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.