FLINT3.7.0-dev
/

acb_theta_ql_local_bound

Synopsis#include <flint/acb_theta.h>
void acb_theta_ql_local_bound(arb_t c, arb_t rho, acb_srcptr z, const acb_mat_t tau, slong ord)source

Sets c≥1c \ge 1 and 0<ρ≤10 < \rho \le 1 such that ∣θa,b(⋅,τ)∣≤c|\theta_{a,b}(\cdot,\tau)| \le c for all characteristics on every ball of radius ρ\rho (for the ∞\infty-norm) centered at a point of z, with ρ\rho chosen to make the Cauchy bounds for derivatives up to order ord small.

One can take c=c0exp⁡((c1+c2ρ)2)c = c_0\exp((c_1 + c_2\rho)^2) for any ρ\rho, with

c0=2g∏j=0g−1(1+2γj−1),c1=πyTY−1y,c2=sup⁡∥x∥∞≤1πxTY−1x,c_0 = 2^g \prod_{j=0}^{g-1} (1 + 2\gamma_j^{-1}), \qquad c_1 = \sqrt{\pi y^T Y^{-1} y}, \qquad c_2 = \sup_{\lVert x \rVert_\infty\leq 1} \sqrt{\pi x^T Y^{-1} x},

where an upper bound for c2c_2 comes from the Cholesky decomposition of πY−1\pi Y^{-1}. Then ρ\rho minimises exp⁡((c1+c2ρ)2)/ρ2 ord+1\exp((c_1 + c_2\rho)^2)/\rho^{2\,\mathit{ord}+1}: it is the minimum of 1 and the positive root of 2c2ρ(c1+c2ρ)=2 ord+12c_2\rho (c_1 + c_2\rho) = 2\,\mathit{ord}+1.

Cite this page

The FLINT team, “acb_theta_ql_local_bound (acb_theta.h),” FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev, 2026. https://flintlib.org/doc/acb_theta/acb_theta_ql_local_bound.html (generated 2026-11-02).