arXiv:1008.0825v1 [math.NT] 4 Aug 2010 650,adteElnukFn tIAS. at Fund Ellentuck the and 0635607, is,o yacntn rmtecnetrdasymptotic. conjectured the from constant a by off ties), osdrdtecs ftefl oua ru SL(2 = Γ Let [FI09] group norm-square. Iwaniec modular the and full of Friedlander being the orbits function of on space. case functions affine the for of considered points morphisms prime of produce groups to aims [BGS06], nak ,ta is, that Γ, in hypoe,asmn napoiaint h Elliott-Halberstam the to approximation that an conjecture, assuming proved, They NATERMO RELNE N IWANIEC AND FRIEDLANDER OF THEOREM A ON otrvc sprilyspotdb S rnsDS0098and DMS-0802998 grants NSF by supported DMS-0808042. partially grant NSF is by Kontorovich supported partially is Bourgain Date 1 h ffieLna iv,itoue yBugi,GmudadSar- and Gamburd Bourgain, by introduced Sieve, Linear Affine The nodtoal,oecnesl hwteeitneo -lotprimes 2-almost of existence the show easily can one Unconditionally, S oevrte aeafruafrtecuto omsurs(wit norm-squares of count the for formula a gave they Moreover ned o any for Indeed, . etme 8 2018. 18, September : srpeetd h ro sa plcto fSee’ asformula. mass Siegel’s of application an is proof The represented. is aldHproi rm ubrTerm hc ssfra in- an for asks elements which of Theorem, Number finitude Prime Hyperbolic called Abstract. h nlgu usinrpaigteitgr ihteGaussian odd the every for with that integers stu a unconditionally prove by we the We off note, replacing integers. count, this question their In for analogous asymptotic. formula proved the a conjectured they as the well conjecture, from as constant Elliott-Halberstam such, of the existence Under the prime. a squared sa is S γ := ∈ ENBUGI N LXKONTOROVICH ALEX AND BOURGAIN JEAN SL(2  n S n[I9,FidadradIaicsuidteso- the studied Iwaniec and Friedlander [FI09], In , ∈ otisifiieymn primes. many infinitely contains Z [ Z i )sc that such ]) x + k γ ∈ k : γ 1. 2 Z n = n = h aaoi elements parabolic the , x Introduction = a 2 := b a d c k + k γ γ  b  k k 2 1 2 2 + ∈ 1 0 1 = o some for c SL(2 S 2 n x + npriua,eeyprime every particular, In . etesto omsursin norm-squares of set the be  d , 2 Z = uhta h norm the that such ) γ p, ∈ SL(2 1 n ≥ , Z , ,there 3, ) Z  ,wt the with ), multiplici- h . dy DMS- 2 AND ALEX KONTOROVICH

2 2 are in Γ, and their norm-square is knxk = x + 2. Then Iwaniec’s theorem [Iwa78] produces infinitely many 2-almost primes in S. In this note, we ask an analogous question, replacing the integers in SL(2, Z) by the Gaussian integers, Γ = SL(2, Z[i]). We prove uncondi- tionally the following Theorem 1.1. The set

S := n ∈ Z : n = kγk2 for some γ ∈ SL(2, Z[i])  +  contains all odd integers n ≥ 3. In particular, it contains all primes. The proof, given in the next section, is an application of Siegel’s mass formula [Sie35]. The argument is sufficiently delicate that it cannot replace the Gaussian integers above by the ring of integers of another number field, even an imaginary quadratic extension (as suggested to us by ), see Remark 2.6. We thank Zeev Rudnick and Peter Sarnak for conversations regard- ing this work.

2. Sketch of the Proof

a b For odd n ≥ 3 and γ = c d with a = a1 + ia2, etc., the conditions n = kγk2 and γ ∈ SL(2, Z[i]) imply

2 2 2 2 2 2 2 2 2 kγk = a1 + b1 + c1 + d1 + a2 + b2 + c2 + d2 = n,  Re(det γ)= a1d1 − b1c1 + b2c2 − a2d2 =1 (2.1)  Im(det γ)= a1d2 + a2d1 − b1c2 − b2c1 =0.  Changing variables

a1 → (y1 + y4)/2, b1 → (y3 + y2)/2,

c1 → (y3 − y2)/2, d1 → (y1 − y4)/2,

a2 → (y5 + y8)/2, b2 → (y7 + y6)/2,

c2 → (y7 − y6)/2, d2 → (y5 − y8)/2, the system (2.1) becomes

2 2 2 2 y3 + y4 + y5 + y6 = n − 2,  2 2 2 2 y1 + y2 + y7 + y8 = n +2, (2.2)  y1y5 + y2y6 − y3y7 − y4y8 =0.  ON A THEOREM OF FRIEDLANDERAND IWANIEC 3

Write 1  1  n +2 F = , G = , 1 n  n − 2    1   and y y −y −y X = 1 2 7 8 , y5 y6 y3 y4  so that (2.2) becomes t XF X = Gn. (2.3)

Recall Siegel’s [Sie35] mass formula, cf. [Cas78, Appendix B, equa- tions (3.10) to (3.17)]. Clearly F is positive definite and alone in its genus, and hence the number N (F,Gn) of solutions X to (2.3) is given by

N (F,Gn)= αp(F,Gn), (2.4) pY≤∞ where the local densities αp are given as follows. For p< ∞, they are defined by −5t t t t αp(F,Gn)= p · # X(mod p ): XF X ≡ Gn(p ) , (2.5) for t sufficiently large. Forp = ∞, we have 3 2 1/2 α∞(F,Gn)=2π (n − 4) . Remark 2.6. In complete generality, it is notoriously difficult to com- pute the local densities αp and extract information such as non-vanishing, see e.g. the formulae in [Yan98, Yan04]. The main problem being how large is “sufficiently large” for t in (2.5) with a given p. In our special case of Γ = SL(2, Z[i]), the literature is sufficient to carry out the task. For p =6 2, both the ramified and unramified local densities can be evaluated as in e.g. [Kit83, Theorem 2]. We turn first to the case p is unramified, p ∤ (n2 − 4). Then 1 χ (4 − n2) α (F,G )= 1 − 1+ p , p n  p2   p 

· where χp = p is the quadratic character mod p. For ramified primes   p ≥ 3, write n +2= mpa and n − 2 = kpb with (mk,p) = 1. Assume 0 ≤ a ≤ b (otherwise reverse their roles). Then if a + b ≡ 0(mod 2), a+1 2 (a+b)/2 (p + 1) (p − 1) χp(−mk) − 1 +(a +1)(p − 1) p α (F,G )= . p n p3+(a+b)/2  4 JEAN BOURGAIN AND ALEX KONTOROVICH

Otherwise, if a + b ≡ 1(mod 2), then (p + 1)2 (a + 1)(p − 1)p(a+b+1)/2 − (pa+1 − 1) α (F,G )= . p n p3+(a+b+1)/2  Inspection shows that these terms never vanish.

It remains to evaluate the dyadic density, α2. As shown by Siegel, see [Rei56], for n odd (and hence n2 −4 odd), it is sufficient to evaluate (2.5) for t = 3, that is, compute the number of solutions mod 23 = 8. One can compute explicitly that for any odd n, the number of solutions to (2.5) mod 8 is 49 152. Since 85 = 32 768, we have 3 α (F,G )= . 2 n 2 In conclusion, the αp’s never vanish so there are no local obstructions for odd n to be represented, and hence the set S of norm-squares in SL(2, Z[i]) contains all the primes. (The prime 2 is in S since it is the norm squared of the identity matrix.) References [BGS06] Jean Bourgain, Alex Gamburd, and Peter Sarnak. Sieving and expanders. C. R. Math. Acad. Sci. Paris, 343(3):155–159, 2006. [Cas78] J. W. S. Cassels. Rational Quadratic Forms. Number 13 in London Math- ematical Society Monographs. Academic Press, London-New York-San Francisco, 1978. [FI09] John B. Friedlander and Henryk Iwaniec. Hyperbolic the- orem. Acta Math., 202(1):1–19, 2009. [Iwa78] Henryk Iwaniec. Almost-primes represented by quadratic polynomials. In- vent. Math., 47:171–188, 1978. [Kit83] Yoshiyuki Kitaoka. A note on local densities of quadratic forms. Nagoya Math. J., 92:145–152, 1983. [Rei56] Irma Reiner. On the two-adic density of representations by quadratic forms. Pacific J. Math., 6:753–762, 1956. [Sie35] Carl Ludwig Siegel. Uber¨ die analytische Theorie der quadratischen For- men. Ann. of Math. (2), 36(3):527–606, 1935. [Yan98] Tonghai Yang. An explicit formula for local densities of quadratic forms. J. , 72(2):309–356, 1998. [Yan04] Tonghai Yang. Local densities of 2-adic quadratic forms. J. Number The- ory, 108(2):287–345, 2004. E-mail address: [email protected] IAS, Princeton, NJ

E-mail address: [email protected] IAS and Brown University, Princeton, NJ