CoCalc Public Fileswww / talks / harvard-talk-2004-12-08 / e2heights / deleted
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1intrinsic formal_sigma_in_s2_2(E::CrvEll, prec::RngIntElt) -> RngSerElt
2{}
3   m := 2;
4   rels := Eltseq(sigma_relation(E,m,prec,prec+2));
5   S := Parent(rels[1]);
6   i := 1;
7   seen := [2];
8   numvars := prec;
9   known := [S.i : i in [1..numvars]];
10   for r in rels do
11      mon := Monomials(r);
12      cof := Coefficients(r);
13      z := 0;
14      for j in [1..#mon] do
15         for i in [1..numvars] do
16            if mon[j] eq S.i and not (i in seen) then
17               z := &+[S|-mon[k]*cof[k]/cof[j] : k in [1..#mon] | k ne j];
18               //print "\nfor var s",i," have expression\n",z;
19               phi := hom<S -> S | known>;
20               val := phi(z);
21               //printf "\ns[%o] := %o;\n", i, val;
22               //print Valuation(LCM([1] cat [Denominator(x) : x in Coefficients(val)]) , 5);
23               known[i] := val;
24               Append(~seen,i);
25            end if;
26         end for;
27      end for;
28   end for;
29   R<s2> := PolynomialRing(RationalField());
30   z := [R|0 : i in [1..numvars]];
31   z[2] := s2;
32   phi := hom<S -> R | z>;
33   S<t> := PowerSeriesRing(R);
34   return S!([0,1] cat [phi(c) : c in known]) + O(t^(#known+1));
35end intrinsic;
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