Adds the MobiusRandomize parameter

CHANGELOG.md

@@ -1,6 +1,7 @@
 ### v1.2.0
 - Adds the global proof method, so now the `Proof` parameter can take the values `"Local"` (equivalent to `true`) and `"Global"`.
 - The `UseOgg` parameter works over number fields now.
+- Adds the `MobiusRandomize` parameter.
 - We use the `UseNP` option to `Factorization`, since it's quicker.
 - Removes sloping from a call to `IsHenselLiftable`, since it's quicker.
 

EvenConductorExponent_ExactpAdics.mag

@@ -17,6 +17,7 @@ function find_3torsion(f, p
   , Strategy:=[* MinPrec, <"limit", MaxPrec>, <"randomize", Randomize>, <"double">*]
   , Catch:=false
   , MobiusSkip:=0
+  , MobiusRandomize:=0
   , Proof:=true
   , UseNP:=true
   , Time:=false
@@ -71,20 +72,26 @@ function find_3torsion(f, p
     vprint EvenConductorExponent_Genus2: "finding equations...";
     R<B0, B1, B2, B3, C0, C1, C2> := PolynomialRing(F, 7);
     ntries := 0;
-    mobs := [[1,0,0,1]];
-    mobs_lim := 0;
+    if MobiusRandomize le 0 then
+      mobs := [[1,0,0,1]];
+      mobs_lim := 0;
+    else
+      mobs := [];
+      mobs_lim := MobiusRandomize;
+    end if;
     while true do
       ntries +:= 1;
       if ntries le MobiusSkip then
         continue;
       end if;
       // ensure there exists a mobius transformation
-      while mobs_lim lt ntries do
+      while #mobs lt ntries do
         mobs_lim +:= 1;
-        mobs cat:= SetToSequence({[a,b,c,d] : a,b,c,d in [-mobs_lim..mobs_lim] | a*d-b*c eq 1 and (c eq 0 or Evaluate(f, a/c) ne 0)} diff SequenceToSet(mobs));
+        mobs cat:= SetToSequence({[a,b,c,d] : a,b,c,d in [-mobs_lim..mobs_lim] | Max([Abs(x) : x in [a,b,c,d]]) eq mobs_lim and a*d-b*c eq 1 and (c eq 0 or Evaluate(f, a/c) ne 0)} diff SequenceToSet(mobs));
       end while;
       // get a defining polynomial
       r`moebius := mobs[ntries];
+      vprint EvenConductorExponent_Genus2, 2: "moebius =", r`moebius;
       a,b,c,d := Explode(r`moebius);
       f2 := &+[Coefficient(f, i) * Polynomial([F|b,a])^i * Polynomial([F|d,c])^(6-i) : i in [0..6]];
       assert Degree(f2) in [5,6];
@@ -343,7 +350,7 @@ function prime_ideal(K, p)
   return pp;
 end function;
 
-function evencondexpdata(C, p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true)
+function evencondexpdata(C, p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0)
   error if Genus(C) ne 2, "Argument 1 must have genus 2";
   error if Type(UseRegularModels) ne BoolElt, "UseRegularModels must be a boolean";
   error if Type(UseOgg) ne BoolElt, "UseOgg must be a boolean";
@@ -358,14 +365,14 @@ function evencondexpdata(C, p : UseRegularModels:=true, MaximumPrecision:=Infini
   pp := ideal<OK | p>;
   error if not (IsPrime(pp) and IsDivisibleBy(AbsoluteNorm(pp), 2)), "Arugment 2 must be (coercible to) a prime over Argument 1 above 2";
   f := crvpol(C);
-  r := find_3torsion(f, pp : MaxPrec:=MaximumPrecision, Proof:=Proof);
+  r := find_3torsion(f, pp : MaxPrec:=MaximumPrecision, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
   dues := [<Degree(L,K), Max(UpperBreaks(TransitionFunction(L,K))), RamificationDegree(L,K)> where L:=x`M : x in r`factors] where K:=r`K;
   s := conductor_exponent_data(f, pp, dues : UseRegularModels:=UseRegularModels, ExtraData:=r);
   return s;
 end function;
 
-function evencondexp(C, p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true)
-  r := evencondexpdata(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof);
+function evencondexp(C, p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0)
+  r := evencondexpdata(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
   if assigned r`Exponent then
     return r`Exponent;
   else
@@ -377,34 +384,34 @@ function is_even(p)
   return IsDivisibleBy(AbsoluteNorm(p), 2);
 end function;
 
-intrinsic EvenConductorExponent_Genus2(C :: CrvHyp[FldRat], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> RngIntElt
+intrinsic EvenConductorExponent_Genus2(C :: CrvHyp[FldRat], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> RngIntElt
   {The conductor exponent of C at prime p.}
-  return evencondexp(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof);
+  return evencondexp(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
 end intrinsic;
 
-intrinsic EvenConductorExponent_Genus2(C :: CrvHyp[FldNum], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> RngIntElt
+intrinsic EvenConductorExponent_Genus2(C :: CrvHyp[FldNum], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> RngIntElt
   {"}
-  return evencondexp(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof);
+  return evencondexp(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
 end intrinsic;
 
-intrinsic EvenConductorExponent_Genus2(C :: CrvHyp[FldRat] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> RngIntElt
+intrinsic EvenConductorExponent_Genus2(C :: CrvHyp[FldRat] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> RngIntElt
   {The conductor exponent of C at 2.}
-  return evencondexp(C, 2 : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof);
+  return evencondexp(C, 2 : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
 end intrinsic;
 
-intrinsic EvenConductorExponentData_Genus2(C :: CrvHyp[FldRat], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> Rec
+intrinsic EvenConductorExponentData_Genus2(C :: CrvHyp[FldRat], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> Rec
   {Data relating to the conductor exponent of C at prime p.}
-  return evencondexpdata(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof);
+  return evencondexpdata(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
 end intrinsic;
 
-intrinsic EvenConductorExponentData_Genus2(C :: CrvHyp[FldNum], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> Rec
+intrinsic EvenConductorExponentData_Genus2(C :: CrvHyp[FldNum], p : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> Rec
   {"}
-  return evencondexpdata(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof);
+  return evencondexpdata(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
 end intrinsic;
 
-intrinsic EvenConductorExponentData_Genus2(C :: CrvHyp[FldRat] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> Rec
+intrinsic EvenConductorExponentData_Genus2(C :: CrvHyp[FldRat] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> Rec
   {Data relating to the conductor exponent of C at 2.}
-  return evencondexpdata(C, 2 : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof);
+  return evencondexpdata(C, 2 : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize);
 end intrinsic;
 
 intrinsic OddConductor(C :: CrvHyp[FldRat]) -> RngIntElt
@@ -421,16 +428,16 @@ intrinsic OddConductor(C :: CrvHyp[FldNum]) -> .
   return &*[p^Conductor(C, p) : fac in Factorization(ideal<OK | Numerator(D) * Denominator(D)>) | not is_even(p) where p:=fac[1]];
 end intrinsic;
 
-intrinsic Conductor_Genus2(C :: CrvHyp[FldRat] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> RngIntElt
+intrinsic Conductor_Genus2(C :: CrvHyp[FldRat] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> RngIntElt
   {The conductor of C.}
   D := Discriminant(C);
-  return &*[Z | p^(p eq 2 select EvenConductorExponent_Genus2(C : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof) else Conductor(C, p)) : p in PrimeDivisors(Numerator(D) * Denominator(D))];
+  return &*[Z | p^(p eq 2 select EvenConductorExponent_Genus2(C : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize) else Conductor(C, p)) : p in PrimeDivisors(Numerator(D) * Denominator(D))];
 end intrinsic;
 
-intrinsic Conductor_Genus2(C :: CrvHyp[FldNum] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true) -> RngOrdIdl
+intrinsic Conductor_Genus2(C :: CrvHyp[FldNum] : UseRegularModels:=true, MaximumPrecision:=Infinity(), UseOgg:=true, Proof:=true, MobiusRandomize:=0) -> RngOrdIdl
   {The conductor of C.}
   D := Discriminant(C);
   F := BaseField(C);
   OF := Integers(F);
-  return &*[PowerIdeal(OF) | p^exponent where exponent := is_even(p) select EvenConductorExponent_Genus2(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof) else Conductor(C, p) where p:=fac[1] : fac in Factorization(ideal<OF | Numerator(D)*Denominator(D)>)];
+  return &*[PowerIdeal(OF) | p^exponent where exponent := is_even(p) select EvenConductorExponent_Genus2(C, p : UseRegularModels:=UseRegularModels, MaximumPrecision:=MaximumPrecision, UseOgg:=UseOgg, Proof:=Proof, MobiusRandomize:=MobiusRandomize) else Conductor(C, p) where p:=fac[1] : fac in Factorization(ideal<OF | Numerator(D)*Denominator(D)>)];
 end intrinsic;

README.md

@@ -60,6 +60,7 @@ The conductor of `C`, a hyperelliptic curve defined over the rationals or a numb
 - `UseRegularModels := true`: If necessary, a regular model will be computed in order to compute the tame part of the conductor. Occasionally, this step can take forever and so be skipped by setting this parameter to false. This can cause the intrinsic to raise an error if regular models were really needed.
 - `UseOgg := true`: Use Ogg's formula to compute conductor exponents, if possible (i.e. if the discriminant has valuation less than 12).
 - `MaximumPrecision := Infinity()`: The underlying algorithm works with p-adic numbers up to some precision determined by this. Setting it to a finite value will sometimes cause the intrinsic to raise an error, but avoids a potential source of infinite loop. Mainly for debugging.
+- `MobiusRandomize := 0`: By default, the algorithm first tries the identity Mobius transformation before trying ones with larger coefficients. Setting this to an integer greater than 0 will skip the identity and start with transformations with coefficients of size up to this. If the algorithm ever appears to be stuck at a factorization step, try setting this to a higher value — e.g. first try 2, then 3, and so on.
 
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 ```