psl2z_init g

Initializes g and set it to the identity element.

psl2z_clear g

Clears g.

psl2z_swap f g

Swaps f and g efficiently.

psl2z_set f g

Sets f to a copy of g.

psl2z_one g

Sets g to the identity element.

psl2z_is_one g

Returns nonzero iff g is the identity element.

psl2z_print g

Prints g to standard output.

psl2z_fprint file g

Prints g to the stream file.

psl2z_equal f g

Returns nonzero iff f and g are equal.

psl2z_mul h f g

Sets h to the product of f and g, namely the matrix product with the signs canonicalized.

psl2z_inv h g

Sets h to the inverse of g.

psl2z_is_correct g

Returns nonzero iff g contains correct data, i.e. satisfying \(ad-bc = 1\), \(c \ge 0\), and \(d > 0\) if \(c = 0\).

psl2z_randtest g state bits

Sets g to a random element of \(\text{PSL}(2, \mathbb{Z})\) with entries of bit length at most bits (or 1, if bits is not positive). We first generate a and d, compute their Bezout coefficients, divide by the GCD, and then correct the signs.

psl2z_get_word word x

Decomposes x into a word in S and T.

psl2z__word word x

Compose x from a word in S and T.

perm_set_word x s t n word

Calculate homomorphism for word from permutations s and t.

psl2z_word_fprint word

Outputs word to a file as vector.

psl2z_word_print word

Outputs word to stdout as vector.

psl2z_word_get_str word

Returns as string representation of word as vector.

psl2z_word_fprint_pretty word

Outputs word to a file in tuples of generators with the corresponding power.

psl2z_word_print_pretty word

Outputs word to stdout in tuples of generators with the corresponding power.

psl2z_word_get_str_pretty word

Returns a string representation of word in tuples of generators with the corresponding power.

psl2z_get_perm p s t n x

Returns the permutation \(p\) corresponding to \(x\) by the homomorphism \(\phi:{\textrm PSL}_2\left({\mathbb Z}\right)\rightarrow S_n\) defined by the permutations \(s\) and \(t\):

\[ \begin{align} \begin{pmatrix} 0 &-1 \\ 1 & 0 \end{pmatrix} &\mapsto s \\ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} &\mapsto t. \\ \end{align} \]

acb_modular_transform w g z prec

Applies the modular transformation g to the complex number z, evaluating

\[`\] \[w = g z = \frac{az+b}{cz+d}.\]

acb_modular_fundamental_domain_approx_arf g x y one_minus_eps prec

Attempts to determine a modular transformation g that maps the complex number \(x+yi\) to the fundamental domain or just slightly outside the fundamental domain, where the target tolerance (not a strict bound) is specified by one_minus_eps.

The inputs are assumed to be finite numbers, with y positive.

Uses floating-point iteration, repeatedly applying either the transformation \(z \gets z + b\) or \(z \gets -1/z\). The iteration is terminated if \(|x| \le 1/2\) and \(x^2 + y^2 \ge 1 - \varepsilon\) where \(1 - \varepsilon\) is passed as one_minus_eps. It is also terminated if too many steps have been taken without convergence, or if the numbers end up too large or too small for the working precision.

The algorithm can fail to produce a satisfactory transformation. The output g is always set to some correct modular transformation, but it is up to the user to verify a posteriori that g maps \(x+yi\) close enough to the fundamental domain.

acb_modular_fundamental_domain_approx w g z one_minus_eps prec

Attempts to determine a modular transformation g that maps the complex number \(z\) to the fundamental domain or just slightly outside the fundamental domain, where the target tolerance (not a strict bound) is specified by one_minus_eps. It also computes the transformed value \(w = gz\).

This function first tries to use acb_modular_fundamental_domain_approx_d and checks if the result is acceptable. If this fails, it calls acb_modular_fundamental_domain_approx_arf with higher precision. Finally, \(w = gz\) is evaluated by a single application of g.

The algorithm can fail to produce a satisfactory transformation. The output g is always set to some correct modular transformation, but it is up to the user to verify a posteriori that \(w\) is close enough to the fundamental domain.

acb_modular_is_in_fundamental_domain z tol prec

Returns nonzero if it is certainly true that \(|z| \ge 1 - \varepsilon\) and \(|\operatorname{Re}(z)| \le 1/2 + \varepsilon\) where \(\varepsilon\) is specified by tol. Returns zero if this is false or cannot be determined.

acb_modular_fill_addseq tab len

Builds a near-optimal addition sequence for a sequence of integers which is assumed to be reasonably dense.

As input, the caller should set each entry in tab to \(-1\) if that index is to be part of the addition sequence, and to 0 otherwise. On output, entry i in tab will either be zero (if the number is not part of the sequence), or a value j such that both j and \(i - j\) are also marked. The first two entries in tab are ignored (the number 1 is always assumed to be part of the sequence).

acb_modular_theta_transform R S C g

We wish to write a theta function with quasiperiod \(\tau\) in terms of a theta function with quasiperiod \(\tau' = g \tau\), given some \(g = (a, b; c, d) \in \text{PSL}(2, \mathbb{Z})\). For \(i = 0, 1, 2, 3\), this function computes integers \(R_i\) and \(S_i\) (R and S should be arrays of length 4) and \(C \in \{0, 1\}\) such that

\[`\] \[\theta_{1+i}(z,\tau) = \exp(\pi i R_i / 4) \cdot A \cdot B \cdot \theta_{1+S_i}(z',\tau')\]

where \(z' = z, A = B = 1\) if \(C = 0\), and

\[`\] \[z' = \frac{-z}{c \tau + d}, \quad A = \sqrt{\frac{i}{c \tau + d}}, \quad B = \exp\left(-\pi i c \frac{z^2}{c \tau + d}\right)\]

if \(C = 1\). Note that \(A\) is well-defined with the principal branch of the square root since \(A^2 = i/(c \tau + d)\) lies in the right half-plane.

Firstly, if \(c = 0\), we have \(\theta_i(z, \tau) = \exp(-\pi i b / 4) \theta_i(z, \tau+b)\) for \(i = 1, 2\), whereas \(\theta_3\) and \(\theta_4\) remain unchanged when \(b\) is even and swap places with each other when \(b\) is odd. In this case we set \(C = 0\).

For an arbitrary \(g\) with \(c > 0\), we set \(C = 1\). The general transformations are given by Rademacher [Rad1973]. We need the function \(\theta_{m,n}(z,\tau)\) defined for \(m, n \in \mathbb{Z}\) by (beware of the typos in [Rad1973])

\[`\] \[\theta_{0,0}(z,\tau) = \theta_3(z,\tau), \quad \theta_{0,1}(z,\tau) = \theta_4(z,\tau)\]

\[`\] \[\theta_{1,0}(z,\tau) = \theta_2(z,\tau), \quad \theta_{1,1}(z,\tau) = i \theta_1(z,\tau)\]

\[`\] \[\theta_{m+2,n}(z,\tau) = (-1)^n \theta_{m,n}(z,\tau)\]

\[`\] \[\theta_{m,n+2}(z,\tau) = \theta_{m,n}(z,\tau).\]

Then we may write

\[ \begin{eqnarray*} \theta_1(z,\tau) &=& \varepsilon_1 A B \theta_1(z', \tau')\\ \theta_2(z,\tau) &=& \varepsilon_2 A B \theta_{1-c,1+a}(z', \tau')\\ \theta_3(z,\tau) &=& \varepsilon_3 A B \theta_{1+d-c,1-b+a}(z', \tau')\\ \theta_4(z,\tau) &=& \varepsilon_4 A B \theta_{1+d,1-b}(z', \tau') \end{eqnarray*} \]

where \(\varepsilon_i\) is an 8th root of unity. Specifically, if we denote the 24th root of unity in the transformation formula of the Dedekind eta function by \(\varepsilon(a,b,c,d) = \exp(\pi i R(a,b,c,d) / 12)\) (see acb_modular_epsilon_arg), then:

\[ \begin{eqnarray*} \varepsilon_1(a,b,c,d) &=& \exp(\pi i [R(-d,b,c,-a) + 1]/4)\\ \varepsilon_2(a,b,c,d) &=& \exp(\pi i [-R(a,b,c,d) + (5+(2-c)a)]/4)\\ \varepsilon_3(a,b,c,d) &=& \exp(\pi i [-R(a,b,c,d) + (4+(c-d-2)(b-a))]/4)\\ \varepsilon_4(a,b,c,d) &=& \exp(\pi i [-R(a,b,c,d) + (3-(2+d)b)]/4)\\ \end{eqnarray*} \]

These formulas are easily derived from the formulas in [Rad1973] (Rademacher has the transformed/untransformed variables exchanged, and his "(varepsilon)" differs from ours by a constant offset in the phase).

acb_modular_addseq_theta exponents aindex bindex num

Constructs an addition sequence for the first num squares and triangular numbers interleaved (excluding zero), i.e. 1, 2, 4, 6, 9, 12, 16, 20, 25, 30 etc.

acb_modular_theta_sum theta1 theta2 theta3 theta4 w w_is_unit q len prec

Simultaneously computes the first len coefficients of each of the formal power series

\[`\] \[\theta_1(z+x,\tau) / q_{1/4} \in \mathbb{C}[[x]]\] \[\theta_2(z+x,\tau) / q_{1/4} \in \mathbb{C}[[x]]\] \[\theta_3(z+x,\tau) \in \mathbb{C}[[x]]\] \[\theta_4(z+x,\tau) \in \mathbb{C}[[x]]\]

given \(w = \exp(\pi i z)\) and \(q = \exp(\pi i \tau)\), by summing a finite truncation of the respective theta function series. In particular, with len equal to 1, computes the respective value of the theta function at the point z. We require len to be positive. If w_is_unit is nonzero, w is assumed to lie on the unit circle, i.e. z is assumed to be real.

Note that the factor \(q_{1/4}\) is removed from \(\theta_1\) and \(\theta_2\). To get the true theta function values, the user has to multiply this factor back. This convention avoids unnecessary computations, since the user can compute \(q_{1/4} = \exp(\pi i \tau / 4)\) followed by \(q = (q_{1/4})^4\), and in many cases when computing products or quotients of theta functions, the factor \(q_{1/4}\) can be eliminated entirely.

This function is intended for \(|q| \ll 1\). It can be called with any \(q\), but will return useless intervals if convergence is not rapid. For general evaluation of theta functions, the user should only call this function after applying a suitable modular transformation.

We consider the sums together, alternatingly updating \((\theta_1, \theta_2)\) or \((\theta_3, \theta_4)\). For \(k = 0, 1, 2, \ldots\), the powers of \(q\) are \(\lfloor (k+2)^2 / 4 \rfloor = 1, 2, 4, 6, 9\) etc. and the powers of \(w\) are \(\pm (k+2) = \pm 2, \pm 3, \pm 4, \ldots\) etc. The scheme is illustrated by the following table:

\[`\] \[\begin{aligned} \begin{array}{llll} & \theta_1, \theta_2 & q^0 & (w^1 \pm w^{-1}) \\ k = 0 & \theta_3, \theta_4 & q^1 & (w^2 \pm w^{-2}) \\ k = 1 & \theta_1, \theta_2 & q^2 & (w^3 \pm w^{-3}) \\ k = 2 & \theta_3, \theta_4 & q^4 & (w^4 \pm w^{-4}) \\ k = 3 & \theta_1, \theta_2 & q^6 & (w^5 \pm w^{-5}) \\ k = 4 & \theta_3, \theta_4 & q^9 & (w^6 \pm w^{-6}) \\ k = 5 & \theta_1, \theta_2 & q^{12} & (w^7 \pm w^{-7}) \\ \end{array} \end{aligned}\]

For some integer \(N \ge 1\), the summation is stopped just before term \(k = N\). Let \(Q = |q|\), \(W = \max(|w|,|w^{-1}|)\), \(E = \lfloor (N+2)^2 / 4 \rfloor\) and \(F = \lfloor (N+1)/2 \rfloor + 1\). The error of the zeroth derivative can be bounded as

\[`\] \[2 Q^E W^{N+2} \left[ 1 + Q^F W + Q^{2F} W^2 + \ldots \right] = \frac{2 Q^E W^{N+2}}{1 - Q^F W}\]

provided that the denominator is positive (otherwise we set the error bound to infinity). When len is greater than 1, consider the derivative of order r. The term of index k and order r picks up a factor of magnitude \((k+2)^r\) from differentiation of \(w^{k+2}\) (it also picks up a factor \(\pi^r\), but we omit this until we rescale the coefficients at the end of the computation). Thus we have the error bound

\[`\] \[2 Q^E W^{N+2} (N+2)^r \left[ 1 + Q^F W \frac{(N+3)^r}{(N+2)^r} + Q^{2F} W^2 \frac{(N+4)^r}{(N+2)^r} + \ldots \right]\]

which by the inequality \((1 + m/(N+2))^r \le \exp(mr/(N+2))\) can be bounded as

\[`\] \[\frac{2 Q^E W^{N+2} (N+2)^r}{1 - Q^F W \exp(r/(N+2))},\]

again valid when the denominator is positive.

To actually evaluate the series, we write the even cosine terms as \(w^{2n} + w^{-2n}\), the odd cosine terms as \(w (w^{2n} + w^{-2n-2})\), and the sine terms as \(w (w^{2n} - w^{-2n-2})\). This way we only need even powers of \(w\) and \(w^{-1}\). The implementation is not yet optimized for real \(z\), in which case further work can be saved.

This function does not permit aliasing between input and output arguments.

acb_modular_theta_const_sum_rs theta2 theta3 theta4 q N prec

Computes the truncated theta constant sums \(\theta_2 = \sum_{k(k+1) < N} q^{k(k+1)}\), \(\theta_3 = \sum_{k^2 < N} q^{k^2}\), \(\theta_4 = \sum_{k^2 < N} (-1)^k q^{k^2}\). The basecase version uses a short addition sequence. The rs version uses rectangular splitting. The algorithms are described in [EHJ2016].

acb_modular_theta_const_sum theta2 theta3 theta4 q prec

Computes the respective theta constants by direct summation (without applying modular transformations). This function selects an appropriate N, calls either acb_modular_theta_const_sum_basecase or acb_modular_theta_const_sum_rs or depending on N, and adds a bound for the truncation error.

acb_modular_theta_notransform theta1 theta2 theta3 theta4 z tau prec

Evaluates the Jacobi theta functions \(\theta_i(z,\tau)\), \(i = 1, 2, 3, 4\) simultaneously. This function does not move \(\tau\) to the fundamental domain. This is generally worse than acb_modular_theta, but can be slightly better for moderate input.

acb_modular_theta theta1 theta2 theta3 theta4 z tau prec

Evaluates the Jacobi theta functions \(\theta_i(z,\tau)\), \(i = 1, 2, 3, 4\) simultaneously. This function moves \(\tau\) to the fundamental domain and then also reduces \(z\) modulo \(\tau\) before calling acb_modular_theta_sum.

acb_modular_theta_jet theta1 theta2 theta3 theta4 z tau len prec

Evaluates the Jacobi theta functions along with their derivatives with respect to z, writing the first len coefficients in the power series \(\theta_i(z+x,\tau) \in \mathbb{C}[[x]]\) to each respective output variable. The notransform version does not move \(\tau\) to the fundamental domain or reduce \(z\) during the computation.

acb_modular_theta_series theta1 theta2 theta3 theta4 z tau len prec

Evaluates the respective Jacobi theta functions of the power series z, truncated to length len. Either of the output variables can be NULL.

acb_modular_addseq_eta exponents aindex bindex num

Constructs an addition sequence for the first num generalized pentagonal numbers (excluding zero), i.e. 1, 2, 5, 7, 12, 15, 22, 26, 35, 40 etc.

acb_modular_eta_sum eta q prec

Evaluates the Dedekind eta function without the leading 24th root, i.e.

\[` \exp(-\pi i \tau/12) \eta(\tau) = \sum_{n=-\infty}^{\infty} (-1)^n q^{(3n^2-n)/2}\]

given \(q = \exp(2 \pi i \tau)\), by summing the defining series.

This function is intended for \(|q| \ll 1\). It can be called with any \(q\), but will return useless intervals if convergence is not rapid. For general evaluation of the eta function, the user should only call this function after applying a suitable modular transformation.

The series is evaluated using either a short addition sequence or rectangular splitting, depending on the number of terms. The algorithms are described in [EHJ2016].

acb_modular_epsilon_arg g

Given \(g = (a, b; c, d)\), computes an integer \(R\) such that \(\varepsilon(a,b,c,d) = \exp(\pi i R / 12)\) is the 24th root of unity in the transformation formula for the Dedekind eta function,

\[`\] \[\eta\left(\frac{a\tau+b}{c\tau+d}\right) = \varepsilon (a,b,c,d) \sqrt{c\tau+d} \eta(\tau).\]

acb_modular_eta r tau prec

Computes the Dedekind eta function \(\eta(\tau)\) given \(\tau\) in the upper half-plane. This function applies the functional equation to move \(\tau\) to the fundamental domain before calling acb_modular_eta_sum.

acb_modular_j r tau prec

Computes Klein's j-invariant \(j(\tau)\) given \(\tau\) in the upper half-plane. The function is normalized so that \(j(i) = 1728\). We first move \(\tau\) to the fundamental domain, which does not change the value of the function. Then we use the formula \(j(\tau) = 32 (\theta_2^8+\theta_3^8+\theta_4^8)^3 / (\theta_2 \theta_3 \theta_4)^8\) where \(\theta_i = \theta_i(0,\tau)\).

acb_modular_lambda r tau prec

Computes the lambda function \(\lambda(\tau) = \theta_2^4(0,\tau) / \theta_3^4(0,\tau)\), which is invariant under modular transformations \((a, b; c, d)\) where \(a, d\) are odd and \(b, c\) are even.

acb_modular_delta r tau prec

Computes the modular discriminant \(\Delta(\tau) = \eta(\tau)^{24}\), which transforms as

\[`\] \[\Delta\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{12} \Delta(\tau).\]

The modular discriminant is sometimes defined with an extra factor \((2\pi)^{12}\), which we omit in this implementation.

acb_modular_eisenstein r tau len prec

Computes simultaneously the first len entries in the sequence of Eisenstein series \(G_4(\tau), G_6(\tau), G_8(\tau), \ldots\), defined by

\[`\] \[G_{2k}(\tau) = \sum_{m^2 + n^2 \ne 0} \frac{1}{(m+n\tau )^{2k}}\]

and satisfying

\[`\] \[G_{2k} \left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{2k} G_{2k}(\tau).\]

We first evaluate \(G_4(\tau)\) and \(G_6(\tau)\) on the fundamental domain using theta functions, and then compute the Eisenstein series of higher index using a recurrence relation.

acb_modular_hilbert_class_poly res D

Sets res to the Hilbert class polynomial of discriminant D, defined as

\[`\] \[H_D(x) = \prod_{(a,b,c)} \left(x - j\left(\frac{-b+\sqrt{D}}{2a}\right)\right)\]

where \((a,b,c)\) ranges over the primitive reduced positive definite binary quadratic forms of discriminant \(b^2 - 4ac = D\).

The Hilbert class polynomial is only defined if \(D < 0\) and D is congruent to 0 or 1 mod 4. If some other value of D is passed as input, res is set to the zero polynomial.