theta.c

/*
    Copyright (C) 2014 Fredrik Johansson

    This file is part of Arb.

    Arb is free software: you can redistribute it and/or modify it under
    the terms of the GNU Lesser General Public License (LGPL) as published
    by the Free Software Foundation; either version 2.1 of the License, or
    (at your option) any later version.  See <http://www.gnu.org/licenses/>.
*/

#include "acb_modular.h"

static void
acb_mul_4th_root(acb_t y, const acb_t x, int r, slong prec)
{
    r &= 7;

    if (r == 0)
    {
        acb_set(y, x);
    }
    else if (r == 4)
    {
        acb_neg(y, x);
    }
    else if (r == 2)
    {
        acb_mul_onei(y, x);
    }
    else if (r == 6)
    {
        acb_mul_onei(y, x);
        acb_neg(y, y);
    }
    else
    {
        fmpq_t t;
        fmpq_init(t);
        fmpq_set_si(t, r, 4);
        arb_sin_cos_pi_fmpq(acb_imagref(y), acb_realref(y), t, prec);
        acb_mul(y, y, x, prec);
        fmpq_clear(t);
    }
}

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)
{
    fmpq_t t;
    psl2z_t g;
    arf_t one_minus_eps;
    acb_t z_prime, tau_prime, q, q4, w, A, B;
    acb_struct thetas[4];
    int w_is_unit, R[4], S[4], C;
    int t1r, t1i, t2r, t2i, t3r, t4r;

    if (!acb_is_finite(z) || !acb_is_finite(tau) ||
        !arb_is_positive(acb_imagref(tau)))
    {
        acb_indeterminate(theta1);
        acb_indeterminate(theta2);
        acb_indeterminate(theta3);
        acb_indeterminate(theta4);
        return;
    }

    /*
    special cases when real(tau) is an integer n:

    z is real:
        theta1    real      if n mod 4 = 0
        theta1    imaginary if n mod 4 = 2
        theta2    real      if n mod 4 = 0
        theta2    imaginary if n mod 4 = 2
        theta3    real      always
        theta4    real      always

    z is imaginary:
        theta1    real      if n mod 4 = 2
        theta1    imaginary if n mod 4 = 0
        theta2    real      if n mod 4 = 0
        theta2    imaginary if n mod 4 = 2
        theta3    real      always
        theta4    real      always
    */
    t1r = t1i = t2r = t2i = t3r = t4r = 0;

    if (arb_is_int(acb_realref(tau)))
    {
        int val;

        if (arb_is_int_2exp_si(acb_realref(tau), 2))
            val = 2;
        else if  (arb_is_int_2exp_si(acb_realref(tau), 1))
            val = 1;
        else
            val = 0;

        if (arb_is_zero(acb_imagref(z)))
        {
            t3r = t4r = 1;
            if (val == 2) t1r = t2r = 1;
            if (val == 1) t1i = t2i = 1;
        }

        if (arb_is_zero(acb_realref(z)))
        {
            t3r = t4r = 1;
            if (val == 2) t1i = t2r = 1;
            if (val == 1) t1r = t2i = 1;
        }
    }

    psl2z_init(g);
    fmpq_init(t);
    arf_init(one_minus_eps);
    acb_init(z_prime);
    acb_init(tau_prime);
    acb_init(q);
    acb_init(q4);
    acb_init(w);
    acb_init(thetas + 0);
    acb_init(thetas + 1);
    acb_init(thetas + 2);
    acb_init(thetas + 3);
    acb_init(A);
    acb_init(B);

    /* move tau to the fundamental domain */
    arf_set_ui_2exp_si(one_minus_eps, 63, -6);
    acb_modular_fundamental_domain_approx(tau_prime, g, tau,
        one_minus_eps, prec);

    /* compute transformation parameters */
    acb_modular_theta_transform(R, S, &C, g);

    if (C == 0)
    {
        acb_set(z_prime, z);
        acb_one(A);
    }
    else
    {
        /* B = 1/(c*tau+d) (temporarily) */
        acb_mul_fmpz(B, tau, &g->c, prec);
        acb_add_fmpz(B, B, &g->d, prec);
        acb_inv(B, B, prec);

        /* -z/(c*tau+d) */
        acb_mul(z_prime, z, B, prec);
        acb_neg(z_prime, z_prime);

        /* A = sqrt(i/(c*tau+d)) */
        acb_mul_onei(A, B);
        acb_sqrt(A, A, prec);

        /* B = exp(-pi i c z^2/(c*tau+d)) */
        /* we first compute the argument here */
        if (acb_is_zero(z))
        {
            acb_zero(B);
        }
        else
        {
            acb_mul(B, z_prime, z, prec);
            acb_mul_fmpz(B, B, &g->c, prec);
        }
    }

    /* reduce z_prime modulo tau_prime if the imaginary part is large */
    if (arf_cmpabs_2exp_si(arb_midref(acb_imagref(z_prime)), 4) > 0)
    {
        arb_t nn;
        arb_init(nn);
        arf_div(arb_midref(nn), arb_midref(acb_imagref(z_prime)),
            arb_midref(acb_imagref(tau_prime)), prec, ARF_RND_DOWN);
        arf_mul_2exp_si(arb_midref(nn), arb_midref(nn), 1);
        arf_add_ui(arb_midref(nn), arb_midref(nn), 1, prec, ARF_RND_DOWN);
        arf_mul_2exp_si(arb_midref(nn), arb_midref(nn), -1);
        arf_floor(arb_midref(nn), arb_midref(nn));

        /* transform z_prime further */
        acb_submul_arb(z_prime, tau_prime, nn, prec);

        /* add -tau n^2 - 2nz to B */
        arb_mul_2exp_si(nn, nn, 1);
        acb_submul_arb(B, z_prime, nn, prec);
        arb_mul_2exp_si(nn, nn, -1);
        arb_sqr(nn, nn, prec);
        acb_submul_arb(B, tau_prime, nn, prec);

        /* theta1, theta4 pick up factors (-1)^n */
        if (!arf_is_int_2exp_si(arb_midref(nn), 1))
        {
            int i;
            for (i = 0; i < 4; i++)
            {
                if (S[i] == 0 || S[i] == 3)
                    R[i] += 4;
            }
        }

        C = 1;

        arb_clear(nn);
    }

    if (C != 0)
        acb_exp_pi_i(B, B, prec);

    /* compute q_{1/4}, q */
    acb_mul_2exp_si(q4, tau_prime, -2);
    acb_exp_pi_i(q4, q4, prec);
    acb_pow_ui(q, q4, 4, prec);

    /* compute w */
    acb_exp_pi_i(w, z_prime, prec);
    w_is_unit = arb_is_zero(acb_imagref(z_prime));

    /* evaluate theta functions of transformed variables */
    acb_modular_theta_sum(thetas + 0, thetas + 1, thetas + 2, thetas + 3,
        w, w_is_unit, q, 1, prec);
    acb_mul(thetas + 0, thetas + 0, q4, prec);
    acb_mul(thetas + 1, thetas + 1, q4, prec);

    /* multiply by roots of unity */
    acb_mul_4th_root(theta1, thetas + S[0], R[0], prec);
    acb_mul_4th_root(theta2, thetas + S[1], R[1], prec);
    acb_mul_4th_root(theta3, thetas + S[2], R[2], prec);
    acb_mul_4th_root(theta4, thetas + S[3], R[3], prec);

    if (C != 0)
    {
        acb_mul(A, A, B, prec);
        acb_mul(theta1, theta1, A, prec);
        acb_mul(theta2, theta2, A, prec);
        acb_mul(theta3, theta3, A, prec);
        acb_mul(theta4, theta4, A, prec);
    }

    if (t1r) arb_zero(acb_imagref(theta1));
    if (t1i) arb_zero(acb_realref(theta1));
    if (t2r) arb_zero(acb_imagref(theta2));
    if (t2i) arb_zero(acb_realref(theta2));
    if (t3r) arb_zero(acb_imagref(theta3));
    if (t4r) arb_zero(acb_imagref(theta4));

    psl2z_clear(g);
    fmpq_clear(t);
    arf_clear(one_minus_eps);
    acb_clear(z_prime);
    acb_clear(tau_prime);
    acb_clear(q);
    acb_clear(q4);
    acb_clear(w);
    acb_clear(thetas + 0);
    acb_clear(thetas + 1);
    acb_clear(thetas + 2);
    acb_clear(thetas + 3);
    acb_clear(A);
    acb_clear(B);
}

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)
{
    acb_t q, q4, w;
    int w_is_unit;

    acb_init(q);
    acb_init(q4);
    acb_init(w);

    /* compute q_{1/4}, q */
    acb_mul_2exp_si(q4, tau, -2);
    acb_exp_pi_i(q4, q4, prec);
    acb_pow_ui(q, q4, 4, prec);

    /* compute w */
    acb_exp_pi_i(w, z, prec);
    w_is_unit = arb_is_zero(acb_imagref(z));

    /* evaluate theta functions */
    acb_modular_theta_sum(theta1, theta2, theta3, theta4,
        w, w_is_unit, q, 1, prec);
    acb_mul(theta1, theta1, q4, prec);
    acb_mul(theta2, theta2, q4, prec);

    acb_clear(q);
    acb_clear(q4);
    acb_clear(w);
}