postgres_schema.md

Table name: av_fqisog

This table represents unpolarized abelian varieties, up to isogeny.

Things to add for the isogeny class

Column	Type	Notes
label	text
g	smallint
q	integer
poly_str	text	Space separated string of coefficients, for searching
p_rank	smallint
dim1_factors	smallint	Number of dimension 1 factors
dim2_factors	smallint	Number of dimension 2 factors
dim3_factors	smallint	Number of dimension 3 factors
dim4_factors	smallint	Number of dimension 4 factors
dim5_factors	smallint	Number of dimension 5 factors
dim1_distinct	smallint	Number of distinct dimension 1 factors
dim2_distinct	smallint	Number of distinct dimension 2 factors
dim3_distinct	smallint	Number of distinct dimension 3 factors
dim4_distinct	smallint	Number of distinct dimension 4 factors
dim5_distinct	smallint	Number of distinct dimension 5 factors
poly	integer[]	Coefficients of the Weil polynomial. The first will always be 1 and the last q^g
angles	float8[]	Angles corresponding to roots in the closure of the upper half plane, divided by pi. All will be in the interval [0, 1], and there will be g of them unless 0 or 1 is included.
angle_rank	smallint	The dimension of the Q-span of the angles (see knowl for complete definition)
slopes	text[]	The sorted list of slopes, as string representations of rational numbers. Duplicated slopes will have "A", "B", etc appended.
abvar_counts	numeric[]	The list of counts #A(F_{q^i}) for i=1..10, for A in this isogeny class
abvar_counts_str	text	A space separated string of abelian variety counts, for searching
curve_counts	numeric[]	The list of curve counts #C(F_{q^i}) for i=1..10 for any curve C of genus g with J(C) in this isogeny class
curve_counts_str	text	A space separated string of curve counts, for searching
point_count	integer	The count #C(F_q), duplicated for searching purposes
has_jacobian	smallint	1 if it is known that this isogeny class contains a Jacobian; -1 if it is known that it does not; 0 otherwise
has_principal_polarization	smallint	1 if it is known that this isogeny class contains a principally polarizable abelian variety; -1 if it is known that it does not; 0 otherwise
is_simple	boolean
simple_factors	text[]	A list of labels of simple factors. Duplicated factors will have "A", "B", etc appended.
simple_distinct	text[]	A list of distinct labels of simple factors.
simple_multiplicities	smallint[]	For each distinct simple factor, the multiplicity in the decomposition.
number_field_degrees	smallint[]	For each distinct simple factor, the degree of the the corresponding number field.
divalg_dimensions	smallint[]	For each distinct simple factor, the dimension of the division algebra over the number field.
number_fields	text[]	The number fields associated to the irreducible factors of the Weil polynomial
galois_groups	text[]	The Galois groups of the number fields associated to the irreducible factors of the Weil polynomial, e.g. "4T3"
places	text[]	A list of lists of lists of rational numbers stored as strings, giving the prime ideals above p. The terms in the outer list correspond to distinct simple factors, the terms in the middle lists correspond to places in the corresponding number field, and each inner list gives coefficients for a two-element generator of that prime ideal (along with p) as coefficients of powers of F.
brauer_invariants	text[]	A list of lists of rational numbers stored as strings. The terms in the outer list correspond to distinct simple factors, and the terms in each inner list correspond to the places in the corresponding number field.
geometric_extension_degree	smallint	The smallest degree extension of the base field over which the endomorphism algebra becomes the full endomorphism algebra
geometric_simple_factors	text[]	A list of labels of simple factors after base changing by the geometric extension degree. Duplicated factors will have "A", "B", etc appended. NULL if geometric_extension_degree is 1.
geometric_simple_distinct	text[]	A list of distinct labels of simple factors after geometric base change. NULL if geometric_extension_degree is 1.
geometric_multiplicities	smallint[]	For each distinct geometric simple factor, the multiplicity in the decomposition. NULL if geometric_extension_degree is 1.
geometric_number_field_degrees	smallint[]	For each distinct geometric simple factor, the degree of the corresponding number field. NULL if geometric_extension_degree is 1.
geometric_divalg_dimensions	smallint[]	For each distinct geometric simple factor, the dimension of the division algebra over the number field. NULL if geometric_extension_degree is 1.
geometric_number_fields	text[]	The number fields associated to the irreducible factors of the base-changed Weil polynomial. NULL if geometric_extension_degree is 1.
geometric_galois_groups	text[]	The Galois groups of the geometric number fields. NULL if geometric_extension_degree is 1.
geometric_places	text[]	Places, after base changing to the geometric field.
geometric_brauer_invariants	text[]	Brauer invariants, after bas changing to the geometric field.
primitive_models	text[]	A list of labels giving primitive models for this isogeny class (ie, this class arises from base change from the model). If primitive, NULL.
is_primitive	boolean
twists	jsonb	A list of triples (label, geom_label, r) where label is the label of a twist, r is an extension degree where the twists become isomorphic, and geom_label is the label of the common base change to that degree.
size	integer	number of isomorphism classes within the isogeny class
zfv_is_bass	boolean	whether all the over-orders for the order Z[F,V] are Gorenstein
zfv_is_maximal	boolean	whether the order Z[F,V] is maximal
zfv_index	numeric	the index of the order Z[F,V] in the maximal order
zfv_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_index	numeric	the index of the order Z[F+V] in the maximal order of the real subfield
zfv_plus_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_norm	numeric	The absolute value of the norm of F-V to Z
isogeny_graphs	jsonb	list of pairs (p, G), where p is a degree (or maybe list of degrees) and G is a list of pairs (u,v) representing the directed edge from u to v. Each of u and v is the isom_letter for the corresponding isogeny class
ideal_class_generators	text[]	A list of isom_letters for isomorphism classes that generate the ideal monoid
ideal_class_relations	integer[]	A matrix of positive integers giving relations between the ideal class generators

Table name: av_fqisom

This table represents unpolarized abelian varieties, up to isomorphism.

Column	Type	Notes
label	text	g.q.weil.enum, where g is the dimension, q is the cardinality of the base field, weil is the encoding of the Weil polynomial and enum is isom_letter
isom_num	integer	A 0-based enumeration of the isomorphism classes within an isogeny class, TBD
isom_letter	text	Base 26 a-z encoding of isom_num
isog_label	text	label for isogeny class
frac_ideal_numerators	numeric[]	numerators for a basis for the fractional ideal, expressed in terms of V^{g-1},...,V,1,F,F^2,...,F^g
frac_ideal_denominators	numeric[]	denominators for a basis for the fractional ideal, expressed in terms of V^{g-1},...,V,1,F,F^2,...,F^g. NULL if all 1.
over_order	text	isom_letter for over order associated to this ideal MAYBE???
is_over_order	boolean	MAYBE???
weak_equivalence_class	text	label for the weak equivalence class???
rep_type	smallint	0=ordinary or Centeleghe-Stix,...
is_reduced	boolean	Whether the fractional ideal is reduced (HNF, minimal norm, lexicographic within same norm)
cm_type	boolean[]	Whether the +imaginary embedding is a p-adic non-unit, for embeddings sorted by real part
cm_elt	numeric[]	An element of Q[F] that is positive imaginary under each embedding in the CM type
can_be_principally_polarized	boolean	Whether this abelian variety has a principal polarization
rational_invariants	numeric[]	Invariant factors of A(F_q)
is_product	boolean	Whether this isomorphism class is a product of smaller dimensional abelian varieties
product_factorization	jsonb	List of pairs (label, e) expressing this as a product of smaller dimensional abelian varities (NULL if not)
endo_ring	jsonb	Some kind of description....
related_objects	text[]	List of URLs

Table name: av_fqpol

This table represents polarized abelian varieties, up to isomorphism.

Column	Type	Notes
label	text	?????
isom_label	text
degree	smallint	degree of the polarization
kernel	smallint[]	invariant factors for the kernel of the isogeny (cokernel of the map of lattices)
is_decomposible	boolean	Whether this polarized abelian variety is a product
decomposition	jsonb	List of pairs (label, e) expressing this polarized abelian variety as a product (NULL if not)
aut_group	text	GAP id
geom_aut_group	text	GAP id
is_serre_obstructed	smallint	-1 if not a Jacobian, 0 if a hyperelliptic Jacobian, 1 if a nonhyperelliptic Jacobian
invariants	jsonb	For small genus, a list of geometric invariants (e.g. Igusa). Only possible in the principal case

Things to add for the isogeny class

Column	Type	Notes
size	integer	number of isomorphism classes within the isogeny class
zfv_is_bass	boolean	whether all the over-orders for the order Z[F,V] are Gorenstein
zfv_is_maximal	boolean	whether the order Z[F,V] is maximal
zfv_index	numeric	the index of the order Z[F,V] in the maximal order
zfv_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_index	numeric	the index of the order Z[F+V] in the maximal order of the real subfield
zfv_plus_index_factorization	numeric[]	A list of pairs (p, e) giving the factorization of the index
zfv_plus_norm	numeric	The absolute value of the norm of F-V to Z
isogeny_graphs	jsonb	list of pairs (p, G), where p is a degree (or maybe list of degrees) and G is a list of pairs (u,v) representing the directed edge from u to v. Each of u and v is the isom_letter for the corresponding isogeny class
ideal_class_generators	text[]	A list of isom_letters for isomorphism classes that generate the ideal monoid
ideal_class_relations	integer[]	A matrix of positive integers giving relations between the ideal class generators

• Whether or not two isogeny classes come together after base extension, and what the degree is

• Change brauer_invs from a string to list of lists of strings

• Change jsonb types to arrays

• Change nf to text[]: a list of number fields in the non-simple case

• Update galois_t when not set

• Add missing isogeny classes (t^2-p)

• Rename/retype some columns:

• poly (jsonb -> integer[])
• angles (jsonb -> float8[])
• ang_rank -> angle_rank (smallint)
• slps -> slopes (jsonb -> text[])
• A_cnts -> abvar_counts (jsonb -> numeric[])
• A_cnts_str -> abvar_counts_str (text)
• C_cnts -> curve_counts (jsonb -> numeric[])
• C_cnts_str -> curve_counts_str (text)
• pt_cnt -> point_count (integer)
• is_jac -> has_jacobian (smallint -> boolean)
• is_pp -> has_principal_polarization (smallint -> boolean)
• decomp -> decomposition (jsonb)
• is_simp -> is_simple (boolean)
• simple_factors (jsonb -> text[])
• simple_distinct (jsonb -> text[])
• brauer_invs -> brauer_invariants (text -> text[])
• places (jsonb -> text[])
• prim_models -> primitive_models (jsonb -> text[])
• is_prim -> is_primitive (boolean)
• nf -> number_fields (text -> text[])
• galois_t -> galois_groups (smallint -> text[])
• galois_n -> XXX

Unchanged:

• label (text)
• g (smallint)
• q (integer)
• poly_str (text)
• p_rank (smallint)
• dim1_factors (smallint)
• dim2_factors (smallint)
• dim3_factors (smallint)
• dim4_factors (smallint)
• dim5_factors (smallint)
• dim1_distinct (smallint)
• dim2_distinct (smallint)
• dim3_distinct (smallint)
• dim4_distinct (smallint)
• dim5_distinct (smallint)

June Ju and Everett Howe: telling whether an abelian variety is absolutely simple Look at primes dividing discriminant of Weil field, products at most 4g^2. Find pairwise r, hash on multiple r Supersingular if and only if the ultimate field is Q

For Stefano

We'll be doing isogeny classes that Stefano has already computed, plus:

• Any g, q
• ordinary or C-S (q=p, no real roots), squarefree
• Z[F,V] = maximal order

For each isogeny class, write lines to two files

• isomorphism_classes.txt (one line per ideal)

isog_label:frac_ideal:rep_type:is_reduced:cm_elt:is_product
e.g.
1.251.v:{{1,0},{0,1}}:0:f:{21,2}:f

• isogeny_classes.txt (one line per class)

isog_label:order_index:order_is_bass:order_is_maximal:size
e.g.
1.251.v:1:t:t:9

Use \N for null.

["1.251.v", [251,21,1]]