Supersingular Divisors on Modular Curves# Introduction Categories Verbose Output Creation Functions Ambient Spaces SupersingularModule(p,N : parameters): RngIntElt, RngInt → ModSS SupersingularModule(p): RngIntElt → ModForm Example: Creating Supersingular Modules Elements M . i: ModSS, RngIntElt → ModSSElt M ! x: ModSS, . → ModSSElt Example: Elements of a Supersingular Module Subspaces CuspidalSubspace(M): ModSS → ModSS EisensteinSubspace(M): ModSS → ModSS OrthogonalComplement(M): ModSS → ModSS Kernel(I, M): [Tup], ModSS → ModSS Decomposition(M, n): ModSS, RngIntElt → [ModSS] Example: Cuspidal and Eisenstein Subspaces Basis Basis(M): ModSS → SeqEnum Properties AuxiliaryLevel(M): ModSS → RngIntElt BaseRing(M): ModSS → Rng Degree(P): ModSSElt → RngElt Dimension(M): ModSS → RngIntElt Eltseq(P): ModSSElt → SeqEnum Level(M): ModSS → RngIntElt ModularEquation(M): ModSS → RngMPolElt Prime(M): ModSS → RngIntElt Example: Properties of a Supersingular Module Associated Spaces BrandtModule(M): ModSS → ModBrdt ModularSymbols(M : parameters): ModSS → ModSym ModularSymbols(M, sign : parameters): ModSS, RngIntElt → ModSym RSpace(M): ModSS → ModTupRng, Map Example: Spaces Associated to a Supersingular Module Predicates IsAmbientSpace(M): ModSS → BoolElt M1 eq M2: ModSS, ModSS → BoolElt P eq Q: ModSSElt, ModSSElt → BoolElt M1 subset M2: ModSS, ModSS → BoolElt UsesBrandt(M): ModSS → BoolElt UsesMestre(M): ModSS → BoolElt Example: Predicates for Supersingular Modules Arithmetic P + Q: ModSSElt, ModSSElt → ModSSElt P - Q: ModSSElt, ModSSElt → ModSSElt a * P: RngElt, ModSSElt → ModSSElt M1 + M2: ModSS, ModSS → ModSS M1 meet M2: ModSS, ModSS → ModSS Example: Arithmetic on Supersingular Divisors Operators HeckeOperator(M, n): ModSS, RngIntElt → AlgMatElt AtkinLehnerOperator(M, q): ModSS, RngIntElt → AlgMatElt Example: Hecke Operators on a Supersingular Module The Monodromy Pairing MonodromyPairing(P, Q): ModSSElt, ModSSElt → RngIntElt MonodromyWeights(M): ModSS → SeqEnum Example: Monodromy Pairing on a Supersingular Module Bibliography