# Divisors in OM representation OM representations of ideals can be used to construct divisors without computing maximal orders which are required for the divisors of type `DivFunElt`. For detailed information please see [[Bauch, 2014](../../references.md#cite-bauch-thesis), [Bauch, 2016](../../references.md#cite-bauch2016lattices)]. ## Constructing Divisors in OM representation ### `OMDivisor(If, Ii): OMIdl, OMIdl -> OMDiv` Given an ideal $If$ in OM representation whose function field is marked as finite and an ideal $Ii$ in OM representation whose function field is marked as infinite, return the divisor in OM representation whose support is the places associated to the prime ideals which divide either $If$ or $Ii$ with coefficients the negative of valuation of $If$ and $Ii$ at the prime ideals. ### `OMDivisor(I): OMIdl -> OMDiv` Given an ideal $I$ in OM representation return the divisor in OM representation whose support is the places associated to the prime ideals which divide $I$ with coefficients the negative of the valuation of $I$ at the prime ideals. ### `OMDivisor(z): FldFunElt -> OMDiv` Given a non zero element $z$ of a function field, return the principal divisor $\sum_P v_P(z) P$ in OM representation. ### `PoleDivisor(a): FldFunElt -> OMDiv` Given a non zero element $a$ of a function field, return the divisor $\sum_{v_P(a)<0} -v_P(a) P$ in OM representation. ### `ZeroDivisor(a): FldFunElt -> OMDiv` Given a non zero element $a$ of a function field, return the divisor $\sum_{v_P(a)>0} v_P(a) P$ in OM representation. ### `OMDivisorOfDegreeOne(F): FldFun -> OMDiv` Given a global function field $F$ returns a divisor of $F$ having degree 1. ### `ReferenceDivisor(F): FldFun -> OMDiv` Given a function field $F$, return, if set, the divisor $B$ in OM representation used to reduce a divisor $D$ of $F$ to $\tilde D$ such $D = \tilde D+kB + (a)$. ### `OMDivisor(D): DivFunElt -> OMDiv` Given a divisor $D$ of a function field of type `DivFunElt` compute an OM representation of $D$. ### `Example: Om Div Constr (ex-42c816)` ```magma > Fq := GF(13); > A := PolynomialRing(Fq); > Ax := PolynomialRing(A); > f := x^5+(t+1)^16*x^4+ x^2+t^3*x+t^6; > F := FunctionField(f); > Montes(F,t); > Ifin := F`PrimeIdeals[t,1]; > D := OMDivisor(Ifin); > D; Free repr. of divisor: -1*P(t,1) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1) Infinite ideal = ---------------------------------------- > Iinf := PrimesAtInfinity(F)[1]; > D := OMDivisor(Ifin,Iinf); > D; Free repr. of divisor: -1*P(t,1)+( -1*P(1/t,1) ) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1) Infinite ideal = P(t,1) ---------------------------------------- > D := OMDivisor(F.1); > D; Principal divisor generated by: 12/t^6*$.1^4 + (12*t^16 + 10*t^15 + 10*t^14 + 12*t^13 + 12*t^3 + 10*t^2 + 10*t + 12)/t^6*$.1^3 + 12/t^6*$.1 + 12/t^3 ---------------------------------------- Free repr. of divisor: 3*P(t,1)+ 3*P(t,2)+( 5*P(1/t,1)+ 5*P(1/t,2)+ -16*P(1/t,3) ) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1)^-3*P(t,2)^-3 Infinite ideal = P(t,1)^-5*P(t,2)^-5*P(t,3)^16 ---------------------------------------- > E := PoleDivisor(F.1); > E; Free repr. of divisor: 16*P(1/t,3) ---------------------------------------- Ideal repr. of divisor: Finite ideal = Infinite ideal = P(t,3)^-16 ---------------------------------------- > C := ZeroDivisor(F.1); > C; Free repr. of divisor: 3*P(t,1)+ 3*P(t,2)+( 5*P(1/t,1)+ 5*P(1/t,2) ) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1)^-3*P(t,2)^-3 Infinite ideal = P(t,1)^-5*P(t,2)^-5 ---------------------------------------- > E := Divisor(F.1); > D := OMDivisor(E); > D; Free repr. of divisor: 3*P(t,1)+ 3*P(t,2)+( 5*P(1/t,1)+ 5*P(1/t,2)+ -16*P(1/t,3) ) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1)^-3*P(t,2)^-3 Infinite ideal = P(t,1)^-5*P(t,2)^-5*P(t,3)^16 ---------------------------------------- ``` ## Arithmetic with Divisors in OM representation ### `k * D: RngIntElt, OMDiv -> OMDiv` Given an integer $k$ and a divisor $D$ in OM representation return the scalar multiple $k D$. ### `D1 + D2: OMDiv, OMDiv -> OMDiv` Given divisors $D1$ and $D2$ in OM representation return their sum also in OM representation. ### `D1 - D2: OMDiv, OMDiv -> OMDiv` Given divisors $D1$ and $D2$ in OM representation return their difference also in OM representation. ### `GCD(D1, D2): OMDiv, OMDiv -> OMDiv` ### `Gcd(D1, D2): OMDiv, OMDiv -> OMDiv` ### `GreatestCommonDivisor(D1, D2): OMDiv, OMDiv -> OMDiv` Given divisors $D1$ and $D2$ in OM representation return their GCD in OM representation. ### `Example: Om Div Arith (ex-31985e)` ```magma > Fq := GF(13); > A := PolynomialRing(Fq); > Ax := PolynomialRing(A); > f := x^5+(t+1)^16*x^4+ x^2+t^3*x+t^6; > F := FunctionField(f); > Montes(F,t); > Ifin := F`PrimeIdeals[t,1]; > D := OMDivisor(Ifin); > E := PoleDivisor(F.1); > C := ZeroDivisor(F.1); > D + E; Free repr. of divisor: -1*P(t,1)+( 16*P(1/t,3) ) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1) Infinite ideal = P(t,3)^-16 ---------------------------------------- > D - C; Free repr. of divisor: -4*P(t,1)+ -3*P(t,2)+( -5*P(1/t,1)+ -5*P(1/t,2) ) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1)^4*P(t,2)^3 Infinite ideal = P(t,1)^5*P(t,2)^5 ---------------------------------------- > 5*C; Free repr. of divisor: 15*P(t,1)+ 15*P(t,2)+( 25*P(1/t,1)+ 25*P(1/t,2) ) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1)^-15*P(t,2)^-15 Infinite ideal = P(t,1)^-25*P(t,2)^-25 ---------------------------------------- > GCD(D, C); Free repr. of divisor: -1*P(t,1) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1) Infinite ideal = ---------------------------------------- > GCD(D, E); Free repr. of divisor: ---------------------------------------- Ideal repr. of divisor: Finite ideal = Infinite ideal = ---------------------------------------- ``` ## Predicates on Divisors in OM representation ### `IsEffective(D): OMDiv -> BoolElt` ### `IsPositive(D): OMDiv -> BoolElt` Given a divisor $D$ in OM representation, return whether the valuation of $D$ at every prime in its support is non negative. ### `IsPrincipal(D): OMDiv -> BoolElt` Given a divisor $D$ in OM representation, return whether $D$ was constructed as the divisor of an element. ### `D1 eq D2: OMDiv, OMDiv -> BoolElt` Given divisors $D1$ and $D2$ in OM representation, return whether they are equal. ### `Example: Om Div Pred (ex-c8e530)` Continuing with the previous example : ```magma > IsPrincipal(OMDivisor(F.1)); true > IsPositive(OMDivisor(F.1)); false > IsPositive(ZeroDivisor(F.1)); true > ZeroDivisor(F.1) eq OMDivisor(F.1); false ``` ## Other Operations on Divisors in OM representation ### `Divisor(D): OMDiv -> DivFunElt` Given a divisor $D$ in OM representation return a divisor of type `DivFunElt`. ### `Valuation(D, P): OMDiv, OMIdl -> RngIntElt` Given a divisor $D$ in OM representation and a prime ideal $P$ in OM representation, return the valuation of $D$ at $P$. ### `Support(D): OMDiv -> [OMIdl], [RngIntElt]` Given a divisor $D$ in OM representation, return a sequence of primes at which $D$ has non zero valuation and a sequence of the valuations that $D$ has at those primes. ### `Height(D): OMDiv -> RngIntElt` ### `CanonicalHeight(D): OMDiv -> RngIntElt` Given a divisor $D$ in OM representation, return the sum of the products of the absolute values of the valuations returned by `Support` with the degree of the primes in the `Support` of $D$ and the maximum degree of the primes in the support of $D$. ### `Degree(D): OMDiv -> RngIntElt` Given a divisor $D$ in OM representation, return the sum of the products of the valuations returned by `Support` with the degrees of the primes in the support of $D$. ### `Basis(D): OMDiv -> SeqEnum` Given a divisor $D$ in OM representation, return a basis of the Riemann–Roch space of $D$. ### `Dimension(D): OMDiv -> RngIntElt` Given a divisor $D$ in OM representation, return the dimension of its Riemann–Roch space. ### `ReducedBasis(D): OMDiv -> SeqEnum` Given a divisor $D$ in OM representation returns a reduced basis for the lattice corresponding to $D$, that is, the lattice given by its finite ideal. ### `SemiReducedBasis(D): OMDiv -> SeqEnum` Given a divisor $D$ in OM representation returns a semi-reduced basis for the lattice corresponding to $D$, that is, the lattice given by its finite ideal. ### `SuccessiveMinima(D): OMDiv -> SeqEnum` Given a divisor $D$ in OM representation returns the successive minima of the lattice given by the finite ideal of $D$. ### `ApproximatedSuccessiveMinima(D): OMDiv -> SeqEnum` Given a divisor $D$ in OM representation returns the ceiling of the successive minima of the lattice given by the finite ideal of $D$. ### `Example: Om Div Ops (ex-e8bf40)` Continuing with the previous example : ```magma > Montes(F, t); > Ifin := F`PrimeIdeals[t,1]; > D := OMDivisor(Ifin); > D; Free repr. of divisor: -1*P(t,1) ---------------------------------------- Ideal repr. of divisor: Finite ideal = P(t,1) Infinite ideal = ---------------------------------------- > Support(D); [ OM prime ideal over t of Algebraic function field defined over Univariate rational function field over GF(13) by x^5 + (t^16 + 3*t^15 + 3*t^14 + t^13 + t^3 + 3*t^2 + 3*t + 1)*x^4 + x^2 + t^3*x + t^6 having residual degree 1 and ramification index 1 Last phi polynomial is x + 10*t^3 ] [ -1 ] > Height(D); 1 1 > Degree(D); -1 > Basis(OMDivisor(F.1)); [ 12/t^6*F.1^4 + (12*t^16 + 10*t^15 + 10*t^14 + 12*t^13 + 12*t^3 + 10*t^2 + 10*t + 12)/t^6*F.1^3 + 12/t^6*F.1 + 12/t^3 ] > Dimension(OMDivisor(F.1)); 1 > ReducedBasis(OMDivisor(F.1)); [ 12/t^6*F.1^4 + (12*t^16 + 10*t^15 + 10*t^14 + 12*t^13 + 12*t^3 + 10*t^2 + 10*t + 12)/t^6*F.1^3 + 12/t^6*F.1 + 12/t^3, 1/t^3*F.1^3 + (t^16 + 3*t^15 + 3*t^14 + t^13 + t^3 + 3*t^2 + 3*t + 1)/t^3*F.1^2 + 1/t^3, 12*F.1^2 + (12*t^16 + 10*t^15 + 10*t^14 + 12*t^13 + 12*t^3 + 10*t^2 + 10*t + 12)*F.1, F.1 + t^16 + 3*t^15 + 3*t^14 + t^13 + t^3 + 3*t^2 + 3*t + 1, F.1^2 + (t^16 + 3*t^15 + 3*t^14 + t^13 + t^3)*F.1 + 10*t^18 + t^17 + 7*t^16 + 11*t^15 + 7*t^14 + 12*t^13 + 10*t^5 + t^4 + 7*t^3 + 11*t^2 + 7*t ] > SemiReducedBasis(OMDivisor(F.1)); [ 12/t^6*F.1^4 + (12*t^16 + 10*t^15 + 10*t^14 + 12*t^13 + 12*t^3 + 10*t^2 + 10*t + 12)/t^6*F.1^3 + 12/t^6*F.1 + 12/t^3, 1/t^3*F.1^3 + (t^16 + 3*t^15 + 3*t^14 + t^13 + t^3 + 3*t^2 + 3*t + 1)/t^3*F.1^2 + 1/t^3, 12*F.1^2 + (12*t^16 + 10*t^15 + 10*t^14 + 12*t^13 + 12*t^3 + 10*t^2 + 10*t + 12)*F.1, F.1 + t^16 + 3*t^15 + 3*t^14 + t^13 + t^3 + 3*t^2 + 3*t + 1, F.1^2 + (t^16 + 3*t^15 + 3*t^14 + t^13 + t^3)*F.1 + 10*t^18 + t^17 + 7*t^16 + 11*t^15 + 7*t^14 + 12*t^13 + 10*t^5 + t^4 + 7*t^3 + 11*t^2 + 7*t ] > SuccessiveMinima(OMDivisor(F.1)); [ 0, 11/2, 11, 27/2, 16 ] > ApproximatedSuccessiveMinima(OMDivisor(F.1)); [ 0, 6, 11, 14, 16 ] ```