acb_siegel_kappa, acb_siegel_kappa2

By [Igu1972, p. 176] and [Mum1983, p. 189], for any symplectic $m$, $(z,\tau)\in \mathbb{C}^g\times \mathcal{H}_g$ and characteristic $(a,b)$,

where $\zeta_8 = e^{i\pi/4}$, $(a', b')$ and the integer $e(m, a, b)$ (Igusa's $\phi_m$) are given by acb_theta_char_table, and the 8th root of unity $\kappa(m)$ is determined up to sign by the choice of branch of $\det(\gamma\tau+\delta)^{1/2}$.

The function decomposes mat with sp2gz_decompose. For trigonal and block-diagonal matrices the action on $\theta_{0,0}$ is explicit; for embedded matrices from $\mathrm{SL}_2(\mathbb{Z})$ it uses acb_modular_theta_transform; and for an embedded $J$ from dimension $2 \le r \le g$, $\kappa(m) \zeta_8^{e(m,0,0)} i^{r/2} \det(\tau_0)^{1/2} = 1$, where $\tau_0$ is the upper left $r \times r$ block of $\tau$ and the square root is the branch that equals $i^{r/2}\det(Y_0)^{1/2}$ for purely imaginary $\tau_0 = iY_0$. To find this branch (when sqr is zero), it picks a purely imaginary matrix $A$ such that $P(t) = \det(A + \tfrac{t+1}{2} (\tau_0 - A))$ has degree $r$ and no roots near the segment $[-1,1]$, and follows the branch of $P(t)^{1/2}$ from $t = -1$ to $t = 1$ as in [MN2019].

@misc{flint-doc-acb-theta-acb-siegel-kappa,
  author       = {{The FLINT team}},
  title        = {{acb\_siegel\_kappa, acb\_siegel\_kappa2 (acb\_theta.h)}},
  howpublished = {FLINT: Fast Library for Number Theory, documentation, version 3.7.0-dev},
  year         = {2026},
  url          = {https://flintlib.org/doc/acb_theta/acb_siegel_kappa.html},
  note         = {Generated 2026-11-02}
}