α, x, y(α, x, y = (αα ζ y x, x + x, y + y Here, we set ( 2πi ζ = exp h The Schrödinger representation S h is realized on the space of functions f : Z/hZ C, cf [4], page 164 The action of the element (α, x, y G h on a function f is given by the new function F : Z/hZ C, F (a = αζ ya f(x + a We will first compute the pullbacks of the Verlinde bundles under the morphism h : X X which multiplies by h on the elliptic curve Using the description of V h as a Fourier-Mukai transform provided by (23, and Theorem 311 in [11], we obtain h V h = OX ( h[o] h In fact, we claim that G h -equivariantly, we have (210 h V h = Sh O X ( h[o]; see also [14], page 162 Indeed, consider the trivial bundle h V h O X (h[o] = V O X, where V is an h-dimensional vector space Both factors of the tensor product on the left carry a G h -action covering the translation X h -action on the base X Therefore, endowing the structure sheaf appearing on the right with the trivial G h -action,

8 8 DRAGOS OPREA we obtain a G h -representation on V Moreover, note that the center of G h acts on V by homotheties Therefore, V = S h, by the uniqueness of the Schrödinger representation This establishes (210 Taking determinants in (210, we obtain (211 h O X ( [o] = Λ h S h O X ( h[o] h This identification is a priori only G h -equivariant, but, since the center of G h acts trivially, the isomorphism is in fact X h -equivariant Similarly, dualizing and taking symmetric powers in (210, we obtain an X h -equivariant identification (212 h Sym k W h = Sym k S h O X (h[o] k = Sym k S h ( Λ h S h q 1 h O X ([o] q 1 Observe that the action of the central elements α of G h on the Heisenberg module is trivial, since M k = Sym k S h ( Λ h S h q 1 (213 α k (α h q 1 = 1 Therefore M k is an X h -module The X h -action splits into eigenspaces indexed by the characters of X h, each appearing with multiplicity m : (214 M k = m Let us write X h for the group of characters of X h For each X h, let L denote the corresponding h-torsion line bundle on X The pullback h L is the trivial bundle endowed with the X h -character Using (212 and (214, we obtain an X h -equivariant identification h Sym k W h = h L m h O X ([o] q 1 This equivariant isomorphism determines the Verlinde bundle on the left, by general considerations about the Picard group of finite quotients We can also give a direct argument as follows Pushing forward the previous equation by h, we obtain the X h -isomorphism (215 Sym k W h h O X = L m O X ([o] q 1 h O X Note that X h -equivariantly (216 h O X = b X h L Comparing (215 and (216, and singling out the X h -invariant part, we conclude that (217 E h,k = Sym k W h = O X ([o] q 1 L m X b h

9 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 9 22 Algebra It remains to determine the multiplicities m appearing in (214 We M k as a representation of the finite group H h Using the standard representation theory for finite groups as in [8] page 17, we obtain (218 m = 1 H h η H h (η 1 Tr Mk (η We will compute this sum explicitly with the aid of the following Lemma 24 Let η H h be an element whose image under the map H h X h has order exactly h/δ in X h The trace of η on M k equals Tr Mk (η = 1 ( qδ, q δ provided that either h or q is odd Proof We pick theta structures, so that G h and S h are given by (29 Consider the basis f 1,, f h of S h given by By definition, the action of is given as f i (j = δ i,j η = (α, x, y µ 2h Z/hZ Z/hZ (219 η f i = αζ y(i x f i x To compute the trace of η on M k, we may assume that α = 1, since the scaling action of the center of H h is trivial, as remarked in (213 We begin by computing the trace Tr Sym k η of the action of η on Sym k S h For simplicity, we will first treat the case x = 0 The eigenvalues of the action of η on S h are 1, ζ y,, ζy h 1 Here, we set Therefore, (220 Tr Sym k η = 1 i 1 i k h ζ y = ζ y ζ (i1++i k y = j 1++j h =k ζ (j1+2j2++hj h y In the above, j r denotes the number of i s which equal r Now, we compute the generating series Write where k Tr Sym k η t k = 1 1 ζ 1 y t h = lm, 1 1 ζ 2 m = gcd(h, y and gcd(l, y = 1 y t 1 1 ζy h t Then ɛ = ζ y is a primitive root of 1 of order l Therefore, the product in the denominator above becomes (1 ζ 1 y t (1 ζ h t = ζ h(h 1/2 ( 1 h ( (t 1(t ɛ (t ɛ l 1 m y y = ( 1 h+y(h 1 (t l 1 m

10 10 DRAGOS OPREA We can extract the coefficient of t k : (221 Tr Sym k η = ( 1 h+y(h 1+m+ k l ( m k l = ( 1 k l ( m k l = 1 ( qm q m In particular, this computation implies that the sum (220 is 1 when m = gcd(h, y = 1 Moreover, the argument shows that the sum (220 vanishes if k is not divisible h by l = gcd(h,y We will now consider the η s in H h for which x 0 For these, the computation is notationally more involved To begin, we write x = x s, and h = h s, where s = gcd(h, x Let u be any constant with u h = ( 1 yx (h +1 Note in particular that u k = 1 for h odd For h even, we have (222 u k = ( 1 xy(q 1 Now, it is easy to see that the eigenvalues of η on S h are (223 λ i,j = u ζ i y σ j, 1 i s, 1 j h, where ( 2πi σ = exp In fact, we can exhibit an eigenvector for λ = λ i,j, namely v λ = h 1 k=0 h ki k(k+1 2 x λ k ζy f i kx We order the indices (i, j lexicographically The trace Tr Sym k η is obtained by summing all products ( ( ( λ 1 λ 1 1,j1 1 1,j λ 1 λ 1 1 2,j1 2 2,j λ 1 2 s,j 1 s λ 1 s,j, s where 1 j i 1 j i 2 j i h Let a 1 be the number of terms in the product whose first index is 1; a 2,, a s have the similar meaning We require a a s = k After substituting (223 in the product above, we sum over the j s, keeping the a s fixed We have seen already in the derivation of (220 that the sum 1 j i 1 ji a i h σ (j i 1 ++ji a i is 0 if h does not divide a i, and it equals 1 otherwise Therefore, writing a i = h a i, we need to evaluate ζ h a 1 y ζ 2h a 2 y ζ sh a s y = γ (a 1 ++sa s y a 1 ++a s = k h a 1 ++a s = k h

11 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 11 Here, we set γ = exp ( 2πi s, so that ζ h y = γ y We have already computed sums of this type in (220 We obtained the answer ( 1 qδ (224 q δ for δ = gcd(s, y = gcd(h, x, y This expression gives the trace Tr Sym k η when h is odd The formula includes the previously considered case x = 0, for which δ = m The sign change (222 is required when h is even Finally, the trace of η on Λ h S h is computed using (219: (225 η f 1 f h = ( 1 x(h+1 h i=1 ζ i x y f 1 f h = ( 1 (h+1(x+y f 1 f h This completes the proof when h is odd When h is even, we take into account the sign corrections of the previous paragraph and (222 We append formula (224 by the overall sign ( 1 xy(q 1 ( 1 (x+y(h+1(q 1 = ( 1 (xy+x+y(q 1 This does not change (224 when q is odd, proving the Lemma When h and q are both even, we note, for further use, that the overall sign of (224 can be rewritten as (226 ( 1 gcd(h,x,y = ( 1 δ We proceed to calculate the sum (218 We claim that the multiplicity m depends only on the order of the character X h To this end, consider the group Aut(H h, µ 2h of automorphisms of H h which restrict to the identity on the center µ 2h As essentially remarked in [2], the characters appearing in the X h - representation M k are exchanged by the action of Aut(H h, µ 2h Beauville s argument is based on the observation that for each F Aut(H h, µ 2h, the standard H h -module structure of S h, ρ : H h GL(S h, is isomorphic to the twisted module structure F ρ : H h GL(S h This follows by examining the character of the center of H h, and by making use of the uniqueness of the Schrödinger representation The same observation applies to the associated H h -module M k With this understood, our claim is a consequence of the Lemma below This result is possibly known, yet for completeness we decided to include the argument Note that the Lemma is not indispensable for the proofs to follow, yet it allows for some simplification of the formulas Lemma 25 Under the action of Aut(H h, µ 2h, two characters of X h belong to the same orbit if and only if they have the same order in X h Proof Fix two characters χ 1, χ 2 of X h : χ i : X h C, (x, y ζ aix+biy, 1 i 2 The condition on the orders of χ 1 and χ 2 translates into gcd(h, a 1, b 1 = gcd(h, a 2, b 2 := τ This implies that we can solve the equations below, with the Greek letters as the unknows: (227 a 1 λ + b 1 µ = a 2 mod h, a 1 ν + b 1 γ = b 2 mod h

12 12 DRAGOS OPREA We claim that we may further achieve (228 λγ µν = 1 mod h This can be seen for instance as follows By the Chinese Remainder Theorem, we may take h to be a power of a prime In this case, assume first that τ = 1 Starting with any solution of (227, define a new quadruple λ = λ + b 1 x, µ = µ a 1 x, ν = ν + b 1 y, γ = γ a 1 y The assumption τ = 1 implies that we can find a pair (x, y such that (228 holds: λ γ µ ν = (λγ µν + b 2 x a 2 y 1 mod h For arbitrary τ, after dividing by τ, and using the case we already proved, we may assume that (227 is satisfied mod h, and that (228 holds true mod h/τ We lift the solution using Hensel s lemma, ensuring that (228 is also satisfied mod h Finally, define F : H h H h by F (α, x, y = (αζ 1 2 (λµx2 +νγy 2 +2µνxy, λx + νy, µx + γy Equation (228 is used to prove that F is an automorphism of H h, while equation (227 shows that F sends χ 1 to χ 2 Henceforth, for the computation of (218, we will take to be the character = λ : Z/hZ Z/hZ (x, y ζ x+y λ = ζ λ(x+y C Here, we assume that λ divides h, so that the character has order δ h ω = h λ Assume that either h or q is odd Using Lemma 24, we rewrite (218 as (229 m = 1 ( 1 qδ h 2 λ (x, y q δ gcd(h,x,y=δ If both h and q are even, each term in (229 is multiplied by the sign ( 1 δ, as it follows from (226 In this case, (230 m = 1 ( 1 δ ( qδ h 2 λ (x, y q δ gcd(h,x,y=δ δ h We will evaluate formulas (229 and (230 in terms of the character (16 defined in the introduction Lemma 26 We have gcd(h,x,y=δ λ (x, y = h2 δ 2 { } h/ω h/δ Proof Replacing h, x and y by h/δ, x/δ and y/δ respectively, we may assume δ = 1 To solve this case, let us set (231 N λ (h = λ (x, y = ζ x+y λ gcd(h,x,y=1 gcd(h,x,y=1

13 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 13 It suffices to show that (232 N λ (h = h 2 { λ h This is immediate when h = p a is a power of a prime In this case, if p a λ, the left hand side of (232 counts the pairs 1 x, y p a such that gcd(p a, x, y = 1 Their number is p 2a 2 (p 2 1, which equals the right hand side Otherwise, since the distinct roots of unity add up to 0, we have = (x,y,p a =1 ζ x+y λ } p (x,y ζ x+y λ If p a 1 λ, then all terms in the last sum are equal to 1, hence giving the answer p 2a 2 Finally, if p a 1 does not divide λ, then replacing ζ λ by ζ pλ, we sum all distinct roots of unity of order p a 1 / gcd(p a 1, λ, each appearing with equal multiplicity This gives the answer 0 The general case follows by induction on the number of prime factors of h, once we establish the multiplicativity in h of the function N λ (h Let h = h 1 h 2 with gcd(h 1, h 2 = 1 Chose integers u, v such that h 1 u + h 2 v = 1 By the Chinese Remainder Theorem, the pairs (x, y mod h are in one-to-one correspondence with pairs (x 1, y 1 mod h 1, (x 2, y 2 mod h 2 such that x x 1 mod h 1, x x 2 mod h 2, Explicitly, we have y y 1 mod h 1, y y 2 mod h 2 x = h 1 ux 2 + h 2 vx 1 mod h, y = h 1 uy 2 + h 2 vy 1 mod h The condition gcd(h, x, y = 1 is equivalent to We compute N λ (h = gcd(h,x,y=1 gcd(h 1, x 1, y 1 = 1, gcd(h 2, x 2, y 2 = 1 ζ x+y λ = = N λv (h 1 N λu (h 2 = h 2 1 gcd(h 1,x 1,y 1=1,gcd(h 2,x 2,y 2=1 { λv h 1 } h 2 2 ζ h2v(x1+y1 λ ζ h1u(x2+y2 λ { } { } { } { λu λ = h 2 = h λh1 λh2 2 h In the last line, we used the fact that the factors u and v do not change the symbol {} since these numbers are prime to h 2 and h 1 respectively h 2 } Putting together (217, (229, (230 and Lemma 26, we complete the proof of Theorem 11

14 14 DRAGOS OPREA 3 Arbitrary numerics 31 Arbitrary rank and degree We will now discuss a variant of Theorem 11, which covers the case of arbitrary rank and degree Let r, d be two integers with h = gcd(r, d Write r = hr, d = hd, where gcd(r, d = 1 We will consider Theta divisors on the moduli space U X (r, d Their definition requires the choice of a twisting vector bundle N of complementary slope We set µ(n = d r (31 Θ r,n = {V U X (r, d, such that h 0 (V N = h 1 (V N 0} To avoid confusion, even though it may be notationally cumbersome, we decorate the Theta s by the twisting bundles N, and by the rank of the bundles in the moduli space It is convenient to assume that N has the minimal possible rank r The level k Verlinde bundle E N r,k = det ( Θ k r,n is obtained by pushing forward the pluri-theta bundle Θ k N on U X(r, d via the morphism det : U X (r, d Jac d (X As before, we have an isomorphism (32 U X (r, d = Sym h X Set-theoretically, this isomorphism is essentially defined twisting (21 by the unique idecomposable vector bundle W r,d of rank r and determinant d [o] on X More precisely, if (p 1,, p h are h points of X, pick (q 1,, q h such that r q i = p i, 1 i h Then, the isomorphism (32 is given by (33 Sym h X (p 1,, p h W r,d O X(q 1 o W r,d O X(q h o U X (r, d Note that the answer on the right hand side of (33 is independent of the choice of q i Indeed, any two q i s must differ by an r -torsion point χ However, by Atiyah s classification, (34 W r,d L χ = W r,d, as both bundles are simple, of the same rank and determinant It was observed in [16], and it is clear from (33, that the determinant becomes the addition morphism Here, we used the identification det : U X (r, d Jac d (X a : Sym h X X, (p 1,, p h p p h X = Jac(X = Jac d (X,

15 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 15 with the second arrow given by twisting degree zero line bundles by O X (d[o] Via this identification, the divisor Θ 1,O( d[o] on Jac d (X corresponds to the canonical Theta on Jac(X Finally, we can easily identify the Theta divisors on U X (r, d There is a natural choice for the twisting bundle N, namely the Atiyah bundle N 0 = W r, d It was shown in [1], and it follows from equation (34, that the tensor product (35 W r, d W r,d = χ L χ splits as the direct sum of all r -torsion line bundles L χ As a consequence of (31, (33, (35, we see that for the bundles V in the Theta divisor, we have q i = χ for some r -torsion point χ, and some 1 i h Thus, Θ r,n0 is the image of the symmetric sum [o] + Sym h 1 X Sym h X We have therefore recovered (25, and thus reduced the computation to the case we already studied Theorem 31 Fix r and d two integers with h = gcd(r, d, and N a vector bundle of slope µ(n = d r, and of minimal rank Then, E N r,k splits as sum of line bundles iff h divides k If k = h(q 1, then E N r,k = ( q 1 Θ 1,(det N h L m X b h Here m are given by the same formulas (14 and (15 as in Theorem 11 Proof When N 0 = W r, d, the statement is a consequence of the above discussion and the proof of Theorem 11 The general case follows from here, since both the Verlinde bundle and the right hand side only change by translations To see this, set L = det N (det N 0 1 On the one hand, formulas of Drezet-Narasimhan [6] imply that E N r,k = det ( Θ k r,n = det (Θ r,n0 det L k = E N0 r,k Lk In the above, we view the degree 0 line bundle L on X, as a line bundle on the Jacobian in the standard way On the other hand, we have Θ 1,(det N h = Θ 1,(det N0 h Lh The Theorem follows by putting these observations together 32 Arbitrary level and rank In this subsection we will prove Theorem 12 We will keep the same notations as in the introduction, writing hr for the rank, and letting hk be the level, with gcd(r, k = 1 We will determine the splitting type of the Verlinde bundle E hr,hk = det ( Θ hk hr = Sym hk W hr,

16 16 DRAGOS OPREA obtained by pushing forward tensor powers of the canonical Theta bundle Θ hr via det : U X (hr, 0 Jac(X = X The case of non-zero degree and arbitrary Theta s is entirely similar, and we will leave the details to the interested reader Proof of Theorem 12 We first consider the case when r is odd The arguments used to prove Theorem 11 go through with only minor changes It suffices to check that the decomposition (17: E hr,hk = W m r,k, holds X hr -equivariantly, after pullback by the morphism hr The pullback of the left hand side is evaluated G hr -equivariantly via (212: (36 (hr E hr,hk = (hr Sym hk W hr = OX (hr[o] hk Sym hk S hk For the right hand side, recall first that W r,k, has rank r and determinant O X (k[o] By comparing ranks and degrees, we see that (37 W r,k, = Wr,k L χ Here L χ is any hr-torsion line bundle with L r χ = L Note χ is uniquely defined only up to r-torsion line bundles The ambiguity inherent in the choice of χ will be shown to be inessential later Observe that the pullback (hr L χ is the trivial bundle, endowed with the X hr -character χ We will determine the pullback of W r,k by the morphism hr As a first step, we show that non-equivariantly (38 r W r,k = OX (kr[o] r The ingredients needed for the proof of (38 are found in Lemma 22 of Atiyah s paper [1] There, it is explained that all indecomposable factors of r W r,k have the same rank r and degree k Therefore, r/r r W r,k = W r,k M i i=1 for some line bundles M i In fact, examining Atiyah s arguments, one can prove a little bit more Using (35, we observe that r W r,k r Wr,k = O X The above tensor product contains W r,k W r,k M i M 1 j as a direct summand, for any i and j Now, applying equation (35 again, we see that r 2 W r,k W r,k = ρ the sum being taken over the r -torsion points ρ This clearly gives a contradiction, unless r = 1 and the bundles M i and M j coincide In conclusion, we proved that (39 r W r,k = r i=1 M, for a suitable line bundle M Taking determinants we obtain that M = O X (kr[o] P, 1 L ρ,

17 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 17 for some r-torsion line bundle P We claim that P is symmetric, ie ( 1 P = P When r is odd, these two facts together imply that P must be trivial, proving (38 The symmetry of P is a consequence of (39 and of the symmetry of W r,k Indeed, ( 1 W r,k = Wr,k, as both bundles are simple, and have the same rank and determinant Having established (38, we compute (310 (hr W r,k = OX (hr[o] hk R, where R is an r-dimensional vector space In fact, R carries a representation of the Theta group G hr, such that the center acts with weight hk However, this does not determine the representation R uniquely, not even as a representation of H hr In fact, one can show that there are precisely h 2 representations R i,j of H hr with central weight hk [17]; they will be indexed by integers i, j Z/hZ Z/hZ To determine R, we will use the following commutative diagram 0 C G hr X hr 0 i 0 C G h X h 0 Here, the morphism i is the r-fold cover α α r, and the middle arrow is the natural morphism of Theta groups G h G hr Via this diagram, we may consider the action of the group G h on both sides of (310 Recall from equation (211 that G h -equivariantly, we have O X (hr[o] hk = OX (h[o] hkr = h O X ([o] kr ( Λ h S h kr Therefore, using (310, we see that G h -equivariantly, R ( Λ h S h kr = h (r W r,k O X ( kr[o] Note that the left hand side is an X h -module, since the center of G h acts trivially; to this end, recall that the morphism i is an r-fold covering of the centers By equation (38, the right hand side is the pullback of a trivial vector bundle, carrying a trivial X h -action Consequently, the X h -representation R ( Λ h kr S h is trivial This latter observation pins down the H hr -representation R Let us again pick theta structures, identifying the Theta group H hr with the Heisenberg The characters of the h 2 representations R i,j were computed in Theorem 3 in [17] There it was proved that the trace of η = (α, x, y H hr = µ2hr Z/hrZ Z/hrZ equals (311 { rα hk ζ ix+jy if (x, y X h, ie if (x, y rz/hrz rz/hrz, Trace Ri,j (η = 0 otherwise Here ζ = exp ( 2πi hr The character of the Hh -representation ( Λ h kr S h was calculated in (225: Trace (η = α hkr ( 1 (h+1(x+ykr Since R ( Λ h kr S h is a trivial Xh -module, we must have i = j = hrk(h+1 2 Then, the trace of R becomes { rα hk ( 1 (h+1kr(x+y if (x, y X h, (312 Trace R (η = 0 otherwise

18 18 DRAGOS OPREA Making use of (36 and (38, we can now check that both sides of (17 agree equivariantly after pullback by hr It remains to prove that H hr -equivariantly: (313 Sym hk S hr = R χ χ mχ In this sum, the χ s are h 2 representatives of the characters of X hr, modulo those characters of X hr which restrict trivially to the subgroup X h X hr Taking representatives is necessary to avoid repetitions Indeed, by comparing characters, we see that R χ = R iff χ restricts trivially to the subgroup X h This equation also takes care of the ambiguity seemingly present in the pullback of (37 by hr Note moreover that each representative χ appearing in the sum (313 restricts to a well-defined character of X h, hence giving rise to an h-torsion line bundle L on X We will write ω for the order of this line bundle In (313, the multiplicities m χ are claimed to have the expressions given in equation (18 of the Theorem Checking (313 amounts to a character calculation For the left hand side, the character was essentially computed in Lemma 24 Going through the proof of the Lemma, we see that the trace of η = (α, x, y H hr on Sym hk S hr is zero, unless (x, y is an h-torsion point, say of order h/δ in X h In the latter case, ( (r + kδ (314 Trace (η = ( 1 xyk(h+1 α hk r r + k It suffices to check that the formula m χ = 1 2h 3 Trace r Sym hk S hr (η Trace R (η 1 χ(η 1 η=(α,x,y µ 2hr X h yields the same answer as (18 Substituting (312 and (314, and recalling that denotes the restriction of χ to X h, we obtain m χ = 1 ( 1 (h+1krδ ( (r + kδ h 2 (x, y r + k rδ δ h By Lemma 26, this expression can be rewritten as m χ = ( 1 (h+1krδ ( (r + kδ (r + kδ 2 rδ δ h rδ (x,y has order h/δ { h/ω h/δ This completes the proof when r is odd When r is even, k must be odd, since gcd(r, k = 1 Therefore, the Theorem holds true for the Verlinde bundle E hk,hr We will now use the level-rank symmetry of the Verlinde bundles under the Fourier-Mukai transform E hr,hk = Êhk,hr, which was explained in item (iv of the introduction We claim that the Atiyah bundles enjoy the analogous symmetry under Fourier-Mukai: W r,k, = Ŵk,r, }

19 A NOTE ON THE VERLINDE BUNDLES ON ELLIPTIC CURVES 19 Indeed, the case of trivial is the following well-known isomorphism generalizing (23: W r,k = Ŵr,k This is a consequence of the fact that both bundles are simple, of the same rank, and same determinant; alternatively, one may argue using the construction of the Atiyah bundles as successive extensions, explained in Section 21 The case of general is an immediate corollary, since the bundles involved differ only by translations To see this, pick any line bundle M with M k =, and let τ M denote the translation induced by M on the elliptic curve We compute Ŵ k,r, = W k,r M = τ M Ŵk,r = τ M W r,k = W r,k, The first and last isomorphism follow as usual by Atiyah s classification, while the second is a general fact about the Fourier-Mukai transform [11] We conclude the proof of the Theorem by collecting the above observations 4 acknowledgements We would like to thank Alina Marian for conversations related to this topic and many useful suggestions, and Mihnea Popa for correspondence References 1 M Atiyah, Vector bundles over an elliptic curve, Proc London Math Soc, 7 (1957, A Beauville, The Cobble hypersurfaces, C R Math Acad Sci Paris, 337 (2003, no 3, P Belkale, The strange duality conjecture for generic curves, to appear in J Amer Math Soc 4 C Birkenhake, H Lange, Complex abelian varieties, Springer-Verlag, Berlin-New York, F Catanese, C Ciliberto, Symmetric products of elliptic curves and surfaces of general type, J of Alg Geom 2 (1993, J M Drezet, MS Narasimhan, Groupe de Picard des varietes de modules de fibres semistables sur les courbes algebriques, Invent Math 97 (1989, no 1, R Donagi, L Tu, Theta functions for SL(n versus GL(n, Math Res Lett 1 (1994, no 3, J Harris, W Fulton, Representation Theory, Springer-Verlag, Berlin-New York, G Hein, D Ploog, Stable bundles and Fourier-Mukai transforms on elliptic curves, Contributions to Algebra and Geometry 46 (2005, A Marian, D Oprea, The level-rank duality for non-abelian theta functions, Invent Math 168 (2007, no 2, S Mukai, Duality between D(X and D( ˆX with its application to Picard sheaves, Nagoya Math J 81 (1981, D Mumford, On equations defining abelian varieties I, Inventiones Math, 1 (1966, D Oprea, The Verlinde bundles and the semihomogeneous Wirtinger duality, submitted, arxiv: A Polishchuk, Abelian varieties, theta functions and the Fourier-Mukai transform, Cambridge University Press, Cambridge, M Popa, Verlinde bundles and generalized theta linear series, Trans Amer Math Soc, 354 (2002, no 5, L Tu, Semistable bundles over an elliptic curve, Adv Math 98 (1993, no 1, J Schulte, Harmonic analysis on finite Heisenberg groups, European J Combin 25 (2004, no 3, Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, address: doprea@mathucsdedu