Download e-book for kindle: 250 problems in elementary number theory by Waclaw Sierpinski

By Waclaw Sierpinski

ISBN-10: 0444000712

ISBN-13: 9780444000712

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Example text

Define the canonical exponent kv = d(v/P ) if v(T ) ≥ 0, v(P ) > 0, d(v/∞) − 2e(v/∞) if v(T ) < 0. 3) In particular, kP = 0 for every finite valuation of q, and k∞ = −2. In general, kv ≥ 0, but above infinity, the canonical exponent may be negative. Moreover, it depends on the choice of the function T in K. 2 we will see that v kv v is a canonical divisor of the curve C. 11 The character x → χv (yx) is trivial on πvn ov if and only if y ∈ πv−kv −n ov . Let μ be a Haar measure for Kv+ . 5]. 3. 12 For a = 0 ∈ Kv and measurable sets M in Kv+ , define μ1 (M ) = μ(aM ).

9), we obtain ζv (f, s)ζv (Fv g, 1 − s) = κ∗+ Kv ˆ Kv∗ ¨ f (a)g(x)χv (abx) dv x dv a |b|1−s d∗v b. v Kv Since f and g occur symmetrically inside the last expression, we obtain the local functional equation, ζv (f, s)ζv (Fv g, 1 − s) = ζv (g, s)ζv (Fv f, 1 − s). It follows that the function ζv (f, s) 1 − qvs−1 = qvkv (s−1/2) ζv (Fv f, 1 − s) 1 − qv−s is independent of f . See [Ta] for its computation in the archimedean case. In Chapter 6, this interplay between the additive group and the multiplicative action will be studied in detail.

Compute the class of P n f modulo P . Thus we find a polynomial fP,n of degree less than deg P such that f − fP,n P −n has the same factors in its denominator as f , to the same power, except that P occurs to a lower power. We continue until f − fP,n P −n has a trivial denominator, and hence is a polynomial f∞ (T ). 3 We compute A/q for q = Fq (T ). Given an adele, for each finite P -adic valuation, we subtract a rational function of the form f /P k to cancel its denominator. Thus we can subtract an element of q so that each finite component becomes a regular function in oP .

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250 problems in elementary number theory by Waclaw Sierpinski

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