Advances in number theory: Proc. 3rd conf. of Canadian - download pdf or read online

By Fernando Q. Gouvea, Noriko Yui

ISBN-10: 0198536682

ISBN-13: 9780198536680

This e-book comprises the complaints of the 3rd convention of the Canadian quantity conception organization. The 38 technical papers provided during this quantity speak about correct and well timed concerns within the fields of analytic quantity conception, arithmetical algebraic geometry, and diophantine approximation. The e-book contains a number of papers honoring Paulo Ribenboim, to whom this convention used to be committed.

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Extra info for Advances in number theory: Proc. 3rd conf. of Canadian Number Theory Association, 1991

Example text

3] by writing out Tn (f ) explicitly and using that 0≤b

Choose a = b = 0, since k ≡ 0 (mod 12). Then g1 = ΔF62 = q − 1032q 2 + 245196q 3 + 10965568q 4 + 60177390q 5 − · · · and g2 = Δ2 = q 2 − 48q 3 + 1080q 4 − 15040q 5 + · · · . We let f2 = g2 and f1 = g1 + 1032g2 = q + 195660q 3 + 12080128q 4 + 44656110q 5 − · · · . 22 2. 23. When k = 36, the Miller basis including f0 is 6218175600q 4 + 15281788354560q 5 + · · · , f0 = 1 + f1 = q+ 57093088q 4 + 37927345230q 5 + · · · , 194184q 4 + 7442432q 5 + · · · , q2 + f2 = q3 − f3 = 2484q 5 + · · · . 24. The SAGE command victor miller basis computes the Miller basis to any desired precision for a given k.

Proof. Suppose γ ∈ SL2 (Z). Since γ induces an automorphism of Z2 , Xn · γ = {δγ : δ ∈ Xn } is also in bijection with the subgroups of Z2 of index n. For each element δγ ∈ Xn · γ, there is σ ∈ SL2 (Z) such that σδγ ∈ Xn (the element σ transforms δγ to Hermite normal form), and the set of elements σδγ is thus equal to Xn . Thus f [σδγ]k = Tn,k (f ) = σδγ∈Xn f [δγ]k = Tn,k (f )[γ]k . δ∈Xn A finite sum of meromorphic function is meromorphic, so Tn,k (f ) is weakly modular. If f is holomorphic on h, then each f [δ]k is holomorphic on h for δ ∈ Xn .

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Advances in number theory: Proc. 3rd conf. of Canadian Number Theory Association, 1991 by Fernando Q. Gouvea, Noriko Yui

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