https://github.com/koffie/mdmagma
Tip revision: 6997866a3bd828f081d181940c7d8804bac81ffc authored by Maarten Derickx on 02 September 2024, 19:58:11 UTC
Correctly set the new flag
Correctly set the new flag
Tip revision: 6997866
strange_characters.m
function NextPrimeInArithmeticProgression(p, a, b);
// the smallest prime q > p such that q mod a = b mod a
assert GCD(a,b) eq 1;
b := b mod a;
q := p;
repeat q := NextPrime(q); until q mod a eq b;
return q;
end function;
function UnitExponent(N)
// the exponent of the group (Z/NZ)^*
return Exponent(UnitGroup(Integers(N)));
end function;
function SuitableRootOfUnity(N : lowerbound := 0)
//
r := UnitExponent(N);
if lowerbound eq 0 then
lowerbound := r;
end if;
p := NextPrimeInArithmeticProgression(lowerbound, r, 1);
zeta := RootOfUnity(r, GF(p));
return zeta, r;
end function;
function NextSuitableRootOfUnity(zeta, r)
//
p := Characteristic(Parent(zeta));
p := NextPrimeInArithmeticProgression(p, r, 1);
zeta := RootOfUnity(r, GF(p));
return zeta, r;
end function;
function IsStrange(label : zeta := 0, r := 0, new := false)
chi := DirichletCharacter(label: zeta:=zeta, r:=r);
S := CuspidalSubspace(ModularSymbols(chi, 2, 1));
if new then
S := NewSubspace(S);
end if;
if Dimension(S) eq 0 then return false, 0, 0; end if;
W := WindingSubmodule(S);
dimS := Dimension(S);
dimW := Dimension(W);
return dimS ne dimW, dimW, dimS;
end function;
function StrangeInfo(N : lowerbound := 2^25, tries := 10, new := false, proof := true)
zeta, r := SuitableRootOfUnity(N : lowerbound := lowerbound);
potentially_strange_characters := [];
for label in ConreyCharacterOrbitReps(N) do
// skip the trivial character
if Order(label) eq 1 then continue; end if;
success := false;
for i in [1..tries] do
try
is_strange, dimW, dimS := IsStrange(label : zeta:=zeta, r:=r, new := new);
success := true;
break;
catch e
zeta, r := NextSuitableRootOfUnity(zeta, r);
end try;
end for;
error if not success, "computation did not succeed, increase the number of tries";
if is_strange and proof then
is_strange, dimW, dimS := IsStrange(label : zeta:=0, r:=0, new := new);
end if;
if is_strange then
Append(~potentially_strange_characters, <CharacterOrbitLabel(label), Order(label), dimS - dimW, dimS>);
end if;
end for;
return potentially_strange_characters;
end function;
function Quote(o)
return Sprintf("\"%o\"",o);
end function;
function AsListOfStrings(l)
return [Quote(x) : x in l];
end function;
function PrintStrangeInfoRange(a, b : lowerbound := 2^25, tries := 10, new := false, proof := true)
print "{";
separator := ",";
for N in [a..b] do
strange_info := StrangeInfo(N : lowerbound := lowerbound, tries := tries, new := new, proof := proof);
if N eq b then separator := ""; end if;
print Quote(N), ":", [AsListOfStrings(l) : l in strange_info], separator;
end for;
print "}";
return "";
end function;
if assigned start and assigned stop then
start := StringToInteger(start);
stop := StringToInteger(stop);
if not assigned new then
new := false;
else
new := new eq "true";
end if;
PrintStrangeInfoRange(start, stop : new := new);
exit;
end if;