An Overview of Witt Vectors

Transcription

1 An Overview of Witt Vectors Daniel Finkel December 7, 2007 Abstract This aer offers a brief overview of the basics of Witt vectors. As an alication, we summarize work of Bartolo and Falcone to rove that Fermat s Last Theorem does not hold true over the -adic integers. 1 Fundamentals of Witt vectors The goal of this aer is to define Witt vectors and develo some of the structure surrounding them. In this section we describe how the Witt vectors generalize the -adics, and how Witt vectors over any commutative ring themselves form a ring. In the second section we introduce the Teichmüller reresentative, show that the integers are a subring of the -adics, and rove De Moivre s formula for Witt vectors. In the third and last section we give necessary and sufficient conditions for a -adic integer to be a k -th ower, and show that that Fermat s Last Theorem is false over the -adic integers. For the following, fix a rime number. Definition. A Witt vector over a commutative ring R is a sequence (X 0, X 1, X 2...) of elements of R. Remark. Witt vectors are a generalization of -adic numbers; indeed, if R = F is the finite field with elements, then any Witt vector over R is just a adic number. The original formulation of -adics integers were ower series a 0 + a a , with a i {0, 1, 2,..., 1}. While the ower series notation is suggestive of analytic insiration, it roved unwieldy for actual calculation. Teichmüller suggested what has become the current notation, wherein each -adic number is reresented as an infinite sequence of elements of F. Witt Vectors have lots of nice roerties, and their alications range through number theory and algebraic geometry. Ernst Witt ( ) showed how to ut a canonical ring structure on the collection of Witt vectors over any ring. To do this, he introduced Witt olynomials. 1

2 Definition. Let be a rime number, and (X 0..., X n,...) be an infinite sequence of indeterminates.for n 0, we define the n-th Witt olynomial W n as W n = n i=0 i X n i i Examle. The first three Witt olynomials are W 0 = X 0, = X n 0 + X n n X n. W 1 = X 0 + X 1, W 2 = X x x 2. The next theorem, which we cite without roof from [1], rovides the key to adding and multilying Witt vectors. Theorem 1. Let (X 0, X 1, X 2...) and (Y 0, Y 1, Y 2...) be two sequences of indeterminates. For every olynomial function Φ Z[X, Y ] there exist a unique sequence (φ 0,..., φ n,...) of elements of Z[X 0,..., X n... ; Y 0,..., Y n,...] such that W n (φ 0,..., φ n,...) = Φ(W n (X 0,..., X n...), W n (Y 0,..., Y n,...)), n = 0, 1,.... Alying Theorem 1, we denote by S i (resectively P i ) the olynomials φ i associated to Φ(X, Y ) = X + Y (resectively Φ(X, Y ) = X.Y ). Letting R be an arbitrary commutative ring, and letting A = (a 0, a 1,...), and B = (b 0, b 1,...) be Witt vectors over R, we have equations for adding and multilying Witt vectors: A + B = (S 0 (A, B), S 1 (A, B),...) A.B = (P 0 (A, B), P 1 (A, B),...) Examle. The first coule values of addition and multilication of Witt vectors are: S 0 (A, B) = a 0 + b 0, S 1 (A, B) = a 1 + b 1 + a 0 + b 0 (a 0 + b 0 ), P 0 (A, B) = a 0 b 0, P 1 (A, B) = b 0 a 1 + a 0 b 1 + a 1 b 1. Note that each term of the numerator of the quotient in S 1 (A, B) is divisible by, so this makes sense even when we are in F. We will deal more with this term in section 3. 2

3 In fact, one can check that these oerations make the Witt vectors into a ring. Corollary 2. The Witt vectors over any commutative ring R form a commutative ring, which we denote by W (R). Examle. (i) If is invertible in R, then W (R) = R N (the roduct of countable number of coies of R). In fact, the Witt olynomials always give a homomorhism from the ring of Witt vectors to R N, and if is invertible this homomorhism is an isomorhism. (ii) The Witt ring of the finite field of order is the ring of -adic integers. (iii) The Witt ring of a finite field of order n is an unramified extension of the ring of -adic integers. The functions P k and S k are actually functions of the first k terms of A and B. In articular, if we truncate all the vectors at the k-th entry, we can still add and multily them. This allows us to define the truncated Witt ring W k (R) := {(a 0, a 1,..., a k 1 ) a i R}. Examle. The truncated Witt ring W k (F ) = Z/ k Z. Definition. The shift ma V : W (R) W (R) is defined by V (A) = (0, a 0, a 1,...). When R is a ring of characteristic, the ma F : W (R) W (R) is defined by F (A) = (a 0, a 1,...). Considering to denote the ma that is multilication by, we also have the identity V F = = F V. This identity can easily be seen to be true when R = F. In this case, W (R) is the ring of -adic integers, F is the identity ma, and = V. Notice that W k (R) = W (R)/(V k W (R)). In articular, when R = F we have W k (F ) = W (F )/( k W (F )) = Z/ k Z. This allows us to seak of Witt vectors being equivalent modulo k when their k-th truncations agree. 2 De Moivre s Formula Definition. The element (ā, 0, 0,...) W (F ) is denoted by a τ and is called the Teichmüller reresentative of a, where ā is the reduction of a modulo. Notice that (a τ ) = (ā, 0, 0,...) = (ā, 0, 0,...) = (ā, 0, 0,...) = a τ 3

4 since ā = ā in F. There is a natural injection Z W (F ) defined by taking 1 1 τ. Following [3] we can describe the reresentatives for n N in W (F ) by the following roosition. Proosition 3. Let n N. For any k = 0, 1,..., let a 0,..., a k Q be elements such that the k-th Witt olynomial W k (a 0,..., a k ) = n. Then (i) a 0 = n and a k+1 = k 0 1 (a k i+1 i k i ak i+1 i ) Z; (ii) n 1 τ = (ā 0, ā 1,...); and (iii) If does not divide n, then n divides each a k. Proof: [3]. Any Witt vector A = (a 0, a 1,...) with a 0 0 (mod ) can be written as the roduct A = a τ 0(1, a 1 /a 0, a 2 /a 0,...). The invertible elements in W (F ) are recisely those A having a 0 0 (mod ). Therefore any element of the quotient field of W (F ), which is the field of -adic numbers, can be written as A = z a τ 0(1, a 1 /a 0, a 2 /a 0,...). This notation is suggestive (if you stare at it long enough) of the comlex number notation z = z e iθ. In fact, we can formulate a similar module/argument notation for Witt vectors over F, and rove that multilication in this case also satisfies De Moivre s formula. In order to do this, we must first define the logarithmic and exonential mas for Witt vectors. Fortunately, we can simly extend the formal ower series definition. Definition. Assume > 2, and let A W k (F ) be a truncated Witt vector. Then we define the functions log(1 + A) = A 1/2(A) 2 + 1/3(A) 3... e A = 1 + A + 1/2!(A) 2 + 1/3!(A) If A is a truncated Witt vectors over F, then log(1 + A) and e A are just (finite) olynomials in W k (F ), since k A = 0 for all A W k (F ). However, since these two mas can be defined for any k, we can define them on the whole of 1 + W (F ) = {A = (1, a 1, a 2,...) a i F } and W (F ) = {A = (0, a 1, a 2,...) a i F }. Since they are defined via their ower series exansion, the two mas are mutually inverse. Now writing an arbitrary Witt vector A = z a τ 0(1, a 1 /a 0.a 2 /a 0,...), we may define the module ρ A := z a τ 0 and the argument θ A := log(1, a 1 /a 0, a 2, a 0,...). This allows us to write, even more suggestively A = ρ A e θ A 4

5 and recover De Moivre s formula for the -adics. ρ AB = ρ A ρ B, θ AB = θ A + θ B 3 An Alication: Fermat s Last Theorem is false for -adic Integers. Armed with this structure, we can determine when -adics have th roots. Letting > 2 as above, take an arbitrary A W (F ), with first term x 0 0 (mod ). Using De Moivre, we have A k = (ρ A e θ A ) k = (ρ A ) k (e θ A ) k = ( z a τ 0) k (e k θ A ) = zk a τ 0(1, 0,..., 0, a }{{} 1 /a 0,...) k where the entries to the right of a 1 /a 0 get more comlicated. From this it is clear that having x i 0 for i = 1,..., k is a necessary condition for a Witt vector to have a k -th ower. It turns out that it is a sufficient condition as well. Let A = (ρ A e θ A ) = z a τ 0(1, a 1 /a 0, a 2 /a 0,...) be a Witt vector such that k divides z and a i 0 (mod ) for i = 1, 2,..., k. Then we can calculate exlicitly that A 1/k = z k a τ oex( 1 k log(1, 0,..., 0, a k+1/a 0,...). But 1 k log(1, 0,..., 0, a k+1/a 0,...) = 1 k (0,..., 0, a k+1/a 0,...) = (0, a k+1 /a 0,...) W (F ). In other words, ( 1 log(1, 0,..., 0, a k k+1 /a 0,...) is in the domain of the exonential function, so A 1/k is well defined, and in fact, uniquely defined. Thus, we have roven Theorem 4. A Witt vector A = z a τ 0(1, a 1 /a 0, a 2 /a 0,...) over F has a k -th root if and only if k divides z and a i = 0 for i = 1, 2,..., k. As an alication of this theorem, we will exhibit an examle of -adics which solve a Fermat equation a n + b n = c n. In articular, we will let n = rime, and we will restrict ourselves to Witt vectors with z = 0 in the language of Theorem 4. Now, to find such a solution, we need only concern ourselves with the truncated Witt vectors W 2 (F ), since a Witt vector A has a -th root according to the theorem when the second term a 1 is 0, and this term in a sum is determined only by the zero-th and first terms of the summands. So the roblem of finding a solution to a + b = c over the -adics reduces to the 5

6 roblem of finding three truncated Witt vectors A, B, C W 2 (F ) such that A = (a 0, 0), B = (b 0, 0), C = (c 0, 0) and A + B = C. This is, of course, a much easier roblem. In articular, we need to choose (a 0, 0), (b 0, 0), and such that (a 0, 0) + (b 0, 0) = (c 0, c 1 ) satisfies c 1 = 0. Looking back to our equations for sums (over F, and hence modulo, in this case) recall that S 1 (A, B) = a 1 + b 1 + a 0 + b 0 (a 0 + b 0 ) = a 0 + b 0 (a 0 + b 0 ) (mod ) because a 1 = b 1 = 0 in this case. Exanding (a 0 + b 0 ) lets us rewrite this as a 0 + b 0 (a 0 + b 0 ) Now notice that ( i) ( 1)( 2)... ( i + 1) = i(i 1)... (1) Which gives us the equivalence S 1 (A, B) = 1 i=1 1 = i=1 ( i) ai 0b i 0. ( 1)( 2)... ( i + 1) (1)(2)... (i) ( 1) i i a i 0b i 0 (mod ). ( 1)i 1 i (mod ). So to find our Fermat trile of -adics is now a matter of finding a 0, b 0, and such that S 1 (A, B) = 0 (mod ). For examle, we check that S 1 (1, 2) 0 (mod 7). S 1 ((1, 0), (2, 0)) = ( 1) (mod7) This means that (1, 0)+(2, 0) = (3, 0) gives a solution to the Fermat equation for n = 7 in W 2 (F ), since all three of the terms in the sum have 7-th roots. Furthermore, any Witt vectors A, B W (F ) with A = (1, 0, a 2,...), B = (2, 0, b 2,...) are seventh owers that sum to a seventh ower. Note also that since 129 = (1, 0) + (2, 0) = (3, 0) (mod 7 2 ) using our association between truncated Witt vectors of length k and integers modulo k, we have that the image of 129 in W (F ) has a seventh root. 6

7 References [1] Serre, Jean-Pierre (1979), Local fields, vol. 67, Graduate Texts in Mathematics, Berlin, New York: Sringer-Verlag, MR554237, ISBN , section II.6 [2] A. Di Bartolo, G. Falcone, On the -th root of a -adic number, htt:// gfalcone/mypapers/ [3] A. Di Bartolo, G. Falcone, Witt vectors and Fermat quotients, htt:// gfalcone/mypapers/ [4] Wikiedia Entry for Witt Vectors, htt://en.wikiedia.org/wiki/witt vectors 7