Darmon Points#
For installation and basic usage instructions please see the main Github repository (https://github.com/mmasdeu/darmonpoints).
The darmonpoints package can compute many different types of what is known as Darmon points. These are known as Stark-Heegner points in some literature, and originated in [Darmon]. Subsequent generalizations were introduced by [Greenberg] and [Trifkovic]. This has been generalized by [GMS1] to elliptic curves defined over number fields of arbitrary signature. Darmon points are attached to triples (F,E,K), where F is a number field, E/F is an elliptic curve defined over F, and K/F is a quadratic extension. These triples must satisfy certain conditions for Darmon points to be attached to them. The article [GMS1] contains an overview of all of this. We include also a variation used in [KP].
The darmonpoints package can also compute equations for some elliptic curves E/F defined over number fields F, as long as certain conditions are satisfied. Namely:
F has narrow class number 1.
if N is the conductor of the elliptic curve, it must admit a factorization of the form N = PDM, where:
P, D and M are relative coprime.
P is a prime ideal of F of prime norm.
D is the discriminant of a quaternion algebra over F which is split at only one infinite place.
Finally, we include the module padicperiods, which allows for the computation of p-adic periods attached to two-dimensional components of the cohomology of the same arithmetic groups, and which has allowed us to find the corresponding abelian surfaces in some cases (see [GM]).
[Darmon]
H.Darmon. Integration on Hp x H and arithmetic applications. Annals of Math.
[Greenberg]
M.Greenberg. Stark-Heegner points and the cohomology of quaternionic Shimura varieties. Duke Math.
[GM]
X.Guitart, M.Masdeu. Periods of modular GL2-type abelian varieties and p-adic integration. Experimental Mathematics.
[GMS1]
(1,2)
X.Guitart, M.Masdeu, M.H.Sengun. Darmon points on elliptic curves over number fields of arbitrary signature. Proc. LMS.
[GMS2]
X.Guitart, M.Masdeu, M.H.Sengun. Uniformization of modular elliptic curves via p-adic methods. Journal of Algebra.
[KP]
A.Pacetti, D.Kohen (with an appendix by M.Masdeu) On Heegner points for primes of additive reduction ramifying in the base field. Transactions of the AMS.
[Trifkovic]
M.Trifkovic. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields. Duke Math.
This work is licensed under a Creative Commons Attribution-Share Alike 3.0 License.
High level functionality#
- Darmon points
- Darmon-Vonk points
- Curve finding
- p-adic periods
HalfPeriodsInTermsOfLambdas()I2_inv_from_xvec()I2_inv_padic_from_half_periods()I2_inv_padic_from_xvec()Theta()Thetas()absolute_igusa_padic_from_half_periods()absolute_igusa_padic_from_xvec()all_possible_ordmats()all_possible_qords()build_Lambdalist_from_AB()change_period_logs()check_absoluteinvs()check_cheatjs()check_generic()check_listI10()compare_AB_periods()compute_lvec_and_Mlist()compute_twisted_jacobian_data()euler_factor_twodim()euler_factor_twodim_tn()evaluate_twisted_jacobian_matrix()find_igusa_invariants()find_igusa_invariants_from_AB()find_initial_approx()find_kadziela_matrices()frobenius_polynomial()generate_listI10()generate_matlists()get_pseudo_orthonormal_homology()guess_equation()igusa_clebsch_absolute_from_half_periods()igusa_clebsch_absolute_from_xvec()igusa_clebsch_from_half_periods()igusa_clebsch_from_xvec()j1_inv_from_xvec()j1_inv_padic_from_half_periods()j1_inv_padic_from_xvec()jacobian_matrix()lambdavec()lambdavec_padic()left_multiply_multiplicative()load_lvec_and_Mlist()multiplicative_scalar_product()normalize_periods()p_adic_l_invariant()p_adic_l_invariant_additive()precompute_powers()qlogs_from_Lp_and_ords()recognize_invariant()right_multiply_multiplicative()take_to_Qp()teichmuller_system()xvec()xvec_padic()
- Plectic points
PlecticGroup()PlecticGroup_classPlecticGroup_class.base_field()PlecticGroup_class.compute_presentation_GG()PlecticGroup_class.compute_presentation_GS()PlecticGroup_class.construct_EV_dict()PlecticGroup_class.construct_VE_dict()PlecticGroup_class.construct_edge_reps()PlecticGroup_class.edge_from_quaternion()PlecticGroup_class.embeddings()PlecticGroup_class.get_BT_reps()PlecticGroup_class.get_Up_reps_bianchi()PlecticGroup_class.get_covering()PlecticGroup_class.get_degeneration()PlecticGroup_class.get_edge_rep()PlecticGroup_class.prime()PlecticGroup_class.quaternion_algebra()PlecticGroup_class.reduce_in_amalgam()PlecticGroup_class.small_group()PlecticGroup_class.use_shapiro()PlecticGroup_class.vert_depth()
additive_integral()check_is_cycle()compute_edge_to_eqs()compute_indeps_parallel()compute_large_system_sparse_parallel()compute_plectic_point()do_twist()evaluate_HC()find_coboundaries()get_cycle()get_edge_list()get_nu0()get_nu1()get_vertex_list()lift_to_HC()op_edge_adj()riemann_sum()riemann_sum_parallel()sample_point()
- Schottky groups
BTTreeBTVertexBallBallsNeighborJoiningTreePreSchottkyGroupSchottkyGroupSchottkyGroup.a_point()SchottkyGroup.balls()SchottkyGroup.find_equivalent_divisor()SchottkyGroup.find_point()SchottkyGroup.gens_extended()SchottkyGroup.in_fundamental_domain()SchottkyGroup.in_which_ball()SchottkyGroup.matrix_to_element()SchottkyGroup.parameters()SchottkyGroup.period()SchottkyGroup.period_matrix()SchottkyGroup.period_naive()SchottkyGroup.theta()SchottkyGroup.to_fundamental_domain()SchottkyGroup.u_function()SchottkyGroup.word_problem()
SchottkyGroup_abstractSchottkyGroup_abstract.all_elements_up_to_length()SchottkyGroup_abstract.base_field()SchottkyGroup_abstract.base_ring()SchottkyGroup_abstract.element_to_matrix()SchottkyGroup_abstract.enumerate_group_elements()SchottkyGroup_abstract.generators()SchottkyGroup_abstract.genus()SchottkyGroup_abstract.group()SchottkyGroup_abstract.inverse_generators()SchottkyGroup_abstract.theta_derivative_naive()SchottkyGroup_abstract.theta_naive()SchottkyGroup_abstract.uniformizer()
all_elements_up_to_length()choose_leaf()enumerate_group_elements()find_eigenvector_matrix()find_midpoint()find_parameter()four_point_configuration()hash_vertex()invert_word()leaf()reduce_word()test_fundamental_domain()uniq()
Arithmetic Groups#
- Arithmetic group elements
ArithGroupElementArithGroupElement.acton()ArithGroupElement.check_consistency()ArithGroupElement.conjugate_by()ArithGroupElement.embed()ArithGroupElement.find_bounding_cycle()ArithGroupElement.is_one()ArithGroupElement.is_scalar()ArithGroupElement.is_trivial_in_abelianization()ArithGroupElement.matrix()ArithGroupElement.quaternion_rep()ArithGroupElement.size()ArithGroupElement.word_rep()
- Non-split Cartan
- Arithmetic group
ArithGroup_fuchsian_genericArithGroup_fuchsian_generic.draw_mat_list()ArithGroup_fuchsian_generic.find_fundom_rep()ArithGroup_fuchsian_generic.fundamental_domain()ArithGroup_fuchsian_generic.fundamental_domain_data()ArithGroup_fuchsian_generic.fundamental_domain_interior_data()ArithGroup_fuchsian_generic.get_archimedean_embedding()ArithGroup_fuchsian_generic.get_embgquats_twisted()ArithGroup_fuchsian_generic.get_word_rep()ArithGroup_fuchsian_generic.is_in_fundom()ArithGroup_fuchsian_generic.is_in_fundom_boundary()ArithGroup_fuchsian_generic.is_in_fundom_exterior()ArithGroup_fuchsian_generic.is_in_fundom_interior()ArithGroup_fuchsian_generic.magma_word_problem()ArithGroup_fuchsian_generic.mat_list()ArithGroup_fuchsian_generic.plot_fundamental_domain()
ArithGroup_nf_fuchsianArithGroup_nf_genericArithGroup_nf_kleinianArithGroup_nf_matrixArithGroup_nf_matrix_newArithGroup_rationalmatrixArithGroup_rationalmatrix.element_of_norm()ArithGroup_rationalmatrix.embed_order()ArithGroup_rationalmatrix.embed_order_legacy()ArithGroup_rationalmatrix.find_fundom_rep()ArithGroup_rationalmatrix.fundamental_domain()ArithGroup_rationalmatrix.fundamental_domain_data()ArithGroup_rationalmatrix.generate_wp_candidates()ArithGroup_rationalmatrix.get_archimedean_embedding()ArithGroup_rationalmatrix.get_word_rep()ArithGroup_rationalmatrix.is_in_fundom()ArithGroup_rationalmatrix.mat_list()ArithGroup_rationalmatrix.non_positive_unit()
ArithGroup_rationalquaternionangle_sign()geodesic_circle()hyperbolic_distance()intersect_geodesic_arcs()is_in_open_interval()moebius_transform()perturb_point_on_arc()sorted_ideal_endpoints()
- S-arithmetic group
ArithGroup()BTEdgeBigArithGroup()BigArithGroup_classBigArithGroup_class.Gpn_Obasis()BigArithGroup_class.Gpn_denominator()BigArithGroup_class.base_field()BigArithGroup_class.base_ring_local_embedding()BigArithGroup_class.clear_cache()BigArithGroup_class.clear_local_splitting()BigArithGroup_class.construct_edge_reps()BigArithGroup_class.coset_reps()BigArithGroup_class.do_tilde()BigArithGroup_class.edge_from_quaternion()BigArithGroup_class.edges_entering_odd_vertex()BigArithGroup_class.edges_leaving_even_vertex()BigArithGroup_class.embed()BigArithGroup_class.embed_wp()BigArithGroup_class.get_BT_reps()BigArithGroup_class.get_BT_reps_twisted()BigArithGroup_class.get_Up_reps()BigArithGroup_class.get_Up_reps_bianchi()BigArithGroup_class.get_Zp_covering()BigArithGroup_class.get_amalgam_reps()BigArithGroup_class.get_coset_ti()BigArithGroup_class.get_covering()BigArithGroup_class.get_edges_upto()BigArithGroup_class.get_embedding()BigArithGroup_class.get_gis()BigArithGroup_class.get_gis_local()BigArithGroup_class.get_gitildes()BigArithGroup_class.get_gitildes_local()BigArithGroup_class.is_in_Gpn_order()BigArithGroup_class.large_group()BigArithGroup_class.local_splitting()BigArithGroup_class.prime()BigArithGroup_class.quaternion_algebra()BigArithGroup_class.reduce_in_amalgam()BigArithGroup_class.save_to_db()BigArithGroup_class.set_wp()BigArithGroup_class.small_group()BigArithGroup_class.subdivide()BigArithGroup_class.use_shapiro()BigArithGroup_class.wp()
MatrixArithGroup()attach_kleinian_code()is_page_initialized()
Cohomology and Homology#
- Abstract cohomology class
CohomologyElementCohomologyGroupCohomologyGroup.ElementCohomologyGroup.GA_to_local()CohomologyGroup.coefficient_module()CohomologyGroup.dimension()CohomologyGroup.eval_at_genpow()CohomologyGroup.fox_gradient()CohomologyGroup.gen()CohomologyGroup.generator_acting_matrix()CohomologyGroup.gens()CohomologyGroup.get_fox_term()CohomologyGroup.get_gen_pow()CohomologyGroup.group()CohomologyGroup.hecke_matrix()CohomologyGroup.space()CohomologyGroup.zero()
- Arithmetic cohomology
ArithActionArithCohArithCohBianchiArithCohElementArithCohOverconvergentBianchiArithActionCohArbitraryget_cocycle_from_elliptic_curve()get_dedekind_rademacher_cocycle()get_overconvergent_class_bianchi()get_overconvergent_class_matrices()get_overconvergent_class_quaternionic()get_rational_cocycle()get_rational_cocycle_from_ap()get_twodim_cocycle()
- Abstract homology class
AbelianizationArithHomologyArithHomologyElementHomologyElementHomologyGroupHomologyGroup.ElementHomologyGroup.coefficient_module()HomologyGroup.free_gens()HomologyGroup.gen()HomologyGroup.gens()HomologyGroup.get_gen_pow()HomologyGroup.get_twisted_fox_term()HomologyGroup.group()HomologyGroup.ngens()HomologyGroup.rank()HomologyGroup.space()HomologyGroup.twisted_fox_gradient()HomologyGroup.zero()
- Homology class
ArithGroupActionOneChainsOneChainsElementOneChainsElement.act_by_hecke()OneChainsElement.act_by_poly_hecke()OneChainsElement.factor_into_generators()OneChainsElement.hecke_smoothen()OneChainsElement.is_cycle()OneChainsElement.is_degree_zero_valued()OneChainsElement.is_zero()OneChainsElement.radius()OneChainsElement.zero_degree_equivalent()
TensorElementTensorProductget_homology_kernel()inverse_shapiro()lattice_homology_cycle()
Integration pairing#
Overconvergent distributions#
- Overconvergent module of distributions
AddMeromorphicFunctionsAddMeromorphicFunctionsElementAddMeromorphicFunctionsElement.evaluate_additive()AddMeromorphicFunctionsElement.evaluate_multiplicative()AddMeromorphicFunctionsElement.left_act_by_matrix()AddMeromorphicFunctionsElement.pair_with()AddMeromorphicFunctionsElement.power_series()AddMeromorphicFunctionsElement.scale_by()AddMeromorphicFunctionsElement.valuation()
OCVnOCVnElementSigma0Actionour_adjusterps_adjuster
- Representations
- Meromorphic functions
Internals#
- Divisors
DivisorsDivisorsElementDivisorsElement.apply_map()DivisorsElement.as_list_of_differences()DivisorsElement.degree()DivisorsElement.gcd()DivisorsElement.is_zero()DivisorsElement.left_act_by_matrix()DivisorsElement.pair_with()DivisorsElement.rational_function()DivisorsElement.restrict()DivisorsElement.scale_by()DivisorsElement.size()DivisorsElement.support()DivisorsElement.value()
- Mixed extensions
QuadExtQuadExt.ElementQuadExt.absolute_degree()QuadExt.base_ring()QuadExt.characteristic()QuadExt.gen()QuadExt.is_finite()QuadExt.polynomial()QuadExt.precision_cap()QuadExt.prime()QuadExt.ramification_index()QuadExt.random_element()QuadExt.relative_degree()QuadExt.residue_class_degree()QuadExt.some_elements()QuadExt.uniformizer()QuadExt.unramified_generator()
QuadExtElementget_word_rep_fast()
- Sparse matrix calculations
- Utility package
BunchFGP_V()FGP_W()JtoP()M2Z()act_H3()act_flt()act_flt_in_disc()affine_transformation()cantor_diagonal()config_section_map()conjugate_quaternion_over_base()covolume()direct_sum_of_maps()direct_sum_of_modules()discover_equation()enumerate_words()field_element_pari_to_sage()find_center()find_containing_affinoid()find_the_unit_of()fwrite()get_C_and_C2()get_c4_and_c6()get_heegner_params()get_j_invariant()getcoords()height_polynomial()hensel_lift()imag_part()is_in_Gamma0loc()is_in_Gamma_1()is_in_principal_affinoid()is_infinity()is_smooth()lift_padic_splitting()magma_F_elt_to_sage()magma_F_ideal_to_sage()magma_integral_quaternion_to_sage()magma_quaternion_to_sage()module_generators()module_generators_new()multiply_out()our_algdep()our_cuberoot()our_lindep()our_nroot()our_sqrt()pari_ordmax_basis_to_sage()period_from_coords()point_radius()polynomial_roots()polynomial_roots_old()print_padic()print_table_latex()quaternion_algebra_invariants_from_ramification()quaternion_to_magma_quaternion()real_part()recognize_DV_algdep()recognize_DV_lindep()recognize_J()recognize_point()reduce_word()reduce_word_tietze()relativize_ATR()sage_F_elt_to_magma()sage_F_ideal_to_magma()sage_order_basis_to_pari()sage_quaternion_to_magma()selmer_group_iterator()set_immutable()simplification_isomorphism()solve_quadratic()syllables_to_tietze()tate_parameter()tietze_to_syllables()translate_into_twosided_list()update_progress()weak_approximation()