Research proof index

The q-expansion of the Weierstrass ℘-function (Silverman, Advanced topics, Theorem I.6.2): for τ in the upper half plane and 0 < im z < im τ (which forces z ∉ Λ_τ), ℘(z; Λ_τ) = (2πi)² (1/12 + Xₐ(e z, e τ)). Proof: group the absolutely convergent sum defining ℘ into rows ω = nτ + m, n : ℤ (Fubini). The condition 0 < im z < im τ guaran...

Evaluation of integral power series commutes with valuative extensions of nonarchimedean local fields: the coefficients are (the same) integers on both sides, and both evaluations are within |q|^N of the common N-th partial sum (valuation_evalInt_sub_sum_le), whose bound transfers along the strictly monotone map of value groups , no continuity argum...

Evaluation of integral power series at a point of the open unit disc is multiplicative. Together with evalInt_add this makes evalInt q a ring homomorphism ℤ⟦X⟧ → k for each |q| < 1. Proof: on the open unit disc, evalInt is the coercion to k of mathlib's topological power-series evaluation over the ring of integers (key below): q is an inte...

The geometric series over a nonarchimedean local field: for |x| < 1, x + x² + x³ + ⋯ = x/(1 - x). (Summability is by the nonarchimedean criterion , the terms tend to zero , and the value is identified through the partial sums x(xⁿ - 1)/(x - 1).)

The convergence criterion for series over a nonarchimedean local field: if each term of f is bounded by |q|^(e i) for an exponent function e with finite sublevel sets, then f is summable , its terms tend to zero cofinitely, which suffices by completeness and the nonarchimedean property (no absolute convergence is needed , contrast the archimedean...

The Lambert series rearrangement over a nonarchimedean local field: for any integer coefficients c and |q| < 1, ∑_{m≥1} c(m)qᵐ/(1 - qᵐ) = ∑_{N≥1} (∑_{d ∣ N} c(d))qᴺ. This is the valuative instantiation of the general tsum_lambert_of_summable (FLT.Slop.NumberTheory.TsumDivisorsAntidiagonal): the geometric row expansions come from `hasSum_geometri...