一些更高级的数学¶
代数几何¶
您可以在Sage中定义任意的代数变体，但有时重要的功能仅限于环 \(\QQ\) 或有限域。例如，我们计算两条仿射平面曲线的并，然后将其恢复为并的不可约分量。
sage: x, y = AffineSpace(2, QQ, 'xy').gens()
sage: C2 = Curve(x^2 + y^2 - 1)
sage: C3 = Curve(x^3 + y^3 - 1)
sage: D = C2 + C3
sage: D
Affine Plane Curve over Rational Field defined by
x^5 + x^3*y^2 + x^2*y^3 + y^5 - x^3 - y^3 - x^2 - y^2 + 1
sage: D.irreducible_components()
[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x^2 + y^2 - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x^3 + y^3 - 1
]
我们还可以通过求两条曲线的交点和计算其不可约分量来求出它们的所有交点。
sage: V = C2.intersection(C3)
sage: V.irreducible_components()
[
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y,
x - 1,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
y - 1,
x,
Closed subscheme of Affine Space of dimension 2 over Rational Field defined by:
x + y + 2,
2*y^2 + 4*y + 3
]
因此，例如， \((1,0)\) 和 \((0,1)\) 都在两条曲线上(明显清晰)，就像某些(二次)点一样 \(y\) 坐标满足 \(2y^2 + 4y + 3=0\) 。
SAGE可以计算射影3空间中扭三次曲面的环理想：
sage: R.<a,b,c,d> = PolynomialRing(QQ, 4)
sage: I = ideal(b^2-a*c, c^2-b*d, a*d-b*c)
sage: F = I.groebner_fan(); F
Groebner fan of the ideal:
Ideal (b^2 - a*c, c^2 - b*d, -b*c + a*d) of Multivariate Polynomial Ring
in a, b, c, d over Rational Field
sage: F.reduced_groebner_bases ()
[[-c^2 + b*d, -b*c + a*d, -b^2 + a*c],
[-b*c + a*d, -c^2 + b*d, b^2 - a*c],
[-c^3 + a*d^2, -c^2 + b*d, b*c - a*d, b^2 - a*c],
[-c^2 + b*d, b^2 - a*c, b*c - a*d, c^3 - a*d^2],
[-b*c + a*d, -b^2 + a*c, c^2 - b*d],
[-b^3 + a^2*d, -b^2 + a*c, c^2 - b*d, b*c - a*d],
[-b^2 + a*c, c^2 - b*d, b*c - a*d, b^3 - a^2*d],
[c^2 - b*d, b*c - a*d, b^2 - a*c]]
sage: F.polyhedralfan()
Polyhedral fan in 4 dimensions of dimension 4
椭圆曲线¶
椭圆曲线的功能包括Pari的大多数椭圆曲线功能、访问Cremona的在线表格中的数据(这需要一个可选的数据库包)、mwrank的功能，即计算完整的Mordell-Weil群的2-Dendents、SEA算法、所有同源的计算、许多关于曲线的新代码 \(\QQ\) ，以及丹尼斯·西蒙的一些代数下降软件。
该命令 EllipticCurve 创建椭圆曲线有多种形式：
椭圆曲线( [\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_6\)] )：返回椭圆曲线
\[y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,\]凡. \(a_i\) 被强迫成为父母的人 \(a_1\) 。如果所有的 \(a_i\) 拥有父级 \(\ZZ\) ，他们被强迫进入 \(\QQ\) 。
椭圆曲线( [\(a_4\), \(a_6\)] )：同上，但 \(a_1=a_2=a_3=0\) 。
EllipticCurve(Label)：返回Cremona数据库中给定的椭圆曲线(new！)克雷莫纳标签。标签是一个字符串，例如
"11a"或"37b2"。字母必须是小写的(以区别于旧的标签)。EllipticCurve(J)：返回带有 \(j\) -不变量 \(j\) 。
椭圆曲线(R， [\(a_1\), \(a_2\), \(a_3\), \(a_4\), \(a_6\)] )：在环上创建椭圆曲线 \(R\) 使用给定的 \(a_i\) 如上所述。
我们将分别对这些构造函数进行说明：
sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve([1,2])
Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field
sage: EllipticCurve('37a')
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: EllipticCurve_from_j(1)
Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field
sage: EllipticCurve(GF(5), [0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
The pair \((0,0)\) is a point on the elliptic curve
\(E\) defined by \(y^2 +
y = x^3 - x\). To create this
point in Sage type E([0,0]). Sage can add points on such an
elliptic curve (recall elliptic curves support an additive group
structure where the point at infinity is the zero element and three
co-linear points on the curve add to zero):
sage: E = EllipticCurve([0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: P = E([0,0])
sage: P + P
(1 : 0 : 1)
sage: 10*P
(161/16 : -2065/64 : 1)
sage: 20*P
(683916417/264517696 : -18784454671297/4302115807744 : 1)
sage: E.conductor()
37
复数上的椭圆曲线由 \(j\) -不变的。Sage计算 \(j\) -不变，如下：
sage: E = EllipticCurve([0,0,0,-4,2]); E
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
sage: E.conductor()
2368
sage: E.j_invariant()
110592/37
如果我们用同样的东西做一条曲线 \(j\) -不变的为 \(E\) ，它不需要同构于 \(E\) 。在下面的示例中，曲线不是同构的，因为它们的导线不同。
sage: F = EllipticCurve_from_j(110592/37)
sage: F.conductor()
37
然而，扭曲的是 \(F\) 乘以2表示同构曲线。
sage: G = F.quadratic_twist(2); G
Elliptic Curve defined by y^2 = x^3 - 4*x + 2 over Rational Field
sage: G.conductor()
2368
sage: G.j_invariant()
110592/37
我们可以计算系数 \(a_n\) 的 \(L\) -系列或模块化形式 \(\sum_{n=0}^\infty a_nq^n\) 附着在椭圆曲线上。此计算使用Pari C库：
sage: E = EllipticCurve([0,0,1,-1,0])
sage: E.anlist(30)
[0, 1, -2, -3, 2, -2, 6, -1, 0, 6, 4, -5, -6, -2, 2, 6, -4, 0, -12, 0, -4,
3, 10, 2, 0, -1, 4, -9, -2, 6, -12]
sage: v = E.anlist(10000)
只需要一秒钟就可以计算出所有的 \(a_n\) 为 \(n\leq 10^5\) ：
sage: %time v = E.anlist(100000)
CPU times: user 0.98 s, sys: 0.06 s, total: 1.04 s
Wall time: 1.06
椭圆曲线可以用它们的Cremona标号来构造。这为椭圆曲线预加载了有关其阶数、玉马川数、调节器等信息。
sage: E = EllipticCurve("37b2")
sage: E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational
Field
sage: E = EllipticCurve("389a")
sage: E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field
sage: E.rank()
2
sage: E = EllipticCurve("5077a")
sage: E.rank()
3
我们还可以直接访问Cremona数据库。
sage: db = sage.databases.cremona.CremonaDatabase()
sage: db.curves(37)
{'a1': [[0, 0, 1, -1, 0], 1, 1], 'b1': [[0, 1, 1, -23, -50], 0, 3]}
sage: db.allcurves(37)
{'a1': [[0, 0, 1, -1, 0], 1, 1],
'b1': [[0, 1, 1, -23, -50], 0, 3],
'b2': [[0, 1, 1, -1873, -31833], 0, 1],
'b3': [[0, 1, 1, -3, 1], 0, 3]}
从数据库返回的对象不是 EllipticCurve 。它们是数据库的元素，有几个字段，仅此而已。Cremona的数据库有一个小版本，默认情况下随Sage一起分发，其中包含有关导体椭圆曲线的有限信息 \(\leq 10000\) 。还有一个较大的可选版本，它包含有关所有导线曲线的广泛数据，最高可达 \(120000\) (截至2005年10月)。Sage还有一个巨大的(2 GB)可选数据库包，其中包含Stein-Watkins数据库中的数亿条椭圆曲线。
狄利克雷特字符¶
A Dirichlet character is the extension of a homomorphism \((\ZZ/N\ZZ)^* \to R^*\), for some ring \(R\), to the map \(\ZZ \to R\) obtained by sending those integers \(x\) with \(\gcd(N,x)>1\) to 0.
sage: G = DirichletGroup(12)
sage: G.list()
[Dirichlet character modulo 12 of conductor 1 mapping 7 |--> 1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1,
Dirichlet character modulo 12 of conductor 12 mapping 7 |--> -1, 5 |--> -1]
sage: G.gens()
(Dirichlet character modulo 12 of conductor 4 mapping 7 |--> -1, 5 |--> 1,
Dirichlet character modulo 12 of conductor 3 mapping 7 |--> 1, 5 |--> -1)
sage: len(G)
4
创建组之后，我们接下来创建一个元素并使用它进行计算。
sage: G = DirichletGroup(21)
sage: chi = G.1; chi
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6
sage: chi.values()
[0, 1, zeta6 - 1, 0, -zeta6, -zeta6 + 1, 0, 0, 1, 0, zeta6, -zeta6, 0, -1,
0, 0, zeta6 - 1, zeta6, 0, -zeta6 + 1, -1]
sage: chi.conductor()
7
sage: chi.modulus()
21
sage: chi.order()
6
sage: chi(19)
-zeta6 + 1
sage: chi(40)
-zeta6 + 1
It is also possible to compute the action of the Galois group \(\text{Gal}(\QQ(\zeta_N)/\QQ)\) on these characters, as well as the direct product decomposition corresponding to the factorization of the modulus.
sage: chi.galois_orbit()
[Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> -zeta6 + 1,
Dirichlet character modulo 21 of conductor 7 mapping 8 |--> 1, 10 |--> zeta6]
sage: go = G.galois_orbits()
sage: [len(orbit) for orbit in go]
[1, 2, 2, 1, 1, 2, 2, 1]
sage: G.decomposition()
[
Group of Dirichlet characters modulo 3 with values in Cyclotomic Field of order 6 and degree 2,
Group of Dirichlet characters modulo 7 with values in Cyclotomic Field of order 6 and degree 2
]
接下来，我们构造Dirichlet字符组mod 20，但是值在 \(\QQ(i)\) ：
sage: K.<i> = NumberField(x^2+1)
sage: G = DirichletGroup(20,K)
sage: G
Group of Dirichlet characters modulo 20 with values in Number Field in i with defining polynomial x^2 + 1
接下来，我们计算以下几个不变量 G ：
sage: G.gens()
(Dirichlet character modulo 20 of conductor 4 mapping 11 |--> -1, 17 |--> 1,
Dirichlet character modulo 20 of conductor 5 mapping 11 |--> 1, 17 |--> i)
sage: G.unit_gens()
(11, 17)
sage: G.zeta()
i
sage: G.zeta_order()
4
在本例中，我们使用数字字段中的值创建一个Dirichlet角色。我们通过第三个参数显式指定了单位根的选择 DirichletGroup 下面。
sage: x = polygen(QQ, 'x')
sage: K = NumberField(x^4 + 1, 'a'); a = K.0
sage: b = K.gen(); a == b
True
sage: K
Number Field in a with defining polynomial x^4 + 1
sage: G = DirichletGroup(5, K, a); G
Group of Dirichlet characters modulo 5 with values in the group of order 8 generated by a in Number Field in a with defining polynomial x^4 + 1
sage: chi = G.0; chi
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> a^2
sage: [(chi^i)(2) for i in range(4)]
[1, a^2, -1, -a^2]
这里 NumberField(x^4 + 1, 'a') 告诉Sage在打印什么时使用符号“a” K IS(具有定义多项式的中的数字字段 \(x^4 + 1\) )。“a”这个名字在这一点上还没有宣布。一次 a = K.0 (或同等条件 a = K.gen() )被求值时，符号“a”表示生成多项式的根 \(x^4+1\) 。
模数形式¶
Sage可以进行一些与模形式相关的计算，包括尺寸、模符号的计算空间、Hecke算子和分解。
有几个函数可用于计算模形式空间的维度。例如,
sage: from sage.modular.dims import dimension_cusp_forms
sage: dimension_cusp_forms(Gamma0(11),2)
1
sage: dimension_cusp_forms(Gamma0(1),12)
1
sage: dimension_cusp_forms(Gamma1(389),2)
6112
接下来，我们说明了Hecke算子在层模符号空间上的计算 \(1\) 和重量 \(12\) 。
sage: M = ModularSymbols(1,12)
sage: M.basis()
([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])
sage: t2 = M.T(2)
sage: t2
Hecke operator T_2 on Modular Symbols space of dimension 3 for Gamma_0(1)
of weight 12 with sign 0 over Rational Field
sage: t2.matrix()
[ -24 0 0]
[ 0 -24 0]
[4860 0 2049]
sage: f = t2.charpoly('x'); f
x^3 - 2001*x^2 - 97776*x - 1180224
sage: factor(f)
(x - 2049) * (x + 24)^2
sage: M.T(11).charpoly('x').factor()
(x - 285311670612) * (x - 534612)^2
我们还可以为 \(\Gamma_0(N)\) 和 \(\Gamma_1(N)\) 。
sage: ModularSymbols(11,2)
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
sage: ModularSymbols(Gamma1(11),2)
Modular Symbols space of dimension 11 for Gamma_1(11) of weight 2 with
sign 0 over Rational Field
让我们计算一些特征多项式和 \(q\) --扩容。
sage: M = ModularSymbols(Gamma1(11),2)
sage: M.T(2).charpoly('x')
x^11 - 8*x^10 + 20*x^9 + 10*x^8 - 145*x^7 + 229*x^6 + 58*x^5 - 360*x^4
+ 70*x^3 - 515*x^2 + 1804*x - 1452
sage: M.T(2).charpoly('x').factor()
(x - 3) * (x + 2)^2 * (x^4 - 7*x^3 + 19*x^2 - 23*x + 11)
* (x^4 - 2*x^3 + 4*x^2 + 2*x + 11)
sage: S = M.cuspidal_submodule()
sage: S.T(2).matrix()
[-2 0]
[ 0 -2]
sage: S.q_expansion_basis(10)
[
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 + O(q^10)
]
我们甚至可以用特征标来计算模符号的空间。
sage: G = DirichletGroup(13)
sage: e = G.0^2
sage: M = ModularSymbols(e,2); M
Modular Symbols space of dimension 4 and level 13, weight 2, character
[zeta6], sign 0, over Cyclotomic Field of order 6 and degree 2
sage: M.T(2).charpoly('x').factor()
(x - zeta6 - 2) * (x - 2*zeta6 - 1) * (x + zeta6 + 1)^2
sage: S = M.cuspidal_submodule(); S
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 4 and level 13, weight 2, character [zeta6], sign 0, over
Cyclotomic Field of order 6 and degree 2
sage: S.T(2).charpoly('x').factor()
(x + zeta6 + 1)^2
sage: S.q_expansion_basis(10)
[
q + (-zeta6 - 1)*q^2 + (2*zeta6 - 2)*q^3 + zeta6*q^4 + (-2*zeta6 + 1)*q^5
+ (-2*zeta6 + 4)*q^6 + (2*zeta6 - 1)*q^8 - zeta6*q^9 + O(q^10)
]
这里是Sage如何计算Hecke算子在模形式空间上的作用的另一个例子。
sage: T = ModularForms(Gamma0(11),2)
sage: T
Modular Forms space of dimension 2 for Congruence Subgroup Gamma0(11) of
weight 2 over Rational Field
sage: T.degree()
2
sage: T.level()
11
sage: T.group()
Congruence Subgroup Gamma0(11)
sage: T.dimension()
2
sage: T.cuspidal_subspace()
Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for
Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: T.eisenstein_subspace()
Eisenstein subspace of dimension 1 of Modular Forms space of dimension 2
for Congruence Subgroup Gamma0(11) of weight 2 over Rational Field
sage: M = ModularSymbols(11); M
Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign
0 over Rational Field
sage: M.weight()
2
sage: M.basis()
((1,0), (1,8), (1,9))
sage: M.sign()
0
让我们 \(T_p\) 表示通常的Hecke算子 (\(p\) 素数)。黑克算符是如何 \(T_2\) ， \(T_3\) ， \(T_5\) 作用于模组符号的空间？
sage: M.T(2).matrix()
[ 3 0 -1]
[ 0 -2 0]
[ 0 0 -2]
sage: M.T(3).matrix()
[ 4 0 -1]
[ 0 -1 0]
[ 0 0 -1]
sage: M.T(5).matrix()
[ 6 0 -1]
[ 0 1 0]
[ 0 0 1]