acb_modular.h

Initialise g to the identity, and free it.

Swap f and g, set f to a copy of g, and set g to the identity.

Return whether g is the identity, whether f and g are equal, and whether g is a valid element in canonical form ($ad - bc = 1$, $c \ge 0$, and $d > 0$ if $c = 0$).

Print g as [a b; c d] to standard output or to file.

Set h to the product $fg$, respectively the inverse $g^{-1}$, in canonical form. Aliasing is allowed.

Sets g to a random element with entries of at most about bits bits (at least 1): random $a$ and $d$ are divided by their GCD, completed with Bézout coefficients, and the signs are canonicalised.

Sets w to $gz = (az + b)/(cz + d)$.

Set g to a transformation that should map $x + yi$ (with $y > 0$) to the fundamental domain or just outside it, by the classical reduction algorithm in double or arf arithmetic at precision prec. The output is always a valid element of $\mathrm{PSL}_2(\mathbb{Z})$, but it is not guaranteed to reduce the point; the caller must check the result.

Sets g to a transformation that should map the midpoint of z to the fundamental domain or just outside it, and w to $gz$ evaluated in ball arithmetic. The output g is always valid, but the caller must check that w is close enough to the domain.

Returns nonzero if it is certain that $|\operatorname{Re}(z)| \le 1/2 + \varepsilon$ and $|z| \ge 1 - \varepsilon$, where $\varepsilon$ is tol, and zero if this is false or cannot be decided.

Set theta1, ..., theta4 to $\theta_1(z, \tau), \ldots, \theta_4(z, \tau)$. acb_modular_theta moves $\tau$ to the fundamental domain, reduces $z$ modulo the transformed lattice when its imaginary part is large, and calls acb_modular_theta_sum. The notransform version sums the series at $\tau$ directly, which is only efficient when $\operatorname{Im}(\tau)$ is not small.

Set each output vector to the first len Taylor coefficients of $\theta_i(z + x, \tau)$ at $x = 0$, so that entry $k$ is $\theta_i^{(k)}(z, \tau)/k!$, with derivatives with respect to $z$. The notransform version does not move $\tau$ to the fundamental domain or reduce $z$.

Set the outputs to $\theta_i(z(x), \tau)$ truncated to length len, for a power series $z(x)$, by composing the theta jets with the nonconstant part of $z$. Any of the outputs may be NULL, in which case that function is not stored.

Computes the data of the transformation formula expressing theta functions at $\tau$ through theta functions at $\tau' = g\tau$: integers $R_i$, $S_i$ (arrays of length 4) and $C \in \{0, 1\}$ such that for $i = 0, 1, 2, 3$

where $z' = z$ and $A = B = 1$ if $C = 0$, and if $C = 1$

Sets the outputs to the first len $\ge 1$ Taylor coefficients of $\theta_1(z+x,\tau)/q_{1/4}$, $\theta_2(z+x,\tau)/q_{1/4}$, $\theta_3(z+x,\tau)$ and $\theta_4(z+x,\tau)$, given $w = e^{\pi i z}$ and $q = e^{\pi i \tau}$, by summing truncated theta series with a rigorous bound for the tail. A nonzero w_is_unit asserts that $|w| = 1$ (real $z$). Aliasing between inputs and outputs is not allowed.

Constructs an addition sequence for the first num squares and pronic numbers interleaved (excluding zero), $1, 2, 4, 6, 9, 12, 16, 20, 25, 30, \ldots$: entry $k$ of exponents is the $k$-th number, and it equals the sum of the entries at aindex[k] and bindex[k], or twice the first plus the second.

Compute the theta constants $\theta_2(0, \tau)/q_{1/4}$, $\theta_3(0, \tau)$ and $\theta_4(0, \tau)$ from $q = e^{\pi i \tau}$ without a modular transformation. The basecase and rs versions sum the truncated series $\sum q^{k(k+1)}$, $\sum q^{k^2}$ and $\sum (-1)^k q^{k^2}$ over the $k \in \mathbb{Z}$ with exponent less than $N$, with a short addition sequence and with rectangular splitting respectively [EHJ2016], without a tail bound. acb_modular_theta_const_sum chooses $N$ from $|q|$ and prec, uses the basecase for $N < 1800$ and adds the truncation error.

Sets z to the Dedekind eta function $\eta(\tau) = e^{\pi i \tau/12} \prod_{n \ge 1} (1 - e^{2\pi i n\tau})$, by moving $\tau$ to the fundamental domain, summing the pentagonal number series with acb_modular_eta_sum, and applying the transformation formula.

Given $g = (a, b; c, d)$, returns an integer $R$ such that $\varepsilon(a,b,c,d) = e^{\pi i R / 12}$ is the 24th root of unity in the transformation formula of the eta function,

Sets eta to $\prod_{n \ge 1} (1 - q^n) = \sum_{k \in \mathbb{Z}} (-1)^k q^{k(3k-1)/2}$, the eta function without the factor $e^{\pi i \tau / 12}$, given $q = e^{2\pi i \tau}$, by summing the pentagonal number series with a rigorous tail bound. Like acb_modular_theta_sum, it is meant for $|q| \ll 1$. The series is summed with a short addition sequence, or with rectangular splitting from 400 terms [EHJ2016].

Constructs an addition sequence for the first num generalised pentagonal numbers (excluding zero), $1, 2, 5, 7, 12, 15, 22, 26, 35, 40, \ldots$, in the format of acb_modular_addseq_theta.

Sets z to Klein's $j$-invariant, normalised so that $j(i) = 1728$, computed on the fundamental domain as $j(\tau) = 32 (\theta_2^8+\theta_3^8+\theta_4^8)^3 / (\theta_2 \theta_3 \theta_4)^8$ with $\theta_i = \theta_i(0, \tau)$. The output is indeterminate unless $\operatorname{Im}(\tau)$ is certainly positive.

Sets r to the modular lambda function $\lambda(\tau) = \theta_2^4(0,\tau) / \theta_3^4(0,\tau)$, which is invariant under the subgroup $\Gamma(2)$ of transformations $(a, b; c, d)$ with $a, d$ odd and $b, c$ even.

Sets z to the modular discriminant $\Delta(\tau) = \eta(\tau)^{24}$, computed as $(\theta_2 \theta_3 \theta_4 / 2)^8$, which transforms as $\Delta(g\tau) = (c\tau+d)^{12} \Delta(\tau)$. The discriminant is sometimes defined with an extra factor $(2\pi)^{12}$, which is omitted here.

Sets the len entries of r to the Eisenstein series $G_4(\tau), G_6(\tau), G_8(\tau), \ldots$, where

$G_4$ and $G_6$ are computed from theta constants on the fundamental domain, the others by the convolution recurrence of [Apostol1997, Theorem 1.13].

Sets res to the Hilbert class polynomial $H_D$ of the negative discriminant $D$, the minimal polynomial over $\mathbb{Q}$ of $j$ at a CM point of discriminant $D$. If $D \ge 0$ or $D \equiv 2, 3 \pmod 4$, res is set to the zero polynomial. See Hilbert class polynomials.

Call acb_elliptic_k, acb_elliptic_k_jet, acb_elliptic_e, acb_elliptic_p and acb_elliptic_p_jet respectively.

Builds an addition sequence for a set of integers that is assumed to be fairly dense. On input, entry $i$ of tab is $-1$ if $i$ is to be part of the sequence and 0 otherwise; on output, a marked entry $i$ holds a value $j$ such that $j$ and $i - j$ are also in the sequence (doubling is preferred). Entries 0 and 1 are ignored; 1 is always assumed to be in the sequence.

Sets z to $xy$ at precision wprec, first rounding the inputs to wprec bits (using the temporaries tmp1, tmp2) when prec exceeds 1024. Used for the powers of $q$ in the series summation, where terms of small magnitude need less precision.