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Tip revision: f69d0e06f67b9ac6f57e7d8e6ba3b3d69e650352 authored by Maarten Derickx on 02 November 2020, 22:43:35 UTC
Quickly lists all non cuspidal places up to diamond operators on X_1(N)
Tip revision: f69d0e0

function X_1_n(n,base_ring : equation_directory:="models_X1_n")
//Input: n - integer
//       base_ring - a ring
//       equation_directory - directory with files FFFc<n>.txt containing models
//Output: C - a curve
//Returns an algebraic model C of the modular curve X_1(m,n) as a curve over base_ring

    n_str := IntegerToString(n);
    file_name := equation_directory cat "/FFFc" cat n_str cat ".txt";
    data := Read(file_name);

    A<x,y> := AffineSpace(base_ring,2);
    X := eval(data);
    C := Curve(A,X);
    return ProjectiveClosure(C);
end function;

function Functions_xyrsbcF2F3(curve)
    //Input: curve - the modular curve X_1(N) as returned by the function X_1
    //Output: x,y,r,s,b,c,F2,F3 - The modular units as in
    FF := FunctionField(curve);
    x := FF.1;
    y := FF.2;
    r := (x^2*y-x*y+y-1)/x/(x*y-1);
    s := (x*y-y+1)/x/y;
    b := r*s*(r-1);
    c := s*(r-1);
    F3 := b;
    F2 := b/(16*b^2+(1-20*c-8*c^2)*b + c*(c-1)^3);
    return x,y,r,s,b,c,F2,F3;
end function;

function TateNormalForm_bc(E,P);
//Return the b,c of the tate normal form of (E,P) as in equation (2) of
    assert P[3] eq 1;



    return [b,c];
end function;

function TateNormalForm_xy(E,P);
//return the x,y of the tate normal form of (E,P) as in section 2.1
    b,c := Explode(TateNormalForm_bc(E,P));
    r := b/c;
    s := c^2/(b-c);
    t := (r*s-2*r+1);
    x := (s-r)/t;
    y := t/(s^2-s-r+1);
    return [x,y];
end function;

function EllipticCurveFromX1Place(P);
//Returns the associated elliptic curve corresponding to a place on X1N
//the elliptic curve is guaranteed to be in tate normal form, so that
//0,0 is the point of order N. The point 0,0 is returned as optional second element
    X1N := Curve(P);
    return E, E ! [0,0];
end function;

function X1PlaceFromEllipticCurve(X1N, E, P)
//Returns a place on X_1(N) given an elliptic curve and a point of order N
    K := BaseRing(E);
    xy := TateNormalForm_xy(E,P);
    dP := Places(X1N(K) ! xy);
    assert #dP eq 1;
    return dP[1];
end function;

function ElementsUpToFrobenius(F)
  orbits := {{Frobenius(x,i) : i in [1..Degree(F)]}: x in F};
  return [Random(orbit) : orbit in orbits];
end function;

function EllipticCurvesOverField(F)
  return &cat[Twists(EllipticCurveFromjInvariant(j)) : j in F];
end function;

function EllipticCurvesOverFieldUpToFrobenius(F)
  return &cat[Twists(EllipticCurveFromjInvariant(j)) : j in ElementsUpToFrobenius(F)];
end function;

function NonCuspidalPlacesUpToDiamond(C,i,N)
  assert IsPrime(N);
  p := Characteristic(BaseRing(C));
  F := GF(p,i);
  ECs := [E: E in EllipticCurvesOverFieldUpToFrobenius(F) | (#E(F) mod N) eq 0];
  places := [];
  for E in ECs do
    gens := [P*(Order(P) div N) : P in Generators(E(F)) | (Order(P) mod N) eq 0];
    assert #gens eq 1;
    time Append(~places,X1PlaceFromEllipticCurve(C,E,gens[1]));
  end for;
  return places;
end function;

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