The Fourier coefficients of metaplectic theta series on GL(2) over rational function fields
aa r X i v : . [ m a t h . N T ] N ov The Fourier coefficients of metaplectictheta series on GL(2) over rational functionfields ∗ S.J.Patterson
To Jeff Hoffstein, to mark his 60 th birthday. Abstract
The idea of the metaplectic theta function was introduced byTomio Kubota in the 1960s. These theta functions are constructedas residues of Eisenstein series and are only known completely in thecase of double covers and, up to the ambiguity of a constant, for triplecovers. In 1992 Jeff Hoffstein published formulæ by which these canbe computed in certain cases over a rational function field. The authorgave an alternative approach in 2007. Both of these methods give thecoefficients in a closed form. The rational function field is unusual inthat it has a large automorphism group. In this paper we show thatthis group has an operation on the coefficients. This operation is notvisible from the explicit formulæ.
The purpose of this paper is to take up the investigation begin in the paper[9]. The main result of this paper is Theorem 1 which describes the action ofthe group of automorphisms of the rational function field on the coefficientsof metaplectic theta series.We fix an integer n and an odd prime power q , q ≡ n ). Thecoefficients of the theta functions are quantities denoted by ρ ( r, ε, i ) where ∗ The results of this papers were presented at a conference held in the Perrotis Agricul-tural School in Thessaloniki from 14 th to 18 th July, 2014 to celebrate Jeff Hoffstein’s 60thbirthday. I would like to thank the organisers for the opportunity to speak at it. is an integer 0 ≤ i < n and r is a non–zero element of F q [ x ]. The quantitiesdescribe the asymptotic distribution of Gauss sums of order n over F q [ x ]. Acrucial property of ρ ( r, ε, i ) is that it depends only on r modulo n th powers.Although the ρ ( r, ε, i ) were constructed only for polynomial r and they canbe extended to arbitrary rational functions r by this property.The automorphism group of F q ( x ) over F q is PGL ( F q ). Let (cid:18) a bc d (cid:19) bea non-degenerate matrix over F q . We shall show that ρ ( r , ε, i ) = ρ ( r , ε, i )where r ( x ) = r (( ax + b ) / ( cx + d ))(( ad − bc ) / ( cx + d ) ) − i . It follows thatthere are a large number of relationships between the ρ ( r, ε, i ). In particularthis very much simplifies the task of computing tables of this function forgiven n and q .Section 2 is dedicated to summarizing and completing the results of [9]and in Section 3 we give the proof of Theorem 1. It should be noted that thefact that such a statement should hold follows from the general discussion ofthe formalism of the ρ given in [7] and [8].In Section 4 we shall discuss the consequences of Theorem 1 when thereare at most three irreducible factors of r and these are of degree 1. Althoughthese are very special they exhibit a number of interesting features which aresuggestive. In particular these coefficients are, as was already noted in [9],related to Selberg sums and some generalizations of these. Selberg sums wereevaluated by G.W. Anderson, [1] , R.J. Evans, [3] and P.B. van Wamelen,[12], and the theory of the metaplectic group throws a new light on thegeneral class of these sums. We shall here both establish the notations we shall need and recall thosere-sults from [9] which we shall make use of here. The notations will, as theresult of experience differ a little from those in the previous paper.We let q and n be as above and let k = F q ( x ) and R = F q [ x ]. We define themap χ : F × q → µ n ( F q ); x x ( q − /n where for any field k µ n ( k ) is the set of n th roots of 1 in k . Let ε be an embedding of µ n ( F q ) in C × . Let e o : F q → C × be a additive character. It is convenient, and involves no loss of generalityfor our purposes, to assume that e o maps a to e πıj/p where the residue class j (mod p ) represents Tr F q / F p ( a ). We define an additive character e on k by e ( f ) = e o ( P v Res v ( f dx )) = e o ( − Res ∞ ( f d x )) where the sum over v is overall finite places of k . Note that in terms of the uniformizer x ∞ = x − at ∞ the latter expression is e o (Res ∞ ( f x − ∞ d x ∞ ). We define the Gauss sumsover R to be g ( r, ε, c ) = P ξ (mod c ) ε ( (cid:0) ξc (cid:1) n ) e o ( rξ/c ) . Here r and c are non-zero2lements of R . For a character ω of F × q we define the finite field Gauss sum τ ( ω ) = P j ∈ F q ω ( j ) e o ( j ).The Davenport-Hasse theorem implies that for r coprime to c one has g ( r, ε, c ) = µ ( c ) ε ( (cid:16) rc (cid:17) − n (cid:18) c ′ c (cid:19) n )( − τ ( εχ )) deg( c ) where µ denotes the M¨obius function in R . The case where c and r are nolonger assumed to be coprime can be reduced to this case. We shall comeback to this later.The functions which concern us here are ψ ( r, ε, η, s ) = (1 − q n − ns ) − X c ∈ R,c ∼ η g ( r, ε, c ) q − deg( c ) s where s ∈ C , Re( s ) > / η ∈ k ×∞ /k × n ∞ where k ∞ denotes the completionof k at the infinite place. The condition c ∼ η means that η/c ∈ R × k × n ∞ .The function ψ ( r, ε, η, s ) has at most one pole modulo π √− n log q Z in Re( s ) > s = 1 + 1 /n . If it exists it is simple and we denote by ρ ( r, ε, η ) theresidue of ψ ( r, ε, η, s ) at s = 1 + 1 /n .The general theory of Eisenstein series over function fields (see [6]) alsoshows that there exists a polynomial Ψ( r, ε, i, T ) so that ψ ( r, ε, π − i ∞ , s ) = q − is (1 − q n +1 − ns ) − Ψ( r, ε, i, q − ns ) . The function Ψ( r, ε, i, T ) depends only on i (mod n ). This leads to ρ ( r, ε, π − i ∞ ) = c q − i ( n +1) /n Ψ( r, ε, i, q − n − )where c = 1 / ( n log q ). The notation in [9] is different; there the residue wasreplaced by the value of (1 − q n − s ) ψ ( r, ε, π − i ∞ , s ) at s = 1 + 1 /n . This isinessential for our purposes. We shall write ρ ( r, ε, i ) = Ψ( r, ε, i, q − n − )which is a much more convenient function to use.We have also for i with 0 ≤ i < n (1 − q n − ns ) ψ ( r, ε, π − i ∞ , s ) = q − n X i ′ ≥ ii ′ ≡ i (mod n ) C ( r, ε, i ′ ) q − i ′ s where C ( r, ε, i ) = X deg( c )= ic monic g ( r, ε, c ) .
3f we let C ∗ ( r, ε, i ) = X deg( c )= ic monic , gcd( r,c )=1 g ( r, ε, c )then C ( r, ε, i ) = X r ∗ g ( r, ε, r ∗ ) ε ( χ ( − ( i −
1) deg( r ∗ ) C ∗ ( rr ∗ ( n − , ε, i − deg( r ∗ ))where r ∗ runs through the set of integers (modulo units) all of whose primefactors divide r and where g ( r, ε, r ∗ ) = 0. If we assume, as we shall do inthis paper, that no non-trivial ( n − st power divides r then we can describethe set of r ∗ as follows. Let Σ be the set of primes dividing r , each ofwhich is to be represented by the corresponding monic polynomial. Denote,for π ∈ S the exponent of π dividing r by e ( π ). Then the set of r ∗ canbe parametrized by the subsets S ⊂ Σ via r ∗ ( S ) = Q π ∈ S π e ( π )+1 with therestriction P π ∈ S ( e ( π ) + 1)deg( π ) ≤ i . One can reduce the g ( r, ε, r ∗ ) to aproduct of Legendre symbols, Gauss sums of the type τ ( εχ j ) and powers of q . The general formula is neither illuminating nor computationally helpfulso that we shall not discuss it here. If we are involved in calculating C ( r, ε, i )and i ≤ e ( π )deg( π ) for all π | r then the only r ∗ in the sum is 1.If we use the Davenport-Hasse theorem to evaluate the Gauss sums inthe definition of C ∗ ( r, ε, i ) then we see that C ∗ ( r, ε, i ) = ( − i τ ( εχ ) i X c µ ( c ) ε ( (cid:16) rc (cid:17) n ) ε ( (cid:18) c ′ c (cid:19) n )where the sum over c is as before. The polynomial Ψ( r, ε, i, T ) satisfies afunctional equation which is given implicitly in [9, top of p. 251]. One canmake this much more useful by a simple observation. Let σ = deg( r ) + 1 andlet i be so that 0 ≤ i < n . Let R = [( σ − i ) /n ] Let i ′ be the least non-negativeresidue modulo n of σ − i . We note first that [( σ − i ′ ) /n ] = R . To see this weobserve that we obtain an equivalent statement if we replace σ by σ where σ ≡ σ (mod n ). We can then assume that i ≤ σ < i + n so that R = 0.Then i ′ = σ − i and σ − i ′ = i . As 0 ≤ i < n we see that [( σ − i ′ ) /n ] = 0as required.Next let R = [( σ − i ) /n ] and R = [( σ − i ′ ) /n ]. Assume that i = i ′ .It is clear that R − ≤ R , R ≤ R . Assume that σ is as above; then σ − i ′ = 2 i − σ and so [( σ − i ) /n ] + [( σ − i ′ ) /n ] = −
1. In the generalcase we therefore have R + R = 2 R −
1. Thus if i < i ′ we have R = R and R = R −
1. 4ith these observations we can now formulate the functional equationfor Ψ( r, ε, i, ∗ ) in two equivalent forms:Ψ( r, ε, i, T ) = ( T p n ) R +1 1 − p − − p n − T Ψ( r, ε, i, ( p n T ) − )+ εχ ( − iσ τ ( εχ i − i ′ )( p n T ) R Ψ( r, ε, i ′ , ( p n T ) − )and Ψ( r, ε, i ′ , T ) = ( T q n ) R − q − − q n − T Ψ( r, ε, i ′ , ( q n T ) − )+ εχ ( − i ′ σ τ ( εχ i ′ − i )( q n T ) R Ψ( r, ε, i, ( q n T ) − ) . Here σ and R are as above and we assume here and henceforth that i < i ′ .With these notations the “Hecke relations at infinity” can be formulatedas follows: ρ ( r, ε, i ) = εχ ( − iσ τ ( εχ i − i ′ ) q − ρ ( r, ε, i ′ )and ρ ( r, ε, i ′ ) = εχ ( − i ′ σ τ ( εχ i ′ − i ) ρ ( r, ε, i ) . If i = i ′ then ρ ( r, ε, i ) = 0 which is the reason for excluding this case in inthe functional equations.The corresponding “Hecke relation” for a finite prime π takes the formfor r o coprime to π : ρ ( r o π j , ε, i ) = εχ ( − i ( j +1) deg( π ) q (( i ) n − ( i − ( j +1)deg( π )) n − (( j +1)deg( π )) n )(1+ n ) q (( j +1)deg( π )) n q − [ ( j +1)deg( π ) n ] g ( − r o , ε j +1 , π ) ρ ( r o π n − − j , ε, i − ( j + 1) deg( π ))for 0 ≤ j < n − ρ ( r o π n − , ε, i ) = 0 . In [9] the relation (7) refers not to ρ ( r, ε, i ) as used here but to the corre-sponding residue of ψ ( r, ε, i ) so that the power of N( π ) has had to be modifiedcorrespondingly. It is convenient to avoid fractional powers of q .We retain the assumption that i < i ′ . Because of the functional equationsfor the Ψ( r, ε, i, T ) we can determine the polynomial from the coefficients ofΨ( r, ε, i, T ) up to that of T R/ together with those of Ψ( r, ε, i ′ , T ) up to T R/ − if R is even; if R is odd these are both to be replaced by T ( R − / .This means that we need only evaluate the corresponding C ( r, ε, j ) We shallgive more details of the cases in Section 4.We should stress here that although the ρ ( r, ε, i ) only depend on r modulo n th powers the same is not true of Ψ( r, ε, i, T ).We saw above that the global Gauss sums for k could be expressed explic-itly in terms of the Gauss sums for the field of constants and certain other5uantities. First of all the connection between the Legendre symbol and theresultant shows that (cid:18) c ′ c (cid:19) n = (cid:26) χ ( D ( c )) if deg( c ) ≡ , χ ( − D ( c )) if deg( c ) ≡ , D ( c ) is the discriminant of the polynomial c (assumed monic).The evaluation of the quadratic Gauss sum is also valid in k ; details aregiven in, for example, [13, XIII, § ω ( D ( c )) = µ ( c )( − deg( c ) , valid of monic polynomials c where ω denotes the quadratic character of F × q .Several proofs are available - see, for example, [10]. It is this formula thatestablishes the connection of the C ∗ ( r, ε, i ) in the special case r = x e ( x − e with a Selberg sum. In [9, §
3] we explained how an analogue of ψ ( r, ε, η, s ) can be defined overa ring obtained from R through the inversion of an arbitrary finite set ofprimes and the relationship of these new funcctions with the original ones.The formulæ given there need a little further explanation. The variable “ η ”in ψ S ( r, ε, η, s ) and ψ S ∪{ π } ( r, ε, η, s ) should strictly speaking be considered aselements of k × S and k × S × k × π respectively. What was written as ψ S ∪{ π } ( r, ε, η, s )should be P θ ∈ r × π /π × n ψ S ∪{ π } ( r, ε, η × θ, s ); here r π denotes the ring of integersof k π , the completion of k at π .Now we can move to the following theorem: Theorem 1
Let g = (cid:18) a bc d (cid:19) ∈ GL ( F q ) and set ∆ = det( g ) . Then if r ∈ k and r gi is defined by r gi ( x ) = r ( ax + bcx + d ) (cid:16) ∆( cx + d ) (cid:17) − i then ρ ( r, ε, i ) = ρ ( r gi , ε, i ) Proof:
The case where c = 0 is elementary so that we can concentrate onthe case c = 0. Let π = x + d/c and π = x − a/c . Let R (resp. R )be the ring obtained from R by inverting π (resp. π ). Consider the map g : k → k ; f ( x ) f g ( x ) = f (( ax + b ) / ( cx + d )). Set π ∞ = x − . Then π g ∞ = − c π / ∆ + O ( π ) (in k π ) and π g = − (∆ /c ) π ∞ + O ( π ∞ ) (in k ∞ ).This means that g maps R to R . We also have d( x g ) = ∆d x/ ( cx + d ) . Wecan conclude that, for θ ∈ F × q , ψ {∞ ,a/c } ( r, ε, π m ∞ × θπ m , s )6s equal to ψ {− d/c, ∞} ( r g ( x )(∆ / ( cx + d ) ) , ε, ( δ − π ) m × θ ( δπ ∞ ) m , s )where δ = − ∆ /c . Using the transformation properties of ψ {∞ ,a/c } (resp. ψ {− d/c, ∞} ) under units of R (resp. R ) we deduce that ψ {∞ ,a/c } ( r, ε, π m ∞ π − m × θ, s ) εχ ( θ ) − m εχ ( − m + m m is equal to ψ {− d/c, ∞} ( r g ( x )(∆ / ( cx + d ) ) , ε, θ − × ( δ − π ) − m ( δπ ∞ ) m , s ) × εχ ( θ ) m + m εχ ( − m + m m εχ ( δ ) ( m + m ) ;consequently ψ {∞ ,a/c } ( r, ε, π m ∞ π − m × θ, s )is equal to ψ {− d/c, ∞} ( r g ( x )(∆ / ( cx + d ) ) , ε, θ − × ( δπ ∞ ) m + m , s ) × εχ ( θ ) m +2 m εχ ( − m + m εχ ( δ ) m + m . Again the behaviour of ψ {− d/c, ∞} in the first variable under multiplication bya unit (in our case this will be (( cx + d ) / ∆) m + m yields that ψ {− d/c, ∞} ((( cx + d ) / ∆) − m − m r g ( x )(∆ / ( cx + d ) ) , ε, θ − × ( δπ ∞ ) m + m , s )is equal to ψ {− d/c, ∞} ( r g ( x )(∆ / ( cx + d ) ) , ε, θ − × ( δπ ∞ ) m + m , s ) εχ ( θ ) m + m ) εχ ( − m + m εχ ( δ ) ( m + m ) . Together these show that ψ {∞ ,a/c } ( r, ε, π m ∞ π − m × θ, s )is equal to ψ {− d/c, ∞} ((( cx + d ) / ∆) − m − m − r g ( x ) , ε, θ − × ( δπ ∞ ) m + m , s ) . We take the residue at s = 1 + 1 /n and sum over θ ∈ F × q . We now get ρ {∞} ( r, ε, π m ∞ π − m ) = ρ {∞} ((( cx + d ) / ∆) − m − m − r g ( x ) , ε, ( δπ ∞ ) m + m ) . The behaviour under units shows that both sides depend on δ in the sameway. We therefore obtain the formula of the theorem if we take residues andwrite i for − m − m . 7 Consequences
We noted in [9] the elementary statement ρ ( r, ε, i ) ∈ τ ( εχ i ) × q − ( n +1)[(1+deg( r ) − i ) /n ] Z [1 /n ]where the estimate for the denominator can probably be improved. The realchallenge is to find the factor lying in Z [1 /n ]. It follows from the resultsof Section 2 that ρ ( r, ε, i ) = 0 if R = [(1 + deg( r ) − i ) /n ] <
0. Let i and i ′ be as in § i < i ′ and 0 ≤ i, i ′ <
1. Recall that we have R = [(1 + deg( r ) − i ′ ) /n ]. The pair ρ ( r, ε, i ) and ρ ( r, ε, i ′ ) are connectedwith one another by the Hecke relation at infinity. In this connection werecall that τ ( εχ a ) i is ( − i − τ ( εχ ai ) times an (integral) Jacobi sum.We shall in this section consider the cases where r has at most three primefactors and that these are of degree 1. By means of a linear transformationwe can assume then that r has the form x e ( x − e ( x − λ ) e λ where λ ∈ F q and λ = 0 ,
1. By the Hecke relations at x , x − x − λ we can take e j ≤ [ n/ − j = 0 , , λ .If just one prime divides we can take r to be x e , the degree of whichis e . It follows that that ρ ( x e , ε, i ) = 0 if i > e + 1. By means of theHecke relations at infinity we only need to determine these coefficients when i < ( e + 1) /
2. This means that C ( x e , ε, i ) = C ∗ ( x e , ε, i ). This sum is adegenerate Selberg sum which we could evaluate by means of the results in[12].We can however proceed in a different way which is, in some respects, illu-minating. We apply Theorem 1 with (cid:18) −
11 0 (cid:19) and we see that ρ ( x e , ε, i ) = ρ ( x − e +2 i − , ε, i ) We can now apply the Hecke relation at x and we see thatthis latter function is a multiple of ρ ( x e − i , ε, e + 1 − i ); the multiplier ismade up of a power of q , a power of εχ ( −
1) and τ ( εχ i − e − ). We note that i = 0 then ρ ( x e , ε, i ) = 1. If i > e − i > − ≤ e − i < e .This means that we can compute ρ ( x e , ε, i ) recursively. We deduce that ρ ( x e , ε, i ) is τ ( εχ i ) times a Jacobi sum, that is, a product of elementaryJacobi sums. This is also what the theory of Selberg sums yields.We now move on to the case where there are two prime factors, thus r is ofthe form x e ( x − e where 1 ≤ e , e ≤ [ n/ −
1. Now R = [( e + e +1 − i ) /n ]and we see that ρ ( x e ( x − e , ε, i ) = 0 if i > e + e + 1. Again theHecke relation at infinity means that we can restrict our attention to thecase i ≤ [( e + e − / e and e without changing the function. We may assume also that e ≤ e . Aswe have already studied the case where one of the exponents is 0 we assumethat e ≥
1. 8ith these restrictions we have R = 0 in the notation used above. Thismeans that, as we shall see below, ρ ( x e ( x − e , ε, i ) is equal to c C ( x e ( x − e , ε, i ) where now i is to be understood not as a residue class but as the leastnon-negative element of that class. There are two cases to be distinguished.If e o + 1 > i then C ( x e ( x − e , ε, i ) = C ∗ ( x e ( x − e , ε, i ) and this meansthat ρ ( x e ( x − e , ε, i ) is equal to τ ( εχ i ) times a Selberg sum which itselfis a Jacobi sum.If e o + 1 ≤ i then there are precisely two elements r ∗ in the terminology ofSection 2, 1 and x e +1 . We can now apply Theorem 1 again but now inverting x − ρ ( x e ( x − e , ε, i ) = ρ (( x − i − − e − e x e , ε, i ).If we now make use of the Hecke relation at ( x −
1) we obtain a multipleof ρ (( x − e + e − i x e , ε, e + e − i + 1). The case i = 0 does not ariseand so e + e − i + 1 < e + e − i + 1 whence it follows that ρ (( x − e + e − i x e , ε, e + e − i + 1) = 0; indeed the corresponding Ψ functionvanishes. It therefore follows that if e o + 1 ≤ iρ ( x e ( x − e , ε, i ) = 0and in the other case, namely e o + 1 > iρ ( x e ( x − e , ε, i )is a power of q times c τ ( εχ i ) times a specific Jacobi sum. We shall comeback to the details in a later publication. What is rather remarkable is thatthere is, in these cases, an “explicit formula” for ρ ( x e ( x − e , ε, i ) and thatthis is in terms of Jacobi sums. I have not found a method of demonstratingthis without the use of the theorem of Anderson, Evans and v. Wamelen.This is not the rule. If we next consider r of the form x e ( x − e ( x − λ ) e λ ,now with 0 < e , e , e λ ≤ [ n/ −
1, then we find a large number of rela-tionships between various ρ ( x e ( x − e ( x − λ ) e λ , ε, i ). The structure ofthis set of relations is that which one knows from the theory of the hy-pergeometric function (see [14]). If we have Min( e , e , e λ ) + 1 > i and e + e + e ≤ n then we see, as before, that ρ ( x e ( x − e ( x − λ ) e λ , ε, i ) = C ∗ ( x e ( x − e ( x − λ ) e λ , ε, i ). The latter sum is an analogue of the hyper-geometric function in the same sense that the standard Selberg sum is ananalogue of the beta function.In certain special cases, when n = 4, or when n = 6 and e , e , e λ areall 2 conjectures of Eckhardt and Patterson and of Chinta, Friedburg andHoffstein respectively suggest that these sums take on a special form andthis is what one finds. It seems as if the corresponding statements can nowbe proved by a new method due to S. Friedberg and D.Ginzburg [4] butthere is still work to be done to complete the proof. In other cases, as one9ould expect, the values are irregular. These evaluations go beyond therange considered in [5] and make it clear that with increasing complexity ofthe r the arithmetical nature of the ρ ( r, ε, i ) also becomes more complex.Furthermore a complete evaluation does not seem to reasonable expectationand one will have to be satisfied with less specific questions. One can, forexample, make estimates for the ρ ( r, ε, i ) in different metrics. One wouldsuspect that one can do better than relatively elementary convexity bounds.We now return to the expression of ρ ( r, ε, i ) in terms of C ( r, ε, i ) or of C ∗ ( r, ε, i ) for R = 0 , , R = [(1 + deg( r ) − i ) /n ]. Let0 ≤ i < n and i ′ = (1 + deg( r ) − i ) n . We assume, as before that i < i ′ Thecondition R ≤ r ) < n . This ismuch more than we need for the purposes of this paper.We recall that by constructionΨ( r, ε, i, T ) = 1 − q n +1 T − q n T q − n X i ′ ≥ ii ′ ≡ i (mod n ) C ( r, ε, i ′ ) T ( i ′ − i ) /n . We shall assume that 0 ≤ i < n .If R = 0 then Ψ( r, ε, i, T ) = C ( r, ε, i ) and there is nothing left to be said.If R = 1 then a direct application of the definition yieldsΨ( r, ε, i, T ) = C ( r, ε, i ) + ( C ( r, ε, i + n ) − ( q − q n C ( r, ε, i )) T. If R = 2 thenΨ( r, ε, i, T ) = C ( r, ε, i ) + ( C ( r, ε, i + n ) − ( q − q n C ( r, ε, i )) T +( C ( r, ε, i + 2 n ) − ( q − q n C ( r, ε, i + n ) − ( q − q n C ( r, ε, i )) T It follows that in the three cases we haveΨ( r, ε, i, q − n − ) = C ( r, ε, i ) , Ψ( r, ε, i, q − n − ) = q − C ( r, ε, i ) + q − n − C ( r, ε, i + n )and Ψ( r, ε, i, q − n − ) = C ( r, ε, i ) q − + C ( r, ε, i + n ) q − n − + C ( r, ε, i + 2 n ) q − n − . respectively. These are perfectly usable expressions but are not as practical asthey might be as the number of summands in C ( r, ε, i + n ) and C ( r, ε, i + 2 n )is large. It is advantageous to exploit the functional equation when R = 1and R = 2; the method is a simple version of the “approximate functionequation”. 10o carry this out we write C j = C ( r, ε, i + jn ) and C ′ j = C ( r, ε, i ′ + jn ).We shall use X with X = q n T rather than T . Let F ( X ) = Ψ( r, ε, i, q − n X )and G ( X ) = Ψ( r, ε, i ′ , q − n X ). Then F ( X ) = P ≤ j ≤ R D j X j and G ( X ) = P ≤ j ≤ R D ′ j X j where D j = C j q − nj − ( q − P ≤ ℓ 11t follows that in the three cases we have F ( X ) = D , F ( X ) = D (1 − ( q − X ) + η ′− D ′ X and F ( X ) = D (1 − ( q − X ) + D X + η ′− D ′ X respectively. We obtain the values of F (1 /q ) in terms of D , of D and D ′ and of D , D and D ′ respectively.It should be noted that whereas on the one hand the theory of Selbergsums allows us to obtain closed expressions for the coefficients of metaplecticsums on the other hand the theory of metaplectic forms, and, in particularTheorem 1, leads to a number of new relations between Selberg sums whichdo not seem to be accessible by elementary methods or those of Anderson.This will be the subject of a future paper. The case discussed above is very straightforward as the structure of the ra-tional curve is explicit. For other curves there is, in general, no “natural”ring of integers, especially if we demand that it be a principal ideal ring.It seems therefore a considerable challenge to gather numerical evidence insuch a case. It is unclear as to whether the nature of the ρ ( r, ε, i ) are typicalof what happens in the case of curves of genus ≥ 1. There seems to be noreason why not but nevertheless one is able to exploit so much in the rationalcase that one is cautious about making any too large extrappolations.One case that is probably deserving of study is that of elliptic curves.To gain some idea of what is needed we consider the function field of anelliptic curve over a field of characteristic = 2 , 3. We can then represent itin Weierstrass form and as usual we make the point at infinity the identityelement of the group. Hasse’s estimate shows that there is at least one furtherrational point on the curve and one knows that the group of points over afinite field is either a (non-trivial) cyclic group or a product of two cyclicgroups; see [2, Prop. 7.1.9]. Let P (resp. P amd P ) be generators ofthe group of rational points in these two cases. Then the ring R of functionsintegral outside {∞ , P } (resp. {∞ , P , P } ) is a principal ideal domain. Themethods of the theory of algebraic curves allow us to represent the ring R asa quotient of a polynomial ring. This comes down to working with a modelof the curve in 3- or 4-dimensional projective space. The determination ofthe elements of this ring and especially of the prime elements is possible butit is no longer as easy as in the case of the rational function field. Alsothe description of the relation ∼ in k × S is considerably more intricate thanbefore. It seems at the moment that the computational effort needed wouldbe considerable and that it would only be justified it there were good reasonto expect that the behaviour of the ψ ( r, ε, η, s ) and ρ ( r, ε, η ) would show12eatures that do not appear in the case of rational function fields. Whetherthis is so is an open question at present. References [1] G.W.Anderson: The evaluation of Selberg sums, Comptes RendusAcad.Sci.Paris,Ser. I , 311(1990)469-472.[2] H. Cohen: A Course in Computational Number Theory , Springer Verlag,1993.[3] R. Evans: The evaluation of Selberg character sums, L’Enseign.Math. n -fold metaplectic cover of SL(2) -the function field case, Inv. Math. 107 (1992) 61-86.[6] D.A. Kazhdan, S.J. Patterson: Metaplectic Forms, Publ.Math. IHES S´em.Th´eorie des nombres, Paris, 1982-1983 , Birkh¨auser, 1984, 199-232.[8] S.J.Patterson, Metaplectic forms and Gauss sums, I, Com-pos.Math. Glasgow Math. J. Pacific J.Math. Modern Algebra, Vol. 1 , (Trans. F. Blum)Fredrick Unger Publishing Co.,New York, 1949.[12] P.B.van Wamelen: Proof of the Evans-Root conjectures for Selberg char-acter sums, J.Lond.Math.Soc.,II.Ser. Basic Number Theory , Springer, 1967.[14] E.T.Whittaker, G.N.Watson, Modern Analysis , 4th edition, CambridgeUniv. Press, 1946 13athematisches InstitutBunsenstr. 3–537073 G¨ottingenGermanye-mail: [email protected]@gwdg.de