db-sato_tate_groups.md

sato_tate_groups

Description	Sato-Tate groups
Status	production
Contact	Andrew Sutherland
Code	sato_tate_groups
Collections	st_groups, st0_groups, small_groups

Todo:

• add rational weight 0 groups of low degree (to support Artin L-functions)
• add generators for weight 1 degree 4 groups

Collection: st_groups

Description: Main collection for Sato-Tate groups

Origin: Code by Andrew Sutherland

Extent: All Sato-Tate groups that arise for elliptic curves (3) and genus 2 curves (52) over a number field. This addresses all self-dual motives with rational coefficients of weight 1 and degree up to 4. Sato-Tate groups of weight 0 and degree 1 (not necessarily rational) are computed on the fly and not stored in the database. The small groups database contains data for all of the 92,804 groups of order less than 512.

Attributes

• _id: ObjectId generated by Mongo DB (includes creation timestamp)
  Example:ObjectId('572bbe5bf9ae3228154a72bd')

• component_group: encoded as GAP id string 'a.b', where a and b are integers; a is the order of the group and b distinguishes groups of the same order.
  Example: u'6.2'

• components: number of components (equal to a in the GAP id of the component group), stored as an integer.
  Example: int(6)

• counts: list of pairs [name,value_list], where x is a class function (a_n denotes the nth elementary symmetric function of the eigenvalues and s_n denotes the nth power sum), and value_list is a list of pairs [v,n] where v is an integer value and n is the number of components for which x=v.
  Example: [[u'a_1',[[0,9]]],[u'a_2',[[-1,,2],[2,1]]].

• degree: degree of the Sato-Tate group (cohomological dimension), a positive integer.
  Example: int(4)

• gens: generators, stored as a list of d-by-d matrices whose entries are strings, where d is the degree; together with the identity component, they generate the group.
  Example: [[[u'0', u'1'], [u'1', u'0']]]

• identity_component: label of the identity component (string).
  Example: 'USp(4)'

• label: label of the form wt.deg.dim.a.bc (string) where wt is the weight, deg is the degree, dim is the real dimension, a.b is the GAP id of the component group, and c is a letter or string of letters used to break ties; uniquely identifies the Sato-Tate group.
  Example: u'1.4.10.1.1a'

• moments: list of lists [x,m_1, m_2,..., ], where x is a class function (elementary symmetric or power sum function of eigenvalues), and m_n is the nth moment of x, stored as s string.
  Example: [[u'a_1',u'1',u'0',u'1',u'0',u'3',u'0',u'14',....,u'4719'],[u'a_2',u'1',u'1',u'2',u'4',u'10',...,u'223412']]

• name: string naming the Sato-Tate group unique within its weight and degree.
  Example: u'USp(4)'

• pretty: pretty-print version of name in latex math mode.
  Example: u'\mathrm{USp}(4)'

• rational: boolean indicating whether the Sato-Tate group satisfies the rationality axiom (currently always True).
  Example: bool(True)

• subgroups: list of labels of maximal proper subgroups.
  Example: [u'1.4.3.1.1a']

• supgroups: list of labels of minimal proper super group.
  Example: [u'1.4.3.1.1a', u'1.4.3.4.2a',u'1.4.3.6.2a']

• trace_histogram: b64 encoded .png file containing 220x124 trace histogram plot.
  Example: u'data:image/png:base64,iVBORw0KGg...II%3D

• trace_zero_density: proportion of components on which the trace is identically zero, rational number encoded as a string.
  Example: u'1/2'

• weight: weight of the Sato-Tate group (nonnegative integer).
  Example: int(1)

Indexes

• {'_id':1} (created by mongo db)
• {'label':1} (for searching by label)
• {'name':1}: (for searching by name)
• {'weight':1}: (for browsing/searching by weight)
• {'degree':1} (for browsing/searching by degree)
• {'weight':1,'degree':1,'real_dimension':1,'components':1} (used to sort search results)

Collection: st0_groups

Description: Collection containing information about the identity components of Sato-Tate groups

Origin: Code by Andrew Sutherland

Extent: Sato-Tate groups that arise for elliptic curves (3) and genus 2 curves (52) over any number field. This addresses all self-dual motives with rational coefficients of weight 1 and degree up to 4.

Attributes

• label: label of the identity component, currently of the form w.d.r, where w is the weight, d is the degree, and r is the real dimension (this is sufficient to uniquely identify the identity component for w=1 and d=0,2,4) but in other cases more information will be required (so the label format may need to vary with w and d).
  Example: u'1.4.6'

• name: text name of the identity component, used for displaying in scrolled input boxes where latex is not supported.
  Example: u'SU(2)xSU(2)'

• pretty: pretty-print version of the name in latex math mode.
  Example: u'\mathrm{SU}(2)\times\mathrm{SU}(2)'

• weight: weight of the corresponding Sato-Tate group (nonnegative integer).
  Example: int(1)

• degree: degree of the corresponding Sato-Tate group (positive integer).
  Example: int(4)

• real_dimension: dimension of the identity component as a connected compact real Lie group (positive integer).
  Example: int(6)

• description: mathematical description of the identity component as a set of d-by-d matrices (latex math mode string) Example: u'\left\{\begin{bmatrix}A&0\\0&B\end{bmatrix}: A,B\in\mathrm{SU}(2)\right\}'

Indexes

• {'_id':1} (created by mongo db)
• {'label':1} (for searching by label)
• {'name':1}: (for searching by name)
• {'weight':1}: (for browsing/searching by weight)
• {'degree':1} (for browsing/searching by degree)

Collection: small_groups

Description: Collection containing information about finite groups of small order used to describe component groups of Sato-Tate groups (could also be used in many other places).

Origin: GAP

Extent: All groups of order up to 255 (up to isomorphism). Easy to add more (up to order 1024).

Attributes

• label: GAP ID encoded as a string u'N.n', where N is the order of the group and n distinguishes non-isomorphic groups of the same order (as determined in GAP).
  Example: u'24.13'

• name: text discription of the group.
  Example: u'C_2*A_4'

• pretty: pretty-print version of the name in latex math mode.
  Example: u'C_2\times A_4'

• order: order of the group, a positive integer.
  Example: int(24)

• exponent: exponent of the group, a positive integer.
  Example: int(6)

• cyclic: true if the group is cyclic, false otherwise.
  Example: bool(False)

• abelian: true if the group is abelian, false otherwise.
  Example: bool(False)

• perfect: true if the group is perfect, false otherwise.
  Example: bool(False)

• simple: true if the group is simple, false otherwise.
  Example: bool(False)

• solvable: true if the group is solvable, false otherwise.
  Example: bool(True)
  Example: [[u'2.1', int(1)], [u'4.2', int(1)], [u'8.5', int(1)], [u'12.3', int(1)]]

• center: small group label identifying the center of the group.
  Example: u'2.1'

• derived_group: small group label identifying its commutator subgroup.
  Example: u'4.2'

• abelian_quotient: small group label identifying the quotient of the group by its commutator subgroup.
  Example: u'6.2'

• maxmimal_subgroups: maximal proper subgroups listed as pairs [label, n], where label is a small group label and n is the number of non-conjugate maximal subgroups in the isomorphism class identified by the label.
  Example: [[u'6.2', int(1)], [u'8.5', int(1)], [u'12.3', int(1)]]

• normal_subgroups: normal proper subgroups listed as pairs [label, n], where label is a small group label and n is the number of non-conjugate normal subgroups in the isomorphism class identified by the label.
  

Indexes

• {'_id':1} (created by mongo db)
• {'label':1} (for searching by label)
• {'name':1}: (for searching by name)
• {'order':1}: (for browsing/searching by order)