Jacobi Motive Functionality#

Creation Functions#

JacobiMotive(A, B): SeqEnum, SeqEnum -> JacketMot#

Kummer: SeqEnum                      Default: [1,0]
Tate  : RngIntElt                    Default: 0
Weight: RngIntElt                    Default: 

Given two sequences of rationals, corresponding to positive and negative elements in the free group on \({\bf Q}/{\bf Z}\), create the resulting Jacobi motive. This requires that the signed sum of the rationals is an integer. The optional Kummer vararg can specify a Kummer twist, and similarly with the Tate vararg. Alternatively, the desired Tate twist can be obtained by giving and integral argument to the Weight vararg, and the effective Tate twist can be obtained by setting Weight as true. A variant with only one argument (\(B\) is empty) is also available.

JacketMotive(A, B, t, rho, j): SeqEnum, SeqEnum, RngQZElt, RngQZElt, RngIntElt -> JacketMot#: Similar to above, this intrinsic spells out the Jacobi summands (\(A,B\)), the Kummer twist \(t^\rho\), and the Tate twist \(j\) explicitly.

KummerTwist(J, t, rho): JacketMot, RngQZElt, RngQZElt -> JacketMot#: Given a Jacobi motive \(J\), return its Kummer twist by \(t^\rho\), where \(\rho\) is rational and \(t\) is a nonzero rational.

TateTwist(J, j): JacketMot, RngIntElt -> JacketMot#: Given a Jacobi motive \(J\) and an integer \(j\), return the \(j\)th Tate twist of \(J\).

Operations#

J1 * J2: JacketMot, JacketMot -> JacketMot#: Given two Jacobi motives, take their tensor product, eliminating any rationals common to the positive and negative parts.

J1 / J2: JacketMot, JacketMot -> JacketMot#: Given two Jacobi motives, take their tensor quotient, eliminating any rationals common to the positive and negative parts.

J1 eq J2: JacketMot, JacketMot -> JacketMot#
J1 ne J2: JacketMot, JacketMot -> JacketMot#: Whether two Jacobi motives are equal, that is, whether they have the same positive and negative parts, their \(t^\rho\) Kummer twists are the same, and they have the same Tate twist parameter.

Scale(J, q): JacketMot, RngQZElt -> JacketMot#: Given a Jacobi motive, scale all the rational numbers defined the datum by the given rational \(q\). The denominator of \(q\) must be coprime to \(m\), and \(q\) must be invertible mod \(m\). The resulting motive will be identical over \({\bf Q}\) but need only be conjugate over \(K_\theta\).

Attributes#

Field(J): JacketMot -> FldNum#: The field of definition of a Jacobi motive.

Weight(J): JacketMot -> RngIntElt#: The motivic weight of a Jacobi motive.

EffectiveWeight(J): JacketMot -> RngIntElt#: The effective motivic weight of a Jacobi motive, that is, the width of its Hodge structure.

HodgeStructure(J): JacketMot -> HodgeStruc#
HodgeVector(J): JacketMot -> HodgeStruc, RngIntElt#
EffectiveHodgeStructure(J): JacketMot -> HodgeStruc#: The Hodge structure of a Jacobi motive.

L-function#

EulerFactor(J, p): JacketMot, RngIntElt -> RngUPolElt#

Degree: RngIntElt                    Default: 
Roots : BoolElt                      Default: false

Given a good prime \(p\), that is, one which is coprime to \(m\) and the Kummer twisting parameter \(t\), compute its Euler factor. The Roots vararg also returns as a second argument the \(p\)-adic approximations to the roots (associated to the prime ideals above \(p\)). The Degree vararg can be used when the full Euler factor is not needed, though it is often still just as easily computed. It is often easier to first identify the Jacobi motive as a Grossencharacter, and then compute its Euler factors.

ComplexEvaluation(J, P): JacketMot, RngOrdIdl -> FldComElt#

Precision: RngIntElt                    Default: 

Given a Jacobi motive and a good degree 1 prime over \(K_\theta\) that splits completely over the cyclotomic field, compute the associated Jacobi sum as a complex number. This is used to identify motives that are equivalent over \({\bf Q}\) but not over \(K_\theta\) in some examples.

Grossencharacter(J): JacketMot -> GrossenChar#: Given a Jacobi motive, identify it as a Grössencharacter. This uses the Weil bound on the conductor, and then tries enough good primes to distinguish the character. This is now the preferred way to compute the LSeries of a Jacobi motive (though the latter still exists).