RESEARCH STATEMENT THOMAS WRIGHT My research interests lie in the field of number theory, articularly in Diohantine equations, rime gas, and ellitic curves. In my thesis, I examined adelic methods for aroximating the number of solutions to a system of Diohantine, Waringtye equations; of articular interest is the asymtotic number of solutions as the arameters move toward infinity. My thesis focused on the question of when the singular series, the series which generally aroximates the number of solutions, converges. My more recent work has focused on altering these adelic methods so that they can be used to garner information about the number of airs of rimes and 2 such that 2 = ν (for examle, if ν = 2, this is the Twin Prime Conjecture. For each even ν, it has been conjectured that the number of such rimes is infinite. We hoe to use adelic methods to show that the conjectures for the various ν are related; more secifically, if we know the density of airs of rimes with difference ν then we can say something about the number of airs of rimes of difference 2ν or ν 2.. Hardy-Littlewood Problems: Introduction The modern roots of Hardy-Littlewood roblems can be traced back to 770, when Joseh Lagrange roved that every natural number can be exressed as the sum of four squares. Insired by this roof, Edward Waring asked if such a theorem could be roven not just for squares but also cubes, quartics, quintics, etc. To ut this more formally, for a natural number k >, does there exist an n such that every natural number can be written as the sum of n k-th owers? Embedded in this query are two actual questions:. Does such an n exist for any k? 2. What is the smallest ossible value for each n? The former question was settled in the affirmative by Hilbert in 909. Modern work, therefore, has focused uon the latter question. The seminal work in the modern aroach to Waring s roblem (and general Diohantine equations was written by Hardy and Littlewood in 99. In this aer, they develoed a method (known as the circle method which uses an integral on [0,] to estimate the number of solutions to the equation x k + x k 2 +... + x k n = ν. While the circle method was of some use in solving the original roblem, it turned out that the roblem to which it was better suited (and the more mathematically interesting roblem was to determine n for which the above equation has solutions
2 THOMAS WRIGHT asymtotically (as ν. As is the case with most such estimates, this Hardy- Littlewood estimate for the number of solutions, known in mathematical literature as the singular series, becomes more effective as ν becomes large. In the intervening 90 years, a number of generalizations and reformulations of the work of Hardy and Littlewood have aeared. Many number theorists currently ursue such roblems; although the techniques for doing so vary greatly, most of them still use some semblance of the circle method. Now, let us ut the initial question another way. Let f : Q n Q m be a olynomial ma, and let ν Q m. The goal is then to study the set f (ν (i.e. the fiber of f which mas down to ν. In articular, we can write N(ν = #{x Z n : f(x = ν} =. Clearly, in the Waring case, m = and f(x = x k + x k 2 +... + x k n. x Z n,f(x=ν The Hardy-Littlewood circle method estimates the size of this fiber f (ν. In my thesis, I let f be a air of quadratic forms, i.e. f = (f, f 2 : Q n Q 2, where f (x = x 2 +... + x 2 n, f 2 (x = λ x 2 +... + λ n x 2 n. Of course, we wish to know the number of solutions to f(x = ν = (ν, ν 2 N 2. I use the adelic (not circle method of T. Ono, Igusa, and Weil to develo a singular series which aroximates the size of f (ν, ν 2 as ν and ν 2 become large. 2. Method We can determine the overall behavior of the singular series on N by examining it locally (i.e. on R = Q and on the -adic numbers Q for each rime and then combining the local information to give us our global solution. In articular, let { if x Z, ϕ (x = 0 otherwise, ϕ (x = e π x 2, where Z is the ring of integers in Q (the elements of Q whose denominators are not divisible by. Moreover, let χ v be a (basic character on Q v (here, v denotes that the subscrit can be either or. Define Gϕ v (ξ = ϕ v (xχ v (< f(x, ξ >dx, and S v (ν = The singular series S(ν is then defined by Q n v Q 2 v Gϕ v (ξ χ v (< ξ, ν >dξ. S(ν = v S v (ν.
RESEARCH STATEMENT 3 3. Convergence The first question of imortance is when the singular series converges. This is answered in [Wr2] by the following theorem: Theorem. If n 6 then the integral S(ν converges. Previous literature (beginning with the Artin Conjectures has set the exectation for convergence of the singular series at n mk 2 for m equations of degree k. In this case, such redictions would give n mk 2 = 8; thus, the fact that n 6 here is surrising. 4. Evaluation The second question is the actual value of the singular series. To this end, we find exlicit results for values of the various S v. If v = then we have the following: Theorem 2. Let K = 2 e π(ν (ν n 2 2, and (λ n ν2 ν L = n i= (λ i λ n u 2 i Additionally, let J 0 denote the 0-th Bessel function of second tye, and let rect denote the usual rectangle function. Moreover, let u i vary over the unit ball, and let dv be the usual measure of this ball in hyersherical coordinates. Then S (ν = Kπ Sn 2 2 rect( 2 + L π(ν n i= (λ i λ n u 2 i ( + 2L 2 dv. Moreover, if v = for a rime then we have the following result: Theorem 3. Let A = { rime: = λ i, = ν j, λ i λ j, or ν λ + ν 2 λ 2 }, and assume A. Then S (ν = n ( + λi u u H i= n + ( where i= G n G n ( λi ν ν2 H = {u F n j= i n,i j ( + λi λ j ( n G n G n G n ( λ i +, : u λ i i= i, u ν ν 2 }, and G is the classical Gauss sum G(, given by { if (mod 4, G = i if 3 (mod 4. ( ν + λ j ν 2
4 THOMAS WRIGHT 5. Current Work on Primes In my more recent work, I have turned my attention toward using these adelic methods to study rimes. This work is joint with Benjamin Weiss and is being done under the suervision of Jeff Lagarias, both of the University of Michigan. In articular, I made the following alteration. Let Y, Z be large real numbers such that Y = Z δ for some small δ. For rimes Y, let if x Z Z, ϕ (x = V if x Z 2 Z, 0 otherwise, for some large V. By way of exlanation, the function ϕ checks to see whether x is divisible by or 2. If x is divisible by, the function attaches a weight of V ; if it is divisible by 2, the function gives weight zero (if does not divide x, the function gives weight. As such, the various ϕ act as counters, telling us how many rime divisors x has and whether it is square-free; the more rime divisors a number has, the smaller its weight will be in this accounting system. For rimes greater than Y, define ϕ (x to be the characteristic function on Z as before. Additionally, define ϕ (x = e πx2 Z. ϕ decays raidly, meaning that we can ignore large values of x. ϕ is also a nice (i.e. Schwartz function, thereby allowing us to use the tools of Fourier analysis. From this, we define φ (x, x 2 = ϕ (x ϕ (x 2, φ (x, x 2 = ϕ (x, and let φ(x, x 2 = v φ v (x, x 2. Now, while x and x 2 must still be integers, φ gives largest weight for x and x 2 rime (or and small, and it gives smaller weight for comosites or large integers. Thus, this function will allow us to use adelic methods to answer questions about rime solutions to equations. Let f(x, x 2 = x x 2. For this equation, N φ (ν then gives an estimate for the number of airs of rimes of difference ν u to some bound (this bound is actually about Z. There has been much conjecture about such a quantity; for examle, Hardy and Littlewood conjectured the following: Hardy-Littlewood Conjecture. Let π ν (y be the number of airs of rimes less than y that have difference ν. For any even ν N, there exists an exlicitly comutable constant C ν such that, as y goes to infinity, y π ν (y C ν log 2 y.
RESEARCH STATEMENT 5 Unlike traditional methods, which attemt to give exlicit comutations for π ν for a secific ν, our adelic construction can be easily altered to comare π ν for different values of ν. It has been conjectured that, once the N s are weighted roerly, the weighted difference between N φ (ν and N φ (2ν is small relative to the actual size of N φ (ν; such a conjecture would rove the equivalence of the Hardy-Littlewood conjectures for π ν and π 2ν. Using these adelic methods, it is hoed that one can make rogress toward roving this equivalence. References [HL] G.H. Hardy, J.E. Littlewood, A new solution to Waring s roblem, Q.J. Math. 48 (99, 272-293. [La] G. Lachaud, Une résentation adélique de la série singuliére et du robléme de Waring, Enseign. Math. 28 (982, 39-69. [On] T. Ono, Lectures on the Hardy-Littlewood singular series (2007-08. Transcribed by T. Wright. [On2] T. Ono, Gauss transformations and zeta functions, Ann. of Math. (2, 9 (970, 332-36. [Wr] T. Wright, An Elementary Aroach to Euler s Concordant Forms, in rearation. [Wr2] T. Wright, On Convergence of Singular Series for a Pair of Quadratic Forms, thesis for Johns Hokins University. [Wr3] T. Wright, Adelic Singular Series and the Goldbach Conjecture, in rearation.