Introduction to the Hardy-Littlewood Circle method Rational Points seminar -Tomer Schlank

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1 Introduction to the Hardy-Littlewood Circle method Rational Points seminar -Tomer Schlank by Asaf Katz, Hebrew Uniersity January 5, 200 Some History and General Remarks The circle method was inented by Hardy and Littlewood during their common work with Sariniasa Ramanujan in the years on the asymptotic estimate of the Partition function. The method gained its reputation when Hardy used it to gie effectie bounds for Waring s Problem a.k.a Waring-Hilbert theorem). Later, works by Vinogrado, Kloosterman, Daenport and others modified the method to work in broader settings 2. We ll present a modern approach of the method, which doesn t use complex analysis, based on [H-]. See [MT] for a simpler treatment of Waring s problem with the circle method. See [IK] for a complete complex analysis based treatment. 2 How To Count y Integration Let F x,..., x n ) Z [x,..., x n ] be an irreducible form of degree d which defines a hypersurface by F = 0 in P n. Assume that R is a small box R = n i= k i, λ i ], define R = { x R n x R}, and N R ) = {x Z n R F x) = 0} ) N R ) counts the number of rational points with bounded nominator and denominator on the hypersurface, note that one can extend that point counting function to any height function one wish to use. We ll be interested in asymptotic behaior of N R ) as. We ll also be interested in p-adic distributions, for that, it s sufficient to choose a ector a,..., a n ) Z n and m N s.t. gcd m, a,..., a n ) = and modify the counting function N R, m, a) = {x Z n R, F x) = 0} If m is composed of primes to high powers, then it s describing the behaior in the adèles A Q). 2. The Heuristic Principle Let N ) denote the number of integer ectors contained in R, easy counting argument shows that N ) = V R) + o )) n 2) Note that for x R, since F is of degree d, F x) = O d). That means that the probability of F x) = 0 for particular x is about d in aerage. So one should expact that N R ) should behae like 3 N ) d = O n d). 2.2 The setup of the Hardy-Littlewood method We ll denote the function f x) = e 2πiαx by e α x) for eery α R. Recall the Fundamental Theorem of Fourier Analysis 4 0 e α x) dα = { x = 0 0 x 0 ) Hardy established the asymptotics of p n) exp π 2n/3 4n, and also asymptotic rate for the error. In 3 937, Rademacher established the full formula. 2 Mainly, changing the focus from complex analysis to exponential sums ia Fourier analysis. 3 Notice that argument makes sense only if n > d. 4 In complex analysis, it corresponds to integrating oer a circle, that s how the circle method got its name originally.

2 Define the generating function S α, R) = S α) = for α [0, ]. Note that N R ) = x Z n R 0 x Z n R e α F x)) = 0 e α F x)), S α) dα 3) So we e effectiely reduced the problem of rational point counting to ealuating or at least obtaining asymptotics) for the integral on the RHS. We ll also define S α, R, m, a) = e α F x)). x Z n R x = a mod m 3 A Probabilistic View Many ideas in modern number theory arised from probabilistic iews 5. Now we ll try to gie estimate the size of S α) as α aries. Assume that k i < λ i, let V denote the olume of R, V R) = i λ i k i ). Assume moreoer that 0 < α <, one might expect that the numbers e α F x)) are randomly scattered on T, as x aries, for almost all α, wrt Lebsgue measure, due to Weyl s Euidistribution criterion. If so, and assume that you take enough samples, the CLT assures you that S α) should hae some sort of normal behaiour. Recall that the CLT also shows that the expected alue should be about the size N, when you tkae N samples. Therefore, according to the suare root cancelling philosophy, we ll get that S α) is about the order of n 2, which is about the size of a random set chosen of n elements. Combine it with the preceding argument, and you ll expect that n > 2d for the method to work. That s about the best there is, altough you can combine the circle method with other facts for specalized cases and obtain results for fewer ariables, i.e. Klossterman achieed results for diagonal uadratics in four ariables. 3. When e α changes its behaior? Apperently when α is near 0 or euialently ), the function S α) is indeed about the expected order of n d, because e α is almost constant, and one can erify that computationaly. When α = 0, the behaior is exactly like n due to the counting argument mentioned before. We might hope that when α is not too close to the endpoints of [0, ], S α) will make a contribution of order n 2 on aerage. one may see that such a simple distinction is not good enough. Assume for instance that all the coefficients of F are diisble by, then we ll actually see a behaiour similiar to the behaior near 0 at fractions of the form α = a, because we ll get artificial reduction of F there. Define S α) = x mod ) e a F x)), note that e a F x)) = e a F y)) if y = x mod, and one gets ) a S = e a F x)) # {y Zn R y = x mod } x mod ) And by simple counting argument # {y Z n R y = x mod } = V R) n +O n ), n so one get ) a S = V R) n n S a) + O n n ) Which means that at α = a, if S a) 0, the behaior is of order n, exactly like in α = 0. Due to the fact is in the denominator, one would like to ealuate S where α is close to fractions of small denominator. 5 i.e. The distribution of the primes and Cramer Model. 2

3 4 Arcs Let δ < 3, denote Q = δ, we ll define I a, ) = that gcd a, ) =. Note that those interals are disjoint assuming 0. We ll define the Major Arcs M as M = Q 0 a a, ) = [ a Q d, a Q d ] for eery Q, such I a, ) 4) We ll define the Minor Arcs m = [0, ] \ M 6. Notice that length I a, )) = 2Q = 2 δ d, which means that the interals I a, ) are d longer than d by factor which tends to infinity. Now we ll approximate S α) on any interal I a, ). 4. Major Arcs Approximation Put θ = α a, and notice that if y = x mod ), then e α F y)) = e a F x)) e θ F y)). So we get S α, R, m, a) = x mod y R y = x mod m) = e a F x)) x mod m e α F y)) y R y = x mod m) e θ F y)) 5) Note that the outer sum, is free of α. α is found implicitly) in θ by θ s defenition. In the major arcs settings, θ < 2 δ d, therefore we hope that the inner sum is well approximated by the integral I, θ) = e θ F r)) dr dr 2 dr n Actually, by calculation, one can deduce that e θ F y)) = m) n I, θ) + O y R y = x mod m) R n +2δ) 6) We won t bother to proof that it s ery technical), but one should notice two things: The shape of the main term - m) n I, θ). The exponential in the error term is strictly less than n, because δ is small. Now plug in 6 in 5 and get S α) = x mod m = We ll define S a, m, a) = e a F x)) n I, θ) + O n +2δ)) x mod m x mod m x = a mod m e a F x)) m) n I, θ) + O n n +2δ) e a F x)). Integrate the euation from aboe oer I a, ) to get S α) dα = S a, m, a) m) n J ) + O Q n+ n d +2δ) Ia,) 6 Notice that M [0, ], but because of periodicity of e α, we can change the integral oer any interal of length one, so we ll disregard that for simplicity. 3

4 Q d Where J ) = I, θ) dθ. Q d Now sum oer all the inerals I a, ) and get the full integral oer the Major Arcs, S α) dα = S Q) J ) + O Q n+3 n d +2δ) = S Q) J ) + O n d +n+5)δ) M Where S Q) = Q 0 a < a, ) = m) n S a, m, a). After some calculations, one can derie that J ) n d c as, where c is the singular integral dw 2 dw c = n ) 7) w R,F w)=0 w F w) Another way to define c is by c = lim ɛ 0 2ɛ) F x) ɛ dx. The singular integral can be thought as real density of the solutions F x) = 0 in R notice that we haen t reuired w R there). Now S Q) captures the local density at any other aluation, apart from the infinity aluation. We ll define p-adic density c p as c p = lim k p n )k { x mod p k) F x) = 0 mod p k}, then the singular seris S Q) becomes p c p. One can show that S a, m, a) factors as a sort of product oer different primes diidng, which can be thought like Chinese Reminder Theorem. If one defines S, m, a) = S a, m, a), then one actually get, due to S factoriazation, 0 a < a, ) = S, m, a) = p S p e, p f, a ). ) S p e, p f, a, which can be thought as partial So we get S Q) = S, m, a) = Q Q p summation oer corresponding Euler product. The main step from here is to take the infinite sum. Estimating such sums in particuler has been a major goal in number theory. I will remark that Deligne s proof of the Riemann Hypothesis for Weil s conjectures yeilds that if is prime, and F is non-singular modulo then S a, m, a) n 2, which is related to Hasse-Weil theorem and its generalizations oer ariaties - Weil s conjectures which Tomer talked about a couple of weeks ago. So assuming suitable estimates, which we won t proe here, one can actually get that S Q) = m n p c p + O Q n ). Notice from the structure of the singular series, if the Hasse principle fails, we won t receie the expected alue. Howeer, if it works, we will reciee the main term of ) c p c p n d, which agrees with our heuristic principle, because, basically the product of the densities tells you what is the probability that a ector is indeed a local solution for all the local places determined by primes and the real numbers. 4.2 Minor Arcs Estimate Our main goal here is to show that m S α, R, m, a) dα is o n d). We ll now impose some restrictions oer F. First we ll assume that F is a diagonal form, i.e. F x) = c x d + c nx d n. Notice that due to the exponent function properties, S α) factors as S α) = n j= S j α), where S j α) are one-dimensional counts, S j α,, m, a j ) = e α c j x d). x Z [k j, λ j ] x = a j mod m) One can then show, using Hölder ineuality that m S α) dα sup S j α) n 4 0 S k α) 4 dα α m )), for some indices j, k. Note that S k α) 4 = c j x d + x d 2 x d 3 x d 4, so x,x 2,x 3,x 4 e α in the integration one gets the number of uadruples x, x 2, x 3, x 4 ) in the specified region s.t. x d + xd 2 = xd 3 + xd 4. One can actually relax the region constraints and count the number of uadruples in Z 4 [, ] 4. 4

5 We e actually reduced the problem to a whole different algebraic ariety! Now we ll use the following number-theoretical lemma: For eery ɛ > 0 there is a constant c ɛ) s.t. # {m N : m N} c ɛ) N ɛ for eery N N When x d xd 3 > 0, apply that to N = xd xd 3 and notice that x d 2 xd 4 will be a positie diisor of N, and we get about O dɛ) solutions. More oer, if x d = xd 3 then xd 2 = xd 3, and one gets O 2) solutions. So in total, 0 S k α) 4 dα 2+dɛ. It remains to bound S j α) in order to bound the supremum in the RHS. From now on, we ll assume that d = 2. Denote β = αc j R \ Q, k = k j, λ = λ j, a = a j, just to ease the notation. S j α,, m, a j ) = e β x 2 ), which looks ery similiar to the statment k x λ of Weyl s criterion. We ll actually bound that in the similiar way of deducing the criterion for polynomials of degree higher than. This method is known as Weyl Differentiation. Look at S j 2 for start, S j 2 = x,y e β x 2 y 2). Set x = y + mz, where z goes oer integer alues in the range z λ k) m. In particuler we get that z 2. Now substitute y = a + mw and scale w appropriately i.e. µ w ν for suitable µ, ν). The substitutions yield x 2 y 2 = m 2 z 2 + 2amz + 2m 2 zw. Then S j 2 becomes e β mz 2 + 2amz ) e β 2βm 2 zw ), so one can get the triial bound z 2 S j 2 µ w ν z 2 µ w ν e β 2m 2 zw ) Now the inside sum is just summation oer a geometric seuence, and one shows that µ w ν e β 2βm 2 zw ) { min, }, so one get that 7 S sinπβ2m 2 ) j 2 { } min, sin πβ2m 2 ) z 2. Note that near zero, sine is almost linear, so we ll introduce the following notation ϕ t) = t max { z Z z t + 2}, due to the almost linearity, ϕ t) = Θ sin πt)). Define Z h = { z Z z 2, h ϕ 2βm 2 z ) < h+ }, where h + 2. If h 2, then sinπ2βm 2 z) + h improe our bound 8 Recall that 8 n k= k, and for h, ) S j 2 max h h h h + h = Θ log n)) and one can get the following bound ) S j 2 max #Z h log h 8), therefore, we can Now one can show that #Z h can be approximated by #Z 0, and we get 9 S j 2 log#z 0. Untill now, we haen t used the Minor Arcs at all. Recall Dirichlet Diophantine Approximation theorem Let α R, fix N > 0, then there exists a rational u s.t. α u < N, gcd u, ) =, N Using Dirichlet s theorem one find a rational u s.t. α u 2 Q. In particular, α u Q 2, but because α / M then > Q. y direct calculation, one gets that ϕ gets that 2m 2 c j uz = l mod, 2m 2 c j uz Q 2 where gcd u, ) =, ) +4m 2 c j, so by definition of ϕ one 7 The sine appears because of the definition of the sine ia Euler s formula. 8 Actually, γ = lim n n k= log n)). k 9 From that one can also deduce a bound for the exponential sums oer finite fields mentioned aboe, although it won t be tight bound, it ll be sufficient for our purpose here. 5

6 where l satisfies l +4m2 c j. So l = O + ), for each l, the number of congruence classes determined by l is finite O ), and the range z 2 containes atmost + 4 numbers from each congruence class 0. So after all that, # { z Z z 2, 0 ϕ 2βm 2 z ) } + ) + ) + +. Now > Q = δ therfore = δ, = δ, and # { z Z z 2, 0 ϕ 2βm 2 z ) } Q = δ. So finally S j 2 Q ) log = 2 δ log, therefore S j δ 2 log) 2. Combine it with the preious estimates m S α) dα δ 2 log) 2 ) n 4 2+2ɛ. Assuming ɛ is small enough, we actually get o n 2), when n 5. So we proed that for diagonal uadratic form of n 5 ariables, there is a constant c R, m, a) s.t. N R, m, a) = c n 2 + o n 2) when. References [Va] Vaughan, R. C., The Hardy Littlewood Method Cambridge Tracts in Mathematics. 25 2nd ed.). Cambridge Uniersity Press, 997. [MT] Steen J. Miller, Ramin Takloo-ighash, An Initation To Modern Number Theory. Princeton Uniersity Press, [IK] Iwaniec, Henryk; Emmanuel Kowalski, Analytic Number Theory. Proidence: American Mathematical Society, [H-] D.R. Heath-rown, Analytic Methods For The Distribution of Rational Points On Algebraic Varieties in Euidistribution in Number Theory, An Introduction, Andrew Granille and Zeé Rudnick, NATO Science Series, [Ts] Yuri Tschinkel, Algebraic arieties with many rational points in Arithmetic Geometry, Henri Darmon, Daid Alexandre Ellwood, rendan Hassett and Yuri Tschinkel, Clay Mathematics Proceedings, To see that, substitute z in the congruence euation for the aailable alues, and due to the congruence, one has to diide by 6