From: kramsay@aol.com (KRamsay) Newsgroups: sci.math Subject: Re: Fermat, G�del, Wile Date: 11 Oct 1998 18:03:28 GMT In article , Jeremy Boden writes: |>Andrew Wiles' accomplishment in using p-adic representations ... |What does p-adic mean??? When p is prime, the field Q_p of p-adics is the completion of the rational numbers Q in the p-adic norm. This construction is analogous to the way the reals are constructed as the completion of the rationals in the usual norm |x|. A real number is an equivalence class of Cauchy sequences of rationals, where a sequence x1,x2,... is Cauchy if for every epsilon>0 there is an N such that if i,j>N, then |xi-xj|0 as i->infinity. To construct the p-adics, we replace the usual norm |x| with the p-adic norm, |x|_p. If x is rational, then |x|_p=p^{-k} where p^k is the power of p for which r can be written as p^k*(a/b) where a and b are both not divisible by p. For instance, in the p-adic norm, high powers of p have small norms, hence are "close" to 0. This p-norm gives a metric where the distance between two rationals x and y is |x-y|_p. More concretely, the p-adics are sometimes described as being numbers represented in base p, but with the proviso that the digits are allowed to run off to infinity *on the left* instead of on the right. In the usual norm, one can always replace a Cauchy sequence with an equivalent one where each term is a number with a terminating base p expansion, and a sequence of such numbers converges if for each N, the expansion up to the N-th position after the decimal never differs by more than addition of 1 to the last place. (One could have, for example, in base 7 the sequence 0.6, 1.1, 0.66, 1.01, .666, 1.001,... which converges to 1. The first N digits after the decimal point settles down to either .6666...6 or 1.0000...0, which differ only by adding 1 to the last considered position. Don't tell the .999 bunch that I said this, though.) In the p-adic norm, the digits to the left of the decimal become less and less "significant", and so the expansion of the limit of the sequence goes off to infinity on the left instead of the right. Every Cauchy sequence in the p-adic norm is equivalent to one where each term has a finite base p expansion. Such a sequence converges just in case for each position, the decimal in that position eventually stabilizes. We can express it in base p by giving the decimal in each place which the expansion stabilizes at (and there is no problem of repeating p-1 versus repeating 0 to worry about). The p-adics are good for analysis of congruences modulo powers of p. The congruence x^3=1 (mod 7^k) has three solutions for each power of k, and they form three Cauchy sequences of rationals in the 7-norm. The associated 7-adic numbers are the three cube roots of 1 in Q_7. One root is just 1, of course. Then we have the roots 2 mod 7, 2+4*7 mod 7^2, 2+4*7+6*7^2 mod 7^3, 2+4*7+6*7^2+3*7^3 mod 7^4,... which converge to a primitive cube root of 1 in Q_7. (It doesn't matter which representative of the congruence classes we pick; the resulting sequence converges to the same element of Q_7.) We would write the limit of the sequence as an infinite series, 2+4*7+6*7^2+3*7^3+..., or you could write it in base 7 as ...3642. There's another sequence which converges to the other root, ...3024. These two elements happen to be the limit of a sequence of (ordinary) integers, and we call elements of Q_p which are the limit of integers "p-adic integers", and denote the set of all such p-adics by Z_p. One standard tool is called (in various guises) "Hensel's lemma". It tells you that if you have a "good enough" root of a polynomial modulo a power of p, then there it can be lifted to a root modulo any higher power of p, and hence that there's an associated p-adic root of the same polynomial. One way to prove Hensel's lemma applies Newton's method for finding roots, and show that under suitable conditions it converges to a p-adic root. It still has the property of approximately doubling the number of digits of "precision" (i.e., running to the left) at each step. As you might guess, I think this is kind of cute. For example, let's say we apply Newton's method to finding a root of 3-1/x=0. The iteration is x-> x-(3-1/x)/(1/x^2)=x(2-3x). If we start with x=0.3 we get the sequence 0.3, 0.33, 0.3333, 0.33333333,... which converges in the usual norm to 1/3, exactly doubling the digits of precision with each step. However, if we start out with x=2 we get the sequence 2, -8, -208, -130208, -50862630208... which looks terrible. Upon closer inspection, we see that it converges to 1/3 in Q_5: for n>1 the n-th term is (1-5^(2^(n-1)))/3, and 5^2^(n-1) converges quickly to zero. I hear that Fernando Gouvea has written a nice textbook on the p-adics. One says that a group representation is p-adic if the vector space on which the group acts has as its coefficient field the p-adics, just as one would say its complex if the coefficient field is the complex numbers. Wiles worked with a kind of p-adic representation which naturally arises from elliptic curves. I think the idea is attributed to Tate. First off, elliptic curves have a commutative group structure. The group structure has the property that the p^k torsion points, those points P which have p^k*P=P+P+P+...+P equal to the identity, are isomorphic to (Z/p^k)x(Z/p^k). Tate considers the sequences of points P1,P2,P3,... where p*P_{k+1}=P_k, and p*P1 is the identity. Thus P_k is a p^k-torsion point of the curve. They form a module over Z_p. If we have x=d0+d1*p+d2*p^2+... an element of Z_p, and {P1,P2,P3,...} an element of the Tate module, we can multiply the one by the other to get another element of the Tate module, {d0*P1,(d0+d1*p)*P2, (d0+d1*p+d2*p^2)*P3,...}. We can extend this module to a Q_p vector space (two dimensional). If the ellptic curve has rational number coefficients, then there is also an action of the Galois group (of the algebraic numbers over the rationals) on this module and this vector space. The coordinates of the torsion points on the elliptic curve are algebraic numbers, so when we apply an element of the Galois group to them, we get the coordinates of a (usually different) torsion point on the curve. One of these sequences {P1,P2,P3,...} of points gets replaced with another of the same type, and this transformation on the Tate module extends to a linear transformation on the 2-dimensional Q_p vector space. This isn't the only way that p-adic representations entered into Wiles' proof, but it's a very important way. Keith Ramsay "Thou Shalt not hunt statistical significance with kramsay@aol.com a shotgun." --Michael Driscoll's 1st commandment ============================================================================== From: NoJunkMail@this.address (Gerry Myerson) Newsgroups: sci.math Subject: Re: p-adics Date: Wed, 04 Feb 1998 11:58:05 +1100 In article <34d1e20e.363774920@firewall>, marek@power-x.demon.co.uk wrote: => Just curious - can anyone explain to a non-mathematician (physics => graduate): => What are p-adic numbers? ...and what are they used for? See G Myerson & A J van der Poorten, Some problems concerning recurrence sequences, American Mathemtical Monthly 102 (1995) 698--705. Gerry Myerson (gerry@mpce.mq.edu.au)