ON CONVERGENCE OF SINGULAR SERIES FOR A PAIR OF QUADRATIC FORMS

Transcription

1 ON CONVEGENCE OF SINGULA SEIES FO A PAI OF QUADATIC FOMS THOMAS WIGHT Abstract. Examiig the system of Diohatie equatios { f (x) x x2 ν, f 2 (x) λ x λx2 ν 2, with λ i λ j ad ν i, λ i Z, we show that the sigular series S(ν) coverges if 6.. Itroductio.. The Classical Warig s Problem. Niety years ago, Hardy ad Littlewood [HL] examied the iteger solutios for the equatio x δ + x δ x δ ν, ν Z for a iteger δ 2. This aer was the semial work i the moder aroach to Diohatie equatios, ad sice that time a umber of geeralizatios ad reformulatios of their work have aeared. (See e.g. [La].) The iitial questio is about a olyomial ma f : Q Q m, where ν Q m. We wish to study the set f (ν) (i.e. the fiber of f which mas dow to ν). I articular, we ca write N(ν) #{x Z : f(x) ν}. x Z,f(x)ν Clearly, i the origial Hardy-Littlewood case, we would let m ad f(x) x δ + x δ x δ. I our case, f will be a air of quadratic forms, i.e. f (f, f 2 ) : Q Q 2, where f (x) x x 2, f 2 (x) λ x λ x 2. We will assume λ i λ j i j ad that is eve. For future ease, we ote that f (x) x 2. Of course, we are iterested i whether a air of atural umbers ca be rereseted by such a air of fuctios. Namely, for ν (ν, ν 2 ) N 2, we examie whe x x 2 ν, λ x λ x 2 ν Mathematics Subject Classificatio. Primary P05; Secodary P55. I would like to thak Takashi Oo for all of his guidace. I would also like to thak ami Takloo-Bighash ad Mike Limarzi for their helful commets ad suggestios.

2 2 T. WIGHT.2. Sigular Series S(ν) as a Itegral Aroximatio for N(ν). Let x (x v ) A, where A is the -sace over the adele rig A over Q. We defie where ϕ(x) v ϕ v (x) ϕ (x) ϕ (x), ϕ (x) the characteristic fuctio o Z, ϕ (x) e π x 2. Sice Z Z, we ote that ϕ (x) is the characteristic fuctio o Z. I articular, we ote that if we defie N ϕ (ν) ϕ(γ) the N ϕ (ν) γ Z,f(γ)ν γ Q,f(γ)ν e πf(γ) e πν N(ν). It is this isight which allows us to write N(ν) i adelic laguage; therefore, the itegral with which we ca aroximate N(ν) ca also be writte i adelic laguage. We will refer to this itegral as the sigular series S(ν), whose urose is aalogous to the Hardy-Littlewood versio of the sigular series. Sice we are dealig adelically, we itroduce the otatio that for x Q, we let {x } deote the fractioal art of x ; i.e. {x } x (mod Z ). Let χ be a basic character of Q, i.e. a o-trivial character o A + which is trivial o Q. I articular, we will let χ(x) v χ v (x v ) be the roduct over the various valuatios of Q, where { e 2πix if v, χ v (x v ) e 2πi{x} if v, v. Usig this iformatio, we defie Gϕ(ξ) ad S(ν) Q A Q 2 A ϕ(x)χ(< f(x), ξ >)dx, Gϕ(ξ) χ(< ξ, ν >)dξ, where S(ν) is the Fourier trasform of Gϕ(ξ). Naturally, sice all of our fuctios defied over the adeles ca be writte as roducts over all the localizatios, we cosider said localizatios of this fuctio for the various laces of Q. I articular, let Gϕ v (ξ) ϕ v (x)χ(< f(x), ξ >)dx, Q v

3 ON CONVEGENCE OF SINGULA SEIES FO A PAI OF QUADATIC FOMS 3 ad It is clear (formally) that S v (ν) Q 2 v Gϕ v (ξ) χ(< ξ, ν >)dξ. S(ν) S v (ν). v We will cosider searately the laces where v is ifiite ad v is fiite. Our goal i this aer is to show that the itegrals which comrise S(ν) coverge locally ad globally; with this covergece, oe ca calculate S(ν) for give ν ad λ. We rove the followig: Theorem A. If 6 the the itegral S(ν) coverges. 2. The Ifiite Place (v ) To evaluate S (ν), we will require reeated use of Fubii s Theorem. Therefore, we must first show that G f ϕ (ξ) L ( 2 ), i.e. 2 G f ϕ (ξ) dξ <. This will allow us to use Fubii s Theorem. Throughout this sectio, we will use G f ϕ, S (ν) ad G f ϕ, S(ν) iterchageably whe the cotext is clear. By our defiitios, 2 G f ϕ(ξ) χ(f (x)ξ + f 2 (x)ξ 2 )dx So e π x i x2 i e 2πi( i x2 i ξ+ i λix2 i ξ2) dx e π e π i x2 i (+2i(ξ+λiξ2)) dx i G f ϕ(ξ) dξ 2 e πx2 i (+2i(ξ+λiξ2)) dx. 2 i e πx2 i (+2i(ξ+λiξ2)) dx dξ. We kow that e πx2 i (+2i(ξ+λiξ2)) dx e i 2 ta (2(ξ +λ iξ 2)). ( + 4(ξ + λ i ξ 2 ) 2 ) 4 So the above itegral is dξ ( + 4(ξ + λ i ξ 2 ) 2 ) 4 2 i 2 i ( + 4(ξ + λ i ξ 2 ) 2 ) 4 ( + 4(ξ + λ i+ ξ dξ. 2 2) 2 ) 4

4 4 T. WIGHT Note that the itegrad is always ositive. So we ca use Hölder s iequality o the above exressio to fid that 2 G f ϕ(ξ) dξ ( i 2 ( ( + 4(ξ + λ i ξ 2 ) 2 ) ) 8 ( Sice the λ i s are uequal, oe ca use the chage of variables i 2 w i ξ + λ i ξ 2, z i ξ + λ i+ 2 ξ 2, ( + 4(ξ + λ i+ 2 ξ 2) 2 ) to covert the right-had side above ito ( ( ) ( ) ) J ( + 4wi 2) ( + 4zi 2) dwi dz i J ( i ( ) ) 4 8 ( + 4wi 2) dwi, ) 8 dξ ) 2. where J is the costat which results from the chage of measure, which we kow to be o-zero. From here, it is a easy exercise to see that this coverges if 6 ad diverges if 4. Sice we assume is eve, it follows that 6. Now, oce we kow that the itegrals coverge, we ca use Fubii s Theorem to simlify this itegral. I articular, through much maiulatio, we ca reduce the above itegral to the itegral over a comact sace: Theorem. Let K 2 e π(ν) (ν ) 2 2. Additioally, let J 0 deote the 0-th Bessel fuctio of secod tye, ad let rect deote the usual rectagle fuctio. Moreover, let u i vary over the uit ball, ad let dv be the usual measure of this ball i hyersherical coordiates. The S(ν) K π S 2 2 rect( ν 2 + (λ 2 ν ) i (λi λ)u2 i π(ν ) i (λ i λ )u 2 i ( + 3. Fiite Places (v ) ) dv. 2(λ ν 2 ν ) ) 2 i (λi λ)u2 i I the case of v, we ca agai use Hölder s iequality to show that G f ϕ v (ξ) L (Q 2 ). This, combied with the results of the revious sectio, shows that the idividual S v (ν) all coverge locally. Thus, it remais oly to show that the roduct coverges globally. Equivaletly, we ca show that the roduct over all but a fiite umber of S v (ν) coverges; i articular, we defie the set A { rime: λ i, ν j λ i λ j, or ν λ + ν 2 λ 2 } ad rove that the roduct over A coverges. Now, we ote first that < f(x), ξ > i x 2 i ξ + i λ i x 2 i ξ 2 i (ξ + λ i ξ 2 )x 2 i.

5 ON CONVEGENCE OF SINGULA SEIES FO A PAI OF QUADATIC FOMS 5 This allows us to write G f ϕ simly ad exlicitly: S (ν) Q 2 Q 2 Q 2 χ(< ξ, ν >)G f ϕ(ξ)dξ χ(ξ ν + ξ 2 ν 2 ) χ( (ξ + λ i ξ 2 )x 2 i )dxdξ Z i χ(ξ ν + ξ 2 ν 2 ) χ((ξ + λ i ξ 2 )x 2 i )dx i dξ. Z i Now, oe ca use the orthogoality relatios of characters to show that the itegral is zero whe ξ Q 2 ( Z ) 2. Thus, S (ν) χ(ξ ν + ξ 2 ν 2 ) χ((ξ + λ i ξ 2 )x 2 i )dx i dξ. Z Z 2 From here, oe ca utilize the theory of Gauss sums to calculate S (ν) exlicitly: i Theorem 2. Let A. The S (ν) ( + λ iu )G G ( + λ i λ j )( ν + λ u H i j i,i j + ( ) ( λ iν ν2 )G G G ( ( λ i )) +, i where H {u F : u λ i i, u ν ν2 }, ad G is the classical Gauss sum G( ), give by { if (mod 4), G i if 3 (mod 4). i j ν 2 ) 3.. Covergece of S (ν). Fially, we wish to show that the roduct S(ν) v S v (ν) coverges. Let us defie T (ν) ( ( + λ iu )G G ( + λ i λ j )( ν + λ u H i j i,i j + ( ) ( λ iν ν2 )G G G ( ( λ i )) + ). i i j ν 2 Clearly, S(ν) ad T (ν) differ by oly a fiite umber of terms (secifically, the terms where A { }). Thus, showig that the latter coverges is equivalet to showig that the former coverges. Now, we kow that the ifiite roduct ( + a i ) coverges if ad oly if a i does as well. Notig that G, the term i the exressio for S (ν) which is )

6 6 T. WIGHT largest (other tha ) for large is aroximately G (which has absolute value 2 + ). So, i our case, a 2 +. This coverges if 2 2, which haes if 6. efereces [HL] G.H. Hardy, J.E. Littlewood, A ew solutio to Warig s roblem, Q.J. Math. 48 (99), [La] G. Lachaud, Ue résetatio adélique de la série siguliére et du robléme de Warig, Eseig. Math. 28 (982), [O] T. Oo, Lectures o the Hardy-Littlewood sigular series ( ). Trascribed by T. Wright. [O2] T. Oo, Gauss trasformatios ad zeta fuctios, A. of Math. (2), 9 (970), Deartmet of Mathematics, Johs Hokis Uiversity, 3400 North Charles St., Baltimore, MD 228 address: wright@math.jhu.edu