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Finally, Witt Vectors

By ellipticcurves

Now, let us try to do arithmetic operations on p-adic numbers written in terms of Teichmüller representatives. Suppose

a = t(a_0) + t(a_1)p + t(a_2)p^2 + \ldots

and

b = t(b_0) + t(b_1)p + t(b_2)p^2 + \ldots.

In other words,

a \equiv a_0 \pmod{p},

a \equiv a_0^p + a_1 p \pmod{p^2},

a \equiv a_0^{p^2} + a_1^p p + a_2 p^2 \pmod{p^2},

and ditto for b. Then

a + b \equiv a_0 + b_0\pmod{p}\\ \text{\ \ \ \ \ \ } \equiv t(a_0 + b_0) \pmod{p}.

a + b \equiv a_0^p + a_1 p + b_0^p + b_1 p \pmod{p^2}\\ \text{\ \ \ \ \ \ }\equiv (a_0+b_0)^p + \big(a_1 + b_1 - \sum_{i=1}^{p-1} \frac{1}{p}\binom{p}{i} a_0^i b_0^{p-i} \big)p \pmod{p^2}.

a b \equiv a_0 b_0 \pmod{p}\\ \text{\ \ \ \ \ \ } \equiv t(a_0 b_0) \pmod{p}.

a b \equiv (a_0^p + a_1 p )( b_0^p + b_1 p) \pmod{p^2}\\\text{\ \ \ \ \ \ } \equiv (a_0 b_0)^p + (a_0^p b_1 + a_1 b_0^p)p \pmod{p^2}.

So, if we define polynomials in x = (x_0, x_1, \ldots) and y = (y_0, y_1, \ldots)

S_0(x, y) = x_0 + y_0,

S_1(x, y) = x_1 + y_1 - \sum_{i=1}^{p-1} \frac{1}{p}\binom{p}{i} x_0^i y_0^{p-i},

P_0(x, y) = x_0 y_0,

P_1(x, y) = x_0^p y_1 + x_1 y_0^p,

then

a+b \equiv t(S_0(a,b)) \pmod{p},

a+b \equiv t(S_0(a,b)) + t(S_1(a,b))p \pmod{p^2},

ab \equiv t(P_0(a,b)) \pmod{p},

ab \equiv t(P_0(a,b)) + t(P_1(a, b))p \pmod{p^2}.

One can also show that there exist polynomials S_j(x, y) and P_j(x, y) for all integer j\geq 0 such that

a+b \equiv \sum_{j=0}^{i-1} t(S_j(a,b))p^j \pmod{p^i} and

ab \equiv \sum_{j=0}^{i-1} t(P_j(a,b))p^j \pmod{p^i},

for all i\geq 1, or, more curtly,

a+b = \sum_{j=0}^{\infty} t(S_j(a,b))p^j and

ab = \sum_{j=0}^{\infty} t(P_j(a,b))p^j.

In other words, the addition and multiplication of p-adic integers can be written in polynomial form when those numbers are written in terms of Teichmüller representatives.

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This entry was posted on June 1, 2009 at 4:51 pm and is filed under Witt vectors, p-adic numbers. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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