acb_modular.h – modular forms of complex variables¶

This module provides methods for numerical evaluation of modular forms, Jacobi theta functions, and elliptic functions.

In the context of this module, tau or \(\tau\) always denotes an element of the complex upper half-plane \(\mathbb{H} = \{z \in \mathbb{C} : \operatorname{Im}(z) > 0\}\). We also often use the variable \(q\), variously defined as \(q = e^{2 \pi i \tau}\) (usually in relation to modular forms) or \(q = e^{\pi i \tau}\) (usually in relation to theta functions) and satisfying \(|q| < 1\). We will clarify the local meaning of \(q\) every time such a quantity appears as a function of \(\tau\).

As usual, the numerical functions in this module compute strict error bounds: if tau is represented by an acb_t whose content overlaps with the real line (or lies in the lower half-plane), and tau is passed to a function defined only on \(\mathbb{H}\), then the output will have an infinite radius. The analogous behavior holds for functions requiring \(|q| < 1\).

The modular group¶

psl2z_struct¶

psl2z_t¶

Represents an element of the modular group \(\text{PSL}(2, \mathbb{Z})\), namely an integer matrix

\[\begin{split}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\end{split}\]

with \(ad-bc = 1\), and with signs canonicalized such that \(c \ge 0\), and \(d > 0\) if \(c = 0\). The struct members a, b, c, d are of type fmpz.

void psl2z_init(psl2z_t g)¶: Initializes g and set it to the identity element.

void psl2z_clear(psl2z_t g)¶: Clears g.

void psl2z_swap(psl2z_t f, psl2z_t g)¶: Swaps f and g efficiently.

void psl2z_set(psl2z_t f, const psl2z_t g)¶: Sets f to a copy of g.

void psl2z_one(psl2z_t g)¶: Sets g to the identity element.

int psl2z_is_one(const psl2z_t g)¶: Returns nonzero iff g is the identity element.

void psl2z_print(const psl2z_t g)¶: Prints g to standard output.

void psl2z_fprint(FILE * file, const psl2z_t g)¶: Prints g to the stream file.

int psl2z_equal(const psl2z_t f, const psl2z_t g)¶: Returns nonzero iff f and g are equal.

void psl2z_mul(psl2z_t h, const psl2z_t f, const psl2z_t g)¶: Sets h to the product of f and g, namely the matrix product with the signs canonicalized.

void psl2z_inv(psl2z_t h, const psl2z_t g)¶: Sets h to the inverse of g.

int psl2z_is_correct(const psl2z_t g)¶: Returns nonzero iff g contains correct data, i.e. satisfying \(ad-bc = 1\), \(c \ge 0\), and \(d > 0\) if \(c = 0\).

void psl2z_randtest(psl2z_t g, flint_rand_t state, slong 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.

Modular transformations¶

void acb_modular_transform(acb_t w, const psl2z_t g, const acb_t z, slong prec)¶: Applies the modular transformation g to the complex number z, evaluating

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

void acb_modular_fundamental_domain_approx_d(psl2z_t g, double x, double y, double one_minus_eps)¶

void acb_modular_fundamental_domain_approx_arf(psl2z_t g, const arf_t x, const arf_t y, const arf_t one_minus_eps, slong 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.

void acb_modular_fundamental_domain_approx(acb_t w, psl2z_t g, const acb_t z, const arf_t one_minus_eps, slong 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.

int acb_modular_is_in_fundamental_domain(const acb_t z, const arf_t tol, slong 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.

Addition sequences¶

void acb_modular_fill_addseq(slong * tab, slong 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).

Jacobi theta functions¶

Unfortunately, there are many inconsistent notational variations for Jacobi theta functions in the literature. Unless otherwise noted, we use the functions

\[\theta_1(z,\tau) = -i \sum_{n=-\infty}^{\infty} (-1)^n \exp(\pi i [(n + 1/2)^2 \tau + (2n + 1) z]) = 2 q_{1/4} \sum_{n=0}^{\infty} (-1)^n q^{n(n+1)} \sin((2n+1) \pi z)\]

\[\theta_2(z,\tau) = \sum_{n=-\infty}^{\infty} \exp(\pi i [(n + 1/2)^2 \tau + (2n + 1) z]) = 2 q_{1/4} \sum_{n=0}^{\infty} q^{n(n+1)} \cos((2n+1) \pi z)\]

\[\theta_3(z,\tau) = \sum_{n=-\infty}^{\infty} \exp(\pi i [n^2 \tau + 2n z]) = 1 + 2 \sum_{n=1}^{\infty} q^{n^2} \cos(2n \pi z)\]

\[\theta_4(z,\tau) = \sum_{n=-\infty}^{\infty} (-1)^n \exp(\pi i [n^2 \tau + 2n z]) = 1 + 2 \sum_{n=1}^{\infty} (-1)^n q^{n^2} \cos(2n \pi z)\]

where \(q = \exp(\pi i \tau)\) and \(q_{1/4} = \exp(\pi i \tau / 4)\). Note that many authors write \(q_{1/4}\) as \(q^{1/4}\), but the principal fourth root \((q)^{1/4} = \exp(\frac{1}{4} \log q)\) differs from \(q_{1/4}\) in general and some formulas are only correct if one reads “\(q^{1/4} = \exp(\pi i \tau / 4)\)”. To avoid confusion, we only write \(q^k\) when \(k\) is an integer.

void acb_modular_theta_transform(int * R, int * S, int * C, const psl2z_t 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

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

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:

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

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).

void acb_modular_addseq_theta(slong * exponents, slong * aindex, slong * bindex, slong 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.

void acb_modular_theta_sum(acb_ptr theta1, acb_ptr theta2, acb_ptr theta3, acb_ptr theta4, const acb_t w, int w_is_unit, const acb_t q, slong len, slong prec)¶

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

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

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:

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

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.

void acb_modular_theta_const_sum_basecase(acb_t theta2, acb_t theta3, acb_t theta4, const acb_t q, slong N, slong prec)¶

void acb_modular_theta_const_sum_rs(acb_t theta2, acb_t theta3, acb_t theta4, const acb_t q, slong N, slong 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 minimal addition sequence. The rs version uses rectangular splitting.

void acb_modular_theta_const_sum(acb_t theta2, acb_t theta3, acb_t theta4, const acb_t q, slong 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.

void acb_modular_theta_notransform(acb_t theta1, acb_t theta2, acb_t theta3, acb_t theta4, const acb_t z, const acb_t tau, slong 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.

void acb_modular_theta(acb_t theta1, acb_t theta2, acb_t theta3, acb_t theta4, const acb_t z, const acb_t tau, slong prec)¶: Evaluates the Jacobi theta functions \(\theta_i(z,\tau)\), \(i = 1, 2, 3, 4\) simultaneously. This function moves \(\tau\) to the fundamental domain before calling acb_modular_theta_sum().

The Dedekind eta function¶

void acb_modular_addseq_eta(slong * exponents, slong * aindex, slong * bindex, slong 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.

void acb_modular_eta_sum(acb_t eta, const acb_t q, slong 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.

int acb_modular_epsilon_arg(const psl2z_t 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,

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

void acb_modular_eta(acb_t r, const acb_t tau, slong 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().

Modular forms¶

void acb_modular_j(acb_t r, const acb_t tau, slong 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 \(\theta_i = \theta_i(0,\tau)\).

void acb_modular_lambda(acb_t r, const acb_t tau, slong 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.

void acb_modular_delta(acb_t r, const acb_t tau, slong 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.

void acb_modular_eisenstein(acb_ptr r, const acb_t tau, slong len, slong 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.

Elliptic functions¶

void acb_modular_elliptic_p(acb_t wp, const acb_t z, const acb_t tau, slong prec)¶

Computes Weierstrass’s elliptic function

\[\wp(z, \tau) = \frac{1}{z^2} + \sum_{n^2+m^2 \ne 0} \left[ \frac{1}{(z+m+n\tau)^2} - \frac{1}{(m+n\tau)^2} \right]\]

which satisfies \(\wp(z, \tau) = \wp(z + 1, \tau) = \wp(z + \tau, \tau)\). To evaluate the function efficiently, we use the formula

\[\wp(z, \tau) = \pi^2 \theta_2^2(0,\tau) \theta_3^2(0,\tau) \frac{\theta_4^2(z,\tau)}{\theta_1^2(z,\tau)} - \frac{\pi^2}{3} \left[ \theta_3^4(0,\tau) + \theta_3^4(0,\tau)\right].\]

void acb_modular_elliptic_p_zpx(acb_ptr wp, const acb_t z, const acb_t tau, slong len, slong prec)¶: Computes the formal power series \(\wp(z + x, \tau) \in \mathbb{C}[[x]]\), truncated to length len. In particular, with len = 2, simultaneously computes \(\wp(z, \tau), \wp'(z, \tau)\) which together generate the field of elliptic functions with periods 1 and \(\tau\).

Elliptic integrals¶

void acb_modular_elliptic_k(acb_t w, const acb_t m, slong prec)¶: Computes the complete elliptic integral of the first kind \(K(m)\), using the arithmetic-geometric mean: \(K(m) = \pi / (2 M(\sqrt{1-m}))\).

void acb_modular_elliptic_k_cpx(acb_ptr w, const acb_t m, slong len, slong prec)¶: Sets the coefficients in the array w to the power series expansion of the complete elliptic integral of the first kind at the point m truncated to length len, i.e. \(K(m+x) \in \mathbb{C}[[x]]\).

void acb_modular_elliptic_e(acb_t w, const acb_t m, slong prec)¶: Computes the complete elliptic integral of the second kind \(E(m)\), which is given by \(E(m) = (1-m)(2m K'(m) + K(m))\) (where the prime denotes a derivative, not a complementary integral).

Class polynomials¶

void acb_modular_hilbert_class_poly(fmpz_poly_t res, slong 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.