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db-abvar.md

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Database abvar
Description Isogeny classes of abelian varieties over finite fields
Status beta
Contact David Roe
Code abvar/fq
Collections fq_iosg

Todo:

  • Compute and add data on Frobenius angle ranks

Collection fq_isog

  • Content: Isogeny classes of abelian varities over finite fields
  • Contributors: Kiran Kedlaya, Taylor Dupuy, David Roe, Christelle Vincent
  • Extent: Complete for pairs (g, q) where q^(g(g-1)/2) is less than 10^7 (as of October 2016)
Field Description Type of stored data Mathematical type Example of stored data Remarks
_id Mongo id ObjectId - assigned by Mongo; contains creation timestamp
label LMFDB Label string - '2.16.am\_cn' [Labeling Scheme](http://beta.lmfdb.org/Variety/Abelian/Fq/Labels)
g Genus int N 2 The degree of the Weil L-polynomial is 2g.
q Cardinality of Field int prime power 16 All of the roots of the Weil L-polynomial have absolute value $1/\sqrt{q}$.
polynomial Coefficients of the Weil L-polynomial list of ints Z^(2g+1) [1,-12,65,-192,256] The first entry will always be 1 and the last $q^g$. For i between 0 and g, $a_{2g-i} = q^{g-i} a_i$.
angle_numbers Frobenius angle numbers list of python floats R^g [0.0826163580681,0.320878822416] The positive arguments of the roots (considered as complex numbers) of the Weil L-polynomial. There will be g of them unless the list includes 0 or pi.
angle_ranks This is one less than the dimension of the space spanned by the arguments of the roots of the Weil polynomial divided by $\pi$ and one. int N 3 This might be empty if we haven't computed it yet.
p_rank The $p$-rank of the abelian variety int N 2 The rank of the $p$-torsion subgroup of the abelian variety. Equal to the number of occurences of the slope 0 in the Newton slopes.
slopes The slopes of the Newton polygon of the Weil polynomial list of strings Q^(2g+1) ['0', '1/2', '1/2', '1'] The slopes are in increasing order, are symmetric under the involution $s \to 1-s$, and the corresponding Newton polygon has endpoints (0,0) and (2g,g).
A_counts The number of points of the abelian variety over extensions of F\_q list of strings N^g ['118', '62068'] Counts are given for $A(F_{q^n})$ for $1 \le n \le max(g,10)$; counts over larger extension fields can be determined from these using the Weil conjectures.
C_counts The number of points of a corresponding curve list of strings Z^g ['5', '243'] If the variety is a Jacobian, these are the point counts of a genus g curve of which this is the Jacobian. In particular, if any point counts are negative then this abelian variety cannot be a Jacobian.
known_jacobian An integer encoding whether the abelian variety is a Jacobian int - 0 1 means that it is definitely a Jacobian, -1 that it is definitely not, and 0 indicates uncertainty.
principally_polarizable An integer encoding whether the abelian variety is principally polarizable int - 0 1 means that it is definitely principally polarizable, -1 that it is definitely not, and 0 indicates uncertainty.
decomposition The decomposition into simple factors list of pairs (string, int) - [['2.16.am_cn',1], ['1.16.ah',2]] The first entry in each pair is the label of the factor, the second is its multiplicity.
brauer_invariants The Brauer invariants of the endomorphism algebra list of strings Q^k ['0','0','1/2'] For a simple isogeny class, the number of invariants is the number of primes above p in the number field defined by the Weil polynomial. For a non simple class, the Brauer invariants of its simple factors are concatenated, and they appear in the order in which the factors appear in the field decomposition.
places The ideals corresponding to the Brauer invariants of the endomorphism algebra list of lists of strings ((Q^d_i)^e_i)^f [[['0','1'],['1','1/2']],[['0','3']]] The outer set of lists corresponds to the simple factors of the isogeny class (so in the example, this isogeny class is a product of two simple isogeny classes). For each simple factor, the list contains one list per prime above p in the number field defined by the Weil polynomial. This list describes the prime ideal above p by giving the second generator of the ideal (the first generator is p), as a list of the coefficients of the generator when written in terms of a specific basis for the number field. This basis contains the powers of a root of the P-polynomial (which is the Weil polynomial but reversed).
primitive_models Every isogeny class defined over smaller fields such that this isogeny class is a base change of this isogeny class. list of strings - ['2.2.ab_ab','2.2.b_ab'] If the isogeny class is primitive, the list is empty. Otherwise, the list contains the label of every primitive isogeny class that base changes to this class. This list is complete.
number_field The label of the number field defined by the Weil polynomial string - '4.0.27792.2' If the number field was not in the database when the isogeny class was added to the database, this string is empty. If the isogeny class is not simple, this is also an empty string.
galois_n The degree label of the Galois group of the Weil polynomial int - 4 If the number field was not in the database when the isogeny class was added to the database, this string is empty. If the isogeny class is not simple, this is also an empty string.
galois_t The transitive label of the Galois group of the Weil polynomial int - 2 If the number field was not in the database when the isogeny class was added to the database, this string is empty. If the isogeny class is not simple, this is also an empty string.

Index information on collection fq_isog:

  • {'_id': 1} (created by mongo)
  • {'label': 1} (for searching)
  • {'polynomial': 1} (for searching)
  • {'p_rank': 1} (for searching)
  • {'slopes': 1} (for searching)
  • {'A_counts': 1} (for searching)
  • {'C_counts': 1} (for searching)
  • {'known_jacobian': 1} (for searching)
  • {'principally_polarizable': 1} (for searching)