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The Vladimirov Heat Kernel in the Program of Jorgenson-Lang

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The Vladimirov Heat Kernel in the Program of Jorgenson-Lang U.U.D.M. Project Report 2018:25 The Vladimirov Heat Kernel in the Program of Jorgenson-Lang Mårten Nilsson Examensarbete i matematik, 30 hp Handledare: Anders Karlsson Examinator: Denis Gaidashev Juni 2018 Department of Mathematics Uppsala University Introduction This thesis discusses possible p-adic analogues to the classical heat equation. The main motivation for this is the fact that the kernel to the heat equation @u − αr2u = 0 @t combined with the Poisson summation formula 1 1 X X f(x + n) = f^(k) e2πikx n=−∞ k=−∞ yields the theta inversion formula 1 p θ( ) = zθ(z); z which constitutes a main ingredient in many proofs in analytic number theory (for example the functional equation of the Riemann zeta function, by applying the Mellin transform, and quadratic reciprocity [11]). The most direct way to try to model this in a p-adic setting is to consider continuous functions f : Qp ! C; as one then can use the machinery of harmonic analysis on locally compact groups. Some work has been done in this direction, initiated by V.S. Vladimirov [19] with his introduction of a certain pseudo- differential operator on complex-valued functions of a p-adic variable, in some respects analogue to the classical Laplacian. This as well as subsequent research seem to mainly have been motivated by finding p- adic analogues to the mathematical models in modern physics, for example constructing a p-adic quantum theory, as well as p-adic versions of several equations of classical physics (see [20] and references therein). Specifically, an analogue to the heat equation has been proposed in this setting. We will give an account of the one-dimensional version of this, confirming results in [3, 14, 19], and connect it to a Poisson summation formula. We will also introduce a more “regularized” operator, and pass it through the same scheme. On the other hand, p-adic numbers have been used extensively in number theory. Notably, John Tate used the adele ring, a construction considering all p-adic fields and the reals “at once”, in combination with an adelic Poisson summation formula to explain all the components of the functional equation for the Riemann zeta function. As a part of a “global Gaussian”, the indicator function on respective set of p-adic integers was used, mainly due to its properties with respect to the Fourier transform. Nonetheless, attempts to p-adically mimic the intimate relation between the heat equation and the Gaussian apparent in the real case have been few, and on a whole, studying different analogues of the heat equation for number theory’s sake seem to be a rather novel concept, initiated by Lang and Jorgenson [2, 10]. Essentially, the program consists of trying to find analogues to the heat equation, the heat kernel, the Poisson summation formula on different mathematical structures, with the goal of acquiring functional equations for “zeta functions” in each setting. We will try to follow this scheme, using the spectral zeta function one may associate with the Vladimirov operator [15]. Although the attained relation is trivial, we show that it could be viewed as a functional equation for the spectral zeta function. The thesis is largely self-contained, apart from the spectral theory for the Vladimirov operator. [15] and the references within is a good introduction to this theory. Also, the construction of the p-adic numbers as well as their most basic properties will be assumed and very briefly stated. Some important features are summarized and proved in Proposition 1.1.1. For a thorough exposition, see [13]. 1 Contents Introduction . .1 1 Prerequisites 3 1.1 The p-adic numbers Qp .....................................3 1.2 Integration on Qp .........................................5 1.3 The Fourier transform . .7 1.4 Do complex-valued functions on Qp have a derivative? . 10 2 Two p-adic heat equations 11 1 2.1 The function space Cc (Qp) ................................... 11 2.2 The Vladimirov operator Dα and its associated heat equation . 14 2.3 The operator ∆C and its associated heat equation . 20 3 A p-adic Poisson summation formula 25 3.1 Two proofs of the Poisson summation formula . 25 3.2 Transforming the summation formula . 32 Bibliography 33 2 Chapter 1 Prerequisites 1.1 The p-adic numbers Qp The p-adic numbers Qp are for each prime number p the field one gets after carrying out completion on Q with respect to the multiplicative norm ¨p−n if x = pna=b; jxjp= 0 if x = 0; where a; b; n 2 Z, and a; b are co-prime numbers not divisible by p, using the fundamental theorem of arithmetic. Essentially, the norm of an element x 2 Q is small if x contains many factors of p, and −1 large if x contains many factors of p . Further, this norm on Q carries over to Qp and thus induces a non-archimedean metric with an associated topology on Qp in the usual way. A central feature of the p-adic norm is the strong triangle inequality, jx − yjp≤ maxfjxjp; jyjpg; also known as the non-archimedean property. Due to this, some fundamental peculiarities of Qp arises, see Proposition 1.1.1 below. Each element has a unique representation 1 X i aip i=−n with 0 ≤ ai < p [13]. The p-adic integers Zp, i.e. may be characterized in several ways, as the elements of the form 1 X i aip i=0 or equivalently B≤1(0), i.e. the closed ball centered in 0 with radius 1. This follows from the fact that the rational integers lie in the unit ball, so the elements in the ring of integers, satisfying 3 n n−1 x + cn−1nx + :::c1x + c0 = 0 where each ck is a rational integer, also satisfies n n−1 0 j jx jp≤ maxfjcn−1x jp; :::; jc0x jpg = jcjx jp; which implies jxjp≤ 1. We will now prove some interesting facts about Qp [12]. Proposition 1.1.1. The following holds in Qp: (i) If y 2 B<r(x), then B<r(x) = B<r(y). The same is true for closed balls. (ii) All open balls are clopen, i.e. both open and closed. (iii) A sequence faig is Cauchy (and thus converges) iff ai − ai−1 ! 0. (iv) Qp is totally disconnected. (v) A set is compact iff it is closed and bounded. (vi) Qp is locally compact, and Zp is compact. Proof. (i) If z 2 B<r(x), then jy − zjp= jy − x + x − zjp≤ max(jy − xjp; jx − zjp) < r so z 2 B<r(y). Similarly if z 2 B<r(y), then z 2 B<r(x). Similar reasoning for closed balls. (ii) Denote a general open ball B<r(a), and write its complement fx 2 Qp : jx − ajp≥ rg as the union fx 2 Qp : jx − ajp= rg [ fx 2 Qp : jx − ajp> rg. The latter is open in all metric spaces, so it is enough to show that the sphere Sr(a) is open. Take x 2 Sr(a); < r; y 2 B<(x). Then jy − ajp= jy − x + x − ajp≤ maxfjy − xjp; jx − ajpg = jx − ajp but jx − ajp= jx − y + y − ajp≤ maxfjx − yjp; jy − ajpg = jy − ajp so jy − ajp= jx − ajp= r and y 2 Sr(a). (iii) “)” holds for all metric spaces. To prove “(”, assume lim jai − ai−1jp= 0; i!1 i.e. for any > 0 there exists a natural number N such that if n > N then jan − an−1jp< . If m > n, then jam − anjp= jam − am−1 + am−1 − am−2 + ::: − anjp< max(jam − am−1jp; :::; jan+1 − anjp) < due to the strong triangle inequality, so Cauchy. (iv) We show that the only connected sets (i.e. cannot be written as the union of two disjoint non-empty open sets in the induced topology) are the singletons fag; a 2 Qp. Note that for each n 2 N, the ball B<p−n (a) is a clopen neighborhood of a as noted in (i). Now take a set A 6= fag but a 2 A. Then there exists an n such that B<p−n (a) \ A 6= A, and thus A = (B<p−n (a) \ A) [ (Qp n B<p−n (a) \ A): 4 Since (Qp n B<p−n (a);B<p−n (a) are open, this is a decomposition of A into disjoint open sets in the induced topology, and thus A is disconnected. (v) “(”: We will establish sequential compactness, as this is equivalent to compactness in metric spaces due to the Heine-Borel theorem. Denote the sequence fakg. Since the sequence is bounded, all elements n are contained in some ball B≤pn (0), n 2 Z, which may be written p Zp. Using the digit expansion of n P1 n n ak, we may write ak = p n=0 ak;np = p · :::ak;1ak;0. We will call ak;0 the first digit, ak;1 the second digit, and so on. Since there is only finitely many possibilities for the first digit, we may find an integer 0 0 ≤ b0 < p such that there exists sub-sequence fakg where each element has b0 as its first digit. In the 0 1 same way, we may from fakg find a sub sequence fakg where the first two digits in each element are b0; b1. Continuing iteratively, this creates 0 0 0 0 fakg = a1; a2; a3; ::: 1 1 1 1 fakg = a1; a2; a3; ::: 2 2 2 2 fakg = a1; a2; a3; ::: 3 3 3 3 fakg = a1; a2; a3; :::; n taking the diagonal yields a sub-sequence to the original series fakg, converging to p · :::b2b1b0.
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