Hypergeometric motives

This is largely a port of the corresponding package in Magma. One important conventional difference: the motivic parameter $t$ has been replaced with $1/t$ to match the classical literature on hypergeometric series. (E.g., see [BeukersHeckman])

The computation of Euler factors is currently only supported for primes $p$ of good or tame reduction.

AUTHORS:

• Frédéric Chapoton

• Kiran S. Kedlaya

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([30], [1,2,3,5]))
sage: H.alpha_beta()
([1/30, 7/30, 11/30, 13/30, 17/30, 19/30, 23/30, 29/30],
 [0, 1/5, 1/3, 2/5, 1/2, 3/5, 2/3, 4/5])
sage: H.M_value() == 30**30 / (15**15 * 10**10 * 6**6)
True
sage: H.euler_factor(2, 7)
T^8 + T^5 + T^3 + 1

REFERENCES:

• [BeukersHeckman]

• [Benasque2009]

• [Kat1991]

• [MagmaHGM]

• [Fedorov2015]

• [Roberts2017]

• [Roberts2015]

• [RRV2022]

• [BeCoMe]

• [Watkins]

class sage.modular.hypergeometric_motive.HypergeometricData(cyclotomic=None, alpha_beta=None, gamma_list=None)

Bases: object

Creation of hypergeometric motives.

INPUT:

Three possibilities are offered, each describing a quotient of products of cyclotomic polynomials.

• cyclotomic – a pair of lists of nonnegative integers, each integer $k$ represents a cyclotomic polynomial ${\mathrm{\Phi }}_k$

• alpha_beta – a pair of lists of rationals, each rational represents a root of unity

• gamma_list – a pair of lists of nonnegative integers, each integer $n$ represents a polynomial $x^n-1$

In the last case, it is also allowed to send just one list of signed integers where signs indicate to which part the integer belongs to.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(cyclotomic=([2], [1]))
Hypergeometric data for [1/2] and [0]

sage: Hyp(alpha_beta=([1/2], [0]))
Hypergeometric data for [1/2] and [0]
sage: Hyp(alpha_beta=([1/5,2/5,3/5,4/5], [0,0,0,0]))
Hypergeometric data for [1/5, 2/5, 3/5, 4/5] and [0, 0, 0, 0]

sage: Hyp(gamma_list=([5], [1,1,1,1,1]))
Hypergeometric data for [1/5, 2/5, 3/5, 4/5] and [0, 0, 0, 0]
sage: Hyp(gamma_list=([5,-1,-1,-1,-1,-1]))
Hypergeometric data for [1/5, 2/5, 3/5, 4/5] and [0, 0, 0, 0]

E_polynomial(vars=None)

Return the E-polynomial of self.

This is a bivariate polynomial.

The algorithm is taken from [FRV2019].

INPUT:

• vars – (optional) pair of variables (default: $u,v$)

REFERENCES:

[FRV2019]

Fernando Rodriguez Villegas, Mixed Hodge numbers and factorial ratios, arXiv 1907.02722

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData
sage: H = HypergeometricData(gamma_list=[-30, -1, 6, 10, 15])
sage: H.E_polynomial()
8*u*v + 7*u + 7*v + 8

sage: p, q = polygens(QQ,'p,q')
sage: H.E_polynomial((p, q))
8*p*q + 7*p + 7*q + 8

sage: H = HypergeometricData(gamma_list=(-11, -2, 1, 3, 4, 5))
sage: H.E_polynomial()
5*u^2*v + 5*u*v^2 + u*v + 1

sage: H = HypergeometricData(gamma_list=(-63, -8, -2, 1, 4, 16, 21, 31))
sage: H.E_polynomial()
21*u^3*v^2 + 21*u^2*v^3 + u^3*v + 23*u^2*v^2 + u*v^3 + u^2*v + u*v^2 + 2*u*v + 1

H_value(p, f, t, ring=None)

Return the trace of the Frobenius, computed in terms of Gauss sums using the hypergeometric trace formula.

INPUT:

• p – a prime number

• f – integer such that $q=p^f$

• t – a rational parameter

• ring – (default: UniversalCyclotomicfield)

The ring could be also ComplexField(n) or QQbar.

OUTPUT: integer

Warning

This is apparently working correctly as can be tested using ComplexField(70) as the value ring.

Using instead UniversalCyclotomicfield, this is much slower than the $p$-adic version padic_H_value().

Unlike in padic_H_value(), tame and wild primes are not supported.

EXAMPLES:

With values in the UniversalCyclotomicField (slow):

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2]*4, [0]*4))
sage: [H.H_value(3,i,-1) for i in range(1,3)]
[0, -12]
sage: [H.H_value(5,i,-1) for i in range(1,3)]
[-4, 276]
sage: [H.H_value(7,i,-1) for i in range(1,3)]  # not tested
[0, -476]
sage: [H.H_value(11,i,-1) for i in range(1,3)]  # not tested
[0, -4972]
sage: [H.H_value(13,i,-1) for i in range(1,3)]  # not tested
[-84, -1420]

With values in ComplexField:

sage: [H.H_value(5,i,-1, ComplexField(60)) for i in range(1,3)]
[-4, 276]

Check issue from Issue #28404:

sage: H1 = Hyp(cyclotomic=([1,1,1], [6,2]))
sage: H2 = Hyp(cyclotomic=([6,2], [1,1,1]))
sage: [H1.H_value(5,1,i) for i in range(2,5)]
[1, -4, -4]
sage: [H2.H_value(5,1,QQ(i)) for i in range(2,5)]
[-4, 1, -4]

REFERENCES:

• [BeCoMe] (Theorem 1.3)

• [Benasque2009]

M_value()

Return the $M$ coefficient that appears in the trace formula.

OUTPUT: a rational

See also

canonical_scheme()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/6,1/3,2/3,5/6], [1/8,3/8,5/8,7/8]))
sage: H.M_value()
729/4096
sage: Hyp(alpha_beta=(([1/2,1/2,1/2,1/2], [0,0,0,0]))).M_value()
256
sage: Hyp(cyclotomic=([5], [1,1,1,1])).M_value()
3125

alpha()

Return the first tuple of rational arguments.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).alpha()
[1/2]

alpha_beta()

Return the pair of lists of rational arguments.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).alpha_beta()
([1/2], [0])

beta()

Return the second tuple of rational arguments.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).beta()
[0]

canonical_scheme(t=None)

Return the canonical scheme.

This is a scheme that contains this hypergeometric motive in its cohomology.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([3], [4]))
sage: H.gamma_list()
[-1, 2, 3, -4]
sage: H.canonical_scheme()
Spectrum of Quotient of Multivariate Polynomial Ring
in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring
in t over Rational Field by the ideal
(X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^2*X1^3 + 27/64*Y0*Y1^4)

sage: H = Hyp(gamma_list=[-2, 3, 4, -5])
sage: H.canonical_scheme()
Spectrum of Quotient of Multivariate Polynomial Ring
in X0, X1, Y0, Y1 over Fraction Field of Univariate Polynomial Ring
in t over Rational Field by the ideal
(X0 + X1 - 1, Y0 + Y1 - 1, (-t)*X0^3*X1^4 + 1728/3125*Y0^2*Y1^5)

REFERENCES:

[Kat1991], section 5.4

cyclotomic_data()

Return the pair of tuples of indices of cyclotomic polynomials.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).cyclotomic_data()
([2], [1])

defining_polynomials()

Return the pair of products of cyclotomic polynomials.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/4,3/4], [0,0])).defining_polynomials()
(x^2 + 1, x^2 - 2*x + 1)

degree()

Return the degree.

This is the sum of the Hodge numbers.

See also

hodge_numbers()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).degree()
1
sage: Hyp(gamma_list=([2,2,4], [8])).degree()
4
sage: Hyp(cyclotomic=([5,6], [1,1,2,2,3])).degree()
6
sage: Hyp(cyclotomic=([3,8], [1,1,1,2,6])).degree()
6
sage: Hyp(cyclotomic=([3,3], [2,2,4])).degree()
4

euler_factor(t, p, deg=None, cache_p=False)

Return the Euler factor of the motive $H_t$ at prime $p$.

INPUT:

• t – rational number, not 0

• p – prime number of good reduction

• deg – integer or None

OUTPUT: a polynomial

See [Benasque2009] for explicit examples of Euler factors.

For odd weight, the sign of the functional equation is +1. For even weight, the sign is computed by a recipe found in Section 11.1 of [Watkins].

If deg is specified, then the polynomial is only computed up to degree deg (inclusive).

The prime $p$ may be tame, but not wild. When $v_p(t-1)$ is nonzero and even, the Euler factor includes a linear term described in Section 11.2 of [Watkins].

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2]*4, [0]*4))
sage: H.euler_factor(-1, 5)
15625*T^4 + 500*T^3 - 130*T^2 + 4*T + 1

sage: H = Hyp(gamma_list=[-6,-1,4,3])
sage: H.weight(), H.degree()
(1, 2)
sage: t = 189/125
sage: [H.euler_factor(1/t,p) for p in [11,13,17,19,23,29]]
[11*T^2 + 4*T + 1,
13*T^2 + 1,
17*T^2 + 1,
19*T^2 + 1,
23*T^2 + 8*T + 1,
29*T^2 + 2*T + 1]

sage: H = Hyp(cyclotomic=([6,2], [1,1,1]))
sage: H.weight(), H.degree()
(2, 3)
sage: [H.euler_factor(1/4,p) for p in [5,7,11,13,17,19]]
[125*T^3 + 20*T^2 + 4*T + 1,
 343*T^3 - 42*T^2 - 6*T + 1,
 -1331*T^3 - 22*T^2 + 2*T + 1,
 -2197*T^3 - 156*T^2 + 12*T + 1,
 4913*T^3 + 323*T^2 + 19*T + 1,
 6859*T^3 - 57*T^2 - 3*T + 1]

sage: H = Hyp(alpha_beta=([1/12,5/12,7/12,11/12], [0,1/2,1/2,1/2]))
sage: H.weight(), H.degree()
(2, 4)
sage: t = -5
sage: [H.euler_factor(1/t,p) for p in [11,13,17,19,23,29]]
[-14641*T^4 - 1210*T^3 + 10*T + 1,
 -28561*T^4 - 2704*T^3 + 16*T + 1,
 -83521*T^4 - 4046*T^3 + 14*T + 1,
 130321*T^4 + 14440*T^3 + 969*T^2 + 40*T + 1,
 279841*T^4 - 25392*T^3 + 1242*T^2 - 48*T + 1,
 707281*T^4 - 7569*T^3 + 696*T^2 - 9*T + 1]

This is an example of higher degree:

sage: H = Hyp(cyclotomic=([11], [7, 12]))
sage: H.euler_factor(2, 13)
371293*T^10 - 85683*T^9 + 26364*T^8 + 1352*T^7 - 65*T^6 + 394*T^5 - 5*T^4 + 8*T^3 + 12*T^2 - 3*T + 1
sage: H.euler_factor(2, 13, deg=4)
-5*T^4 + 8*T^3 + 12*T^2 - 3*T + 1
sage: H.euler_factor(2, 19) # long time
2476099*T^10 - 651605*T^9 + 233206*T^8 - 77254*T^7 + 20349*T^6 - 4611*T^5 + 1071*T^4 - 214*T^3 + 34*T^2 - 5*T + 1

This is an example of tame primes:

sage: H = Hyp(cyclotomic=[[4,2,2], [3,1,1]])
sage: H.euler_factor(8, 7)
-7*T^3 + 7*T^2 - T + 1
sage: H.euler_factor(50, 7)
-7*T^3 + 7*T^2 - T + 1
sage: H.euler_factor(7, 7)
-T + 1
sage: H.euler_factor(1/7^2, 7)
T + 1
sage: H.euler_factor(1/7^4, 7)
7*T^3 + 7*T^2 + T + 1

This is an example with $t=1$:

sage: H = Hyp(cyclotomic=[[4,2], [3,1]])
sage: H.euler_factor(1, 7)
-T^2 + 1
sage: H = Hyp(cyclotomic=[[5], [1,1,1,1]])
sage: H.euler_factor(1, 7)
343*T^2 - 6*T + 1

REFERENCES:

• [Roberts2015]

• [Watkins]

euler_factor_tame_contribution(t, p, mo, deg=None)

Return a contribution to the Euler factor of the motive $H_t$ at a tame prime.

The output is only nontrivial when $t$ has nonzero $p$-adic valuation. The algorithm is described in Section 11.4.1 of [Watkins].

INPUT:

• t – rational number, not 0 or 1

• p – prime number of good reduction

• mo – integer

• deg – integer (optional)

OUTPUT: a polynomial

If deg is specified, the output is truncated to that degree (inclusive).

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=[[3,7], [4,5,6]])
sage: H.euler_factor_tame_contribution(11^2, 11, 4)
1
sage: H.euler_factor_tame_contribution(11^20, 11, 4)
1331*T^2 + 1
sage: H.euler_factor_tame_contribution(11^20, 11, 4, deg=1)
1
sage: H.euler_factor_tame_contribution(11^20, 11, 5)
1771561*T^4 + 161051*T^3 + 6171*T^2 + 121*T + 1
sage: H.euler_factor_tame_contribution(11^20, 11, 5, deg=3)
161051*T^3 + 6171*T^2 + 121*T + 1
sage: H.euler_factor_tame_contribution(11^20, 11, 6)
1

gamma_array()

Return the dictionary $\{v:{\gamma }_v\}$ for the expression

$$\prod _v(T^v-1{)}^{{\gamma }_v}$$

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).gamma_array()
{1: -2, 2: 1}
sage: Hyp(cyclotomic=([6,2], [1,1,1])).gamma_array()
{1: -3, 3: -1, 6: 1}

gamma_list()

Return a list of integers describing the $x^n-1$ factors.

Each integer $n$ stands for $(x^{|n|}-1{)}^{\mathrm{sgn}(n)}$.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).gamma_list()
[-1, -1, 2]

sage: Hyp(cyclotomic=([6,2], [1,1,1])).gamma_list()
[-1, -1, -1, -3, 6]

sage: Hyp(cyclotomic=([3], [4])).gamma_list()
[-1, 2, 3, -4]

gauss_table(p, f, prec)

Return (and cache) a table of Gauss sums used in the trace formula.

See also

gauss_table_full()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([3], [4]))
sage: H.gauss_table(2, 2, 4)
(4, [1 + 2 + 2^2 + 2^3, 1 + 2 + 2^2 + 2^3, 1 + 2 + 2^2 + 2^3])

gauss_table_full()

Return a dict of all stored tables of Gauss sums.

The result is passed by reference, and is an attribute of the class; consequently, modifying the result has global side effects. Use with caution.

See also

gauss_table()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([3], [4]))
sage: H.euler_factor(2, 7, cache_p=True)
7*T^2 - 3*T + 1
sage: H.gauss_table_full()[(7, 1)]
(2, array('l', [-1, -29, -25, -48, -47, -22]))

Clearing cached values:

sage: H = Hyp(cyclotomic=([3], [4]))
sage: H.euler_factor(2, 7, cache_p=True)
7*T^2 - 3*T + 1
sage: d = H.gauss_table_full()
sage: d.clear() # Delete all entries of this dict
sage: H1 = Hyp(cyclotomic=([5], [12]))
sage: d1 = H1.gauss_table_full()
sage: len(d1.keys()) # No cached values
0

has_symmetry_at_one()

If True, the motive H(t=1) is a direct sum of two motives.

Note that simultaneous exchange of (t,1/t) and (alpha,beta) always gives the same motive.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=[[1/2]*16, [0]*16]).has_symmetry_at_one()
True

REFERENCES:

• [Roberts2017]

hodge_function(x)

Evaluate the Hodge polygon as a function.

See also

hodge_numbers(), hodge_polynomial(), hodge_polygon_vertices()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([6,10], [3,12]))
sage: H.hodge_function(3)
2
sage: H.hodge_function(4)
4

hodge_numbers()

Return the Hodge numbers.

See also

degree(), hodge_polynomial(), hodge_polygon()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([3], [6]))
sage: H.hodge_numbers()
[1, 1]

sage: H = Hyp(cyclotomic=([4], [1,2]))
sage: H.hodge_numbers()
[2]

sage: H = Hyp(gamma_list=([8,2,2,2], [6,4,3,1]))
sage: H.hodge_numbers()
[1, 2, 2, 1]

sage: H = Hyp(gamma_list=([5], [1,1,1,1,1]))
sage: H.hodge_numbers()
[1, 1, 1, 1]

sage: H = Hyp(gamma_list=[6,1,-4,-3])
sage: H.hodge_numbers()
[1, 1]

sage: H = Hyp(gamma_list=[-3]*4 + [1]*12)
sage: H.hodge_numbers()
[1, 1, 1, 1, 1, 1, 1, 1]

REFERENCES:

• [Fedorov2015]

hodge_polygon_vertices()

Return the vertices of the Hodge polygon.

See also

hodge_numbers(), hodge_polynomial(), hodge_function()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([6,10], [3,12]))
sage: H.hodge_polygon_vertices()
[(0, 0), (1, 0), (3, 2), (5, 6), (6, 9)]
sage: H = Hyp(cyclotomic=([2,2,2,2,3,3,3,6,6], [1,1,4,5,9]))
sage: H.hodge_polygon_vertices()
[(0, 0), (1, 0), (4, 3), (7, 9), (10, 18), (13, 30), (14, 35)]

hodge_polynomial()

Return the Hodge polynomial.

See also

hodge_numbers(), hodge_polygon_vertices(), hodge_function()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([6,10], [3,12]))
sage: H.hodge_polynomial()
(T^3 + 2*T^2 + 2*T + 1)/T^2
sage: H = Hyp(cyclotomic=([2,2,2,2,3,3,3,6,6], [1,1,4,5,9]))
sage: H.hodge_polynomial()
(T^5 + 3*T^4 + 3*T^3 + 3*T^2 + 3*T + 1)/T^2

is_primitive()

Return whether this data is primitive.

See also

primitive_index(), primitive_data()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(cyclotomic=([3], [4])).is_primitive()
True
sage: Hyp(gamma_list=[-2, 4, 6, -8]).is_primitive()
False
sage: Hyp(gamma_list=[-3, 6, 9, -12]).is_primitive()
False

lattice_polytope()

Return the associated lattice polytope.

This uses the matrix defined in section 3 of [RRV2022] and section 3 of [RV2019].

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(gamma_list=[-5, -2, 3, 4])
sage: P = H.lattice_polytope(); P
2-d lattice polytope in 2-d lattice M
sage: P.polyhedron().f_vector()
(1, 4, 4, 1)
sage: len(P.points())
7

The Chebyshev example from [RV2019]:

sage: H = Hyp(gamma_list=[-30, -1, 6, 10, 15])
sage: P = H.lattice_polytope(); P
3-d lattice polytope in 3-d lattice M
sage: len(P.points())
19
sage: P.polyhedron().f_vector()
(1, 5, 9, 6, 1)

lfunction(t, prec=53)

Return the $L$-function of self.

The result is a wrapper around a PARI $L$-function.

INPUT:

• prec – precision (default: 53)

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([3], [4]))
sage: L = H.lfunction(1/64); L
PARI L-function associated to Hypergeometric data for [1/3, 2/3] and [1/4, 3/4]
sage: L(4)
0.997734256321692

padic_H_value(p, f, t, prec=None, cache_p=False)

Return the $p$-adic trace of Frobenius, computed using the Gross-Koblitz formula.

If left unspecified, $prec$ is set to the minimum $p$-adic precision needed to recover the Euler factor.

If cache_p is True, then the function caches an intermediate result which depends only on $p$ and $f$. This leads to a significant speedup when iterating over $t$.

INPUT:

• p – a prime number

• f – integer such that $q=p^f$

• t – a rational parameter

• prec – precision (optional)

• cache_p – boolean

OUTPUT: integer

EXAMPLES:

From Benasque report [Benasque2009], page 8:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2]*4, [0]*4))
sage: [H.padic_H_value(3,i,-1) for i in range(1,3)]
[0, -12]
sage: [H.padic_H_value(5,i,-1) for i in range(1,3)]
[-4, 276]
sage: [H.padic_H_value(7,i,-1) for i in range(1,3)]
[0, -476]
sage: [H.padic_H_value(11,i,-1) for i in range(1,3)]
[0, -4972]

From [Roberts2015] (but note conventions regarding $t$):

sage: H = Hyp(gamma_list=[-6,-1,4,3])
sage: t = 189/125
sage: H.padic_H_value(13,1,1/t)
0

REFERENCES:

• [MagmaHGM]

primitive_data()

Return a primitive version.

See also

is_primitive(), primitive_index()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([3], [4]))
sage: H2 = Hyp(gamma_list=[-2, 4, 6, -8])
sage: H2.primitive_data() == H
True

primitive_index()

Return the primitive index.

See also

is_primitive(), primitive_data()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(cyclotomic=([3], [4])).primitive_index()
1
sage: Hyp(gamma_list=[-2, 4, 6, -8]).primitive_index()
2
sage: Hyp(gamma_list=[-3, 6, 9, -12]).primitive_index()
3

sign(t, p)

Return the sign of the functional equation for the Euler factor of the motive $H_t$ at the prime $p$.

For odd weight, the sign of the functional equation is +1. For even weight, the sign is computed by a recipe found in Section 11.1 of [Watkins] (when 0 is not in alpha).

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(cyclotomic=([6,2], [1,1,1]))
sage: H.weight(), H.degree()
(2, 3)
sage: [H.sign(1/4,p) for p in [5,7,11,13,17,19]]
[1, 1, -1, -1, 1, 1]

sage: H = Hyp(alpha_beta=([1/12,5/12,7/12,11/12], [0,1/2,1/2,1/2]))
sage: H.weight(), H.degree()
(2, 4)
sage: t = -5
sage: [H.sign(1/t,p) for p in [11,13,17,19,23,29]]
[-1, -1, -1, 1, 1, 1]

We check that Issue #28404 is fixed:

sage: H = Hyp(cyclotomic=([1,1,1], [6,2]))
sage: [H.sign(4,p) for p in [5,7,11,13,17,19]]
[1, 1, -1, -1, 1, 1]

swap_alpha_beta()

Return the hypergeometric data with alpha and beta exchanged.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2], [0]))
sage: H.swap_alpha_beta()
Hypergeometric data for [0] and [1/2]

trace(p, f, t, prec=None, cache_p=False)

alias of padic_H_value().

twist()

Return the twist of this data.

This is defined by adding $1/2$ to each rational in $\alpha$ and $\beta$.

This is an involution.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/2], [0]))
sage: H.twist()
Hypergeometric data for [0] and [1/2]
sage: H.twist().twist() == H
True

sage: Hyp(cyclotomic=([6], [1,2])).twist().cyclotomic_data()
([3], [1, 2])

weight()

Return the motivic weight of this motivic data.

EXAMPLES:

With rational inputs:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(alpha_beta=([1/2], [0])).weight()
0
sage: Hyp(alpha_beta=([1/4,3/4], [0,0])).weight()
1
sage: Hyp(alpha_beta=([1/6,1/3,2/3,5/6], [0,0,1/4,3/4])).weight()
1
sage: H = Hyp(alpha_beta=([1/6,1/3,2/3,5/6], [1/8,3/8,5/8,7/8]))
sage: H.weight()
1

With cyclotomic inputs:

sage: Hyp(cyclotomic=([6,2], [1,1,1])).weight()
2
sage: Hyp(cyclotomic=([6], [1,2])).weight()
0
sage: Hyp(cyclotomic=([8], [1,2,3])).weight()
0
sage: Hyp(cyclotomic=([5], [1,1,1,1])).weight()
3
sage: Hyp(cyclotomic=([5,6], [1,1,2,2,3])).weight()
1
sage: Hyp(cyclotomic=([3,8], [1,1,1,2,6])).weight()
2
sage: Hyp(cyclotomic=([3,3], [2,2,4])).weight()
1

With gamma list input:

sage: Hyp(gamma_list=([8,2,2,2], [6,4,3,1])).weight()
3

wild_primes()

Return the wild primes.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: Hyp(cyclotomic=([3], [4])).wild_primes()
[2, 3]
sage: Hyp(cyclotomic=([2,2,2,2,3,3,3,6,6], [1,1,4,5,9])).wild_primes()
[2, 3, 5]

zigzag(x, flip_beta=False)

Count alpha’s at most x minus beta’s at most x.

This function is used to compute the weight and the Hodge numbers. With flip_beta set to True, replace each $b$ in $\beta$ with $1-b$.

See also

weight(), hodge_numbers()

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import HypergeometricData as Hyp
sage: H = Hyp(alpha_beta=([1/6,1/3,2/3,5/6], [1/8,3/8,5/8,7/8]))
sage: [H.zigzag(x) for x in [0, 1/3, 1/2]]
[0, 1, 0]
sage: H = Hyp(cyclotomic=([5], [1,1,1,1]))
sage: [H.zigzag(x) for x in [0,1/6,1/4,1/2,3/4,5/6]]
[-4, -4, -3, -2, -1, 0]

sage.modular.hypergeometric_motive.alpha_to_cyclotomic(alpha)

Convert from a list of rationals arguments to a list of integers.

The input represents arguments of some roots of unity.

The output represent a product of cyclotomic polynomials with exactly the given roots. Note that the multiplicity of $r/s$ in the list must be independent of $r$; otherwise, a ValueError will be raised.

This is the inverse of cyclotomic_to_alpha().

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import alpha_to_cyclotomic
sage: alpha_to_cyclotomic([0])
[1]
sage: alpha_to_cyclotomic([1/2])
[2]
sage: alpha_to_cyclotomic([1/5, 2/5, 3/5, 4/5])
[5]
sage: alpha_to_cyclotomic([0, 1/6, 1/3, 1/2, 2/3, 5/6])
[1, 2, 3, 6]
sage: alpha_to_cyclotomic([1/3, 2/3, 1/2])
[2, 3]

sage.modular.hypergeometric_motive.capital_M(n)

Auxiliary function, used to describe the canonical scheme.

INPUT:

• n – integer

OUTPUT: a rational

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import capital_M
sage: [capital_M(i) for i in range(1, 8)]
[1, 4, 27, 64, 3125, 432, 823543]

sage.modular.hypergeometric_motive.characteristic_polynomial_from_traces(traces, d, q, i, sign, deg=None, use_fe=True)

Given a sequence of traces $t_1,\dots ,t_k$, return the corresponding characteristic polynomial with Weil numbers as roots.

The characteristic polynomial is defined by the generating series

$$P(T)=\exp (-\sum _{k\ge 1}{t}_k\frac{T^k}{k})$$

and should have the property that reciprocals of all roots have absolute value $q^{i/2}$.

INPUT:

• traces – list of integers $t_1,\dots ,t_k$

• d – the degree of the characteristic polynomial

• q – power of a prime number

• i – integer; the weight in the motivic sense

• sign – integer; the sign

• deg – integer or None

• use_fe – boolean (default: True)

OUTPUT: a polynomial

If deg is specified, only the coefficients up to this degree (inclusive) are computed.

If use_fe is False, we ignore the local functional equation.

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import characteristic_polynomial_from_traces
sage: characteristic_polynomial_from_traces([1, 1], 1, 3, 0, -1)
-T + 1
sage: characteristic_polynomial_from_traces([25], 1, 5, 4, -1)
-25*T + 1

sage: characteristic_polynomial_from_traces([3], 2, 5, 1, 1)
5*T^2 - 3*T + 1
sage: characteristic_polynomial_from_traces([1], 2, 7, 1, 1)
7*T^2 - T + 1

sage: characteristic_polynomial_from_traces([20], 3, 29, 2, 1)
24389*T^3 - 580*T^2 - 20*T + 1
sage: characteristic_polynomial_from_traces([12], 3, 13, 2, -1)
-2197*T^3 + 156*T^2 - 12*T + 1

sage: characteristic_polynomial_from_traces([36, 7620], 4, 17, 3, 1)
24137569*T^4 - 176868*T^3 - 3162*T^2 - 36*T + 1
sage: characteristic_polynomial_from_traces([-4, 276], 4, 5, 3, 1)
15625*T^4 + 500*T^3 - 130*T^2 + 4*T + 1
sage: characteristic_polynomial_from_traces([4, -276], 4, 5, 3, 1)
15625*T^4 - 500*T^3 + 146*T^2 - 4*T + 1
sage: characteristic_polynomial_from_traces([22, 484], 4, 31, 2, -1)
-923521*T^4 + 21142*T^3 - 22*T + 1

sage: characteristic_polynomial_from_traces([22], 4, 31, 2, -1, deg=1)
-22*T + 1
sage: characteristic_polynomial_from_traces([22, 484], 4, 31, 2, -1, deg=4)
-923521*T^4 + 21142*T^3 - 22*T + 1

sage.modular.hypergeometric_motive.cyclotomic_to_alpha(cyclo)

Convert a list of indices of cyclotomic polynomials to a list of rational numbers.

The input represents a product of cyclotomic polynomials.

The output is the list of arguments of the roots of the given product of cyclotomic polynomials.

This is the inverse of alpha_to_cyclotomic().

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import cyclotomic_to_alpha
sage: cyclotomic_to_alpha([1])
[0]
sage: cyclotomic_to_alpha([2])
[1/2]
sage: cyclotomic_to_alpha([5])
[1/5, 2/5, 3/5, 4/5]
sage: cyclotomic_to_alpha([1, 2, 3, 6])
[0, 1/6, 1/3, 1/2, 2/3, 5/6]
sage: cyclotomic_to_alpha([2, 3])
[1/3, 1/2, 2/3]

sage.modular.hypergeometric_motive.cyclotomic_to_gamma(cyclo_up, cyclo_down)

Convert a quotient of products of cyclotomic polynomials to a quotient of products of polynomials $x^n-1$.

INPUT:

• cyclo_up – list of indices of cyclotomic polynomials in the numerator

• cyclo_down – list of indices of cyclotomic polynomials in the denominator

OUTPUT:

a dictionary mapping an integer $n$ to the power of $x^n-1$ that appears in the given product

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import cyclotomic_to_gamma
sage: cyclotomic_to_gamma([6], [1])
{2: -1, 3: -1, 6: 1}

sage.modular.hypergeometric_motive.enumerate_hypergeometric_data(d, weight=None)

Return an iterator over parameters of hypergeometric motives (up to swapping).

INPUT:

• d – the degree

• weight – (optional) integer; specifies the motivic weight

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import enumerate_hypergeometric_data as enum
sage: l = [H for H in enum(6, weight=2) if H.hodge_numbers()[0] == 1]
sage: len(l)
112

sage.modular.hypergeometric_motive.gamma_list_to_cyclotomic(galist)

Convert a quotient of products of polynomials $x^n-1$ to a quotient of products of cyclotomic polynomials.

INPUT:

• galist – list of integers, where an integer $n$ represents the power $(x^{|n|}-1{)}^{\mathrm{sgn}(n)}$

OUTPUT:

a pair of list of integers, where $k$ represents the cyclotomic polynomial ${\mathrm{\Phi }}_k$

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import gamma_list_to_cyclotomic
sage: gamma_list_to_cyclotomic([-1, -1, 2])
([2], [1])

sage: gamma_list_to_cyclotomic([-1, -1, -1, -3, 6])
([2, 6], [1, 1, 1])

sage: gamma_list_to_cyclotomic([-1, 2, 3, -4])
([3], [4])

sage: gamma_list_to_cyclotomic([8, 2, 2, 2, -6, -4, -3, -1])
([2, 2, 8], [3, 3, 6])

sage.modular.hypergeometric_motive.possible_hypergeometric_data(d, weight=None)

Return the list of possible parameters of hypergeometric motives (up to swapping).

INPUT:

• d – the degree

• weight – (optional) integer; specifies the motivic weight

EXAMPLES:

sage: from sage.modular.hypergeometric_motive import possible_hypergeometric_data as P
sage: [len(P(i,weight=2)) for i in range(1, 7)]
[0, 0, 10, 30, 93, 234]