"""
Theta constants
"""
def _compute_theta_char_poly(char_dict, f):
r"""
Return the coefficient at ``f`` of the Siegel modular form
.. math:
\sum_{l \in\text{char_dict}} \alpha[l] * \prod_i \theta_l[i](8Z).
INPUT:
- ``char_dict`` -- a dictionary whose keys are *tuples* of theta
characteristics and whose values are in some ring.
- ``f`` -- a triple `(a,b,c)` of rational numbers such that the
quadratic form `[a,b,c]` is semi positive definite
(i.e. `a,c \geq 0` and `b^2-4ac \leq 0`).
EXAMPLES::
sage: from sage.modular.siegel.theta_constant import _multiply_theta_char
sage: from sage.modular.siegel.theta_constant import _compute_theta_char_poly
sage: theta_constants = {((1, 1, 0, 0), (0, 0, 1, 1), (1, 1, 0, 0), (0, 0, 1, 1)): 1}
sage: _compute_theta_char_poly(theta_constants, [2, 0, 10])
32
sage: _compute_theta_char_poly(theta_constants, [2, 0, 2])
8
sage: _compute_theta_char_poly(theta_constants, [0, 0, 0])
0
"""
return sum(_multiply_theta_char(list(l), f)*char_dict[l] for l in char_dict)
def _multiply_theta_char(l, f):
r"""
Return the coefficient at ``f`` of the theta series `\prod_t \theta_t`
where `t` runs through the list ``l`` of theta characteristics.
INPUT:
- ``l`` -- a list of quadruples `t` in `{0,1}^4`
- ``f`` -- a triple `(a,b,c)` of rational numbers such that the
quadratic form `[a,b,c]` is semi positive definite
(i.e. `a,c \geq 0` and `b^2-4ac \leq 0`).
EXAMPLES::
sage: from sage.modular.siegel.theta_constant import _multiply_theta_char
sage: from sage.modular.siegel.theta_constant import _compute_theta_char_poly
sage: tc = [(1, 1, 0, 0), (0, 0, 1, 1), (1, 1, 0, 0), (0, 0, 1, 1)]
sage: _multiply_theta_char(tc, [2, 0, 6])
-32
sage: _multiply_theta_char(tc, [2, 0, 10])
32
sage: _multiply_theta_char(tc, [0, 0, 0])
0
"""
if 0 == len(l):
return (1 if (0, 0, 0) == f else 0)
a, b, c = f
m1, m2, m3, m4 = l[0]
if 1 == (m1*m3 + m2*m4)%2:
return 0
coeff = 0
from sage.misc.all import isqrt, xsrange
for u in xsrange(m1, isqrt(a)+1, 2):
for v in xsrange(m2, isqrt(c)+1, 2):
if 0 == u and 0 == v:
coeff += _multiply_theta_char(l[1:], (a, b, c))
continue
ap, bp, cp = (a-u*u, b-2*u*v, c-v*v)
if bp*bp-4*ap*cp <= 0:
val = (2 if 0 == (u*m3 + v*m4)%4 else -2)
coeff += val * _multiply_theta_char(l[1:], (ap, bp, cp))
if u != 0 and v != 0:
ap, bp, cp = (a-u*u, b+2*u*v, c-v*v)
if bp*bp-4*ap*cp <= 0:
val = (2 if 0 == (u*m3 - v*m4)%4 else -2)
coeff += val * _multiply_theta_char(l[1:], (ap, bp, cp))
return coeff
