Research proof index

The leading-term principle: if F = X + O(X²) then |F(q)| = |q| on the punctured open unit disc , ultrametrically the leading term dominates the tail, which has valuation at most |q|² by valuation_evalInt_le_pow.

If the first M coefficients of F vanish, its evaluation at a point of the open unit disc has valuation at most |q|^M: the partial sums satisfy the bound by the nonarchimedean triangle inequality, and it passes to the limit by the ultrametric isosceles principle (if v(σ - T) < v(T) and v(σ) < v(T) then v(T) ≤ max(v(σ), v(σ - T)) < v(T), absurd).

Quantitative tail bound: the evaluation of an integral power series on the open unit disc is within |q|^N of its N-th partial sum.

The point (X(u,q), Y(u,q)) satisfies the Weierstrass equation y² + xy = x³ + a₄x + a₆ of the Tate curve, as an identity in ℚ(u)⟦q⟧.

A pair of measures minimizing $\tau$ exists.

For p.η ≤ 1/8, there exist τ-minimizers X₁, X₂ at zero Rusza distance. For p.η < 1/8, all minimizers are fine, by tau_strictly_decreases'. For p.η = 1/8, we use a limit of minimizers for η < 1/8, which exists by compactness.