/************************************************************************** thickness.m William Stein December 12, 1999 There is a computation that might be easy for you to make, that might shed light on whether or not the topological closure of the set of modular Galois representations (in the universal deformation space) contains an open set. Tell me whether or not this is feasible. Start, say, with the Fourier expansion mod 3 of the newform of weight two of level 11 (call it w) but I want to ignore the Euler factor at 3, i.e., I want to ignore the Fourier coefficents a_n where n is divisible by 3. Now look for eigenforms of arbitrary weight, of level 33, whose Fourier coefficients are in Z_3 and are congruent to those of w modulo 3 (ignoring the Fourier coefficents a_n where n is divisible by 3). Every time you find one, RECORD its Fourier expansion modulo 9 (again ignoring the Fourier coefficents a_n where n is divisible by 3). Count the number of these ( Fourier expansion modulo 9 ignoring the Fourier coefficents a_n where n is divisible by 3) that you get. Do you get 27 of them? ***********************************************************************/ /* Step 1: Start, say, with the Fourier expansion mod 3 of the newform of weight two of level 11 (call it w) but I want to ignore the Euler factor at 3, i.e., I want to ignore the Fourier coefficents a_n where n is divisible by 3. */ prec := 37; "Precision: ", prec; function To_pAdic(g,p,q) return &+[pAdicRing(p)!(Integers()!Coefficient(g,n))*q^n : n in [1..Degree(g)]]; end function; R<q> := PowerSeriesRing(pAdicRing(3)); w := To_pAdic(qEigenform(ModularFactor("11k2A"),prec), 3, q); "w = ", w; /* Step 2: Now look for eigenforms of arbitrary weight, of level 33, whose Fourier coefficients are in Z_3 and are congruent to those of w modulo 3 (ignoring the Fourier coefficents a_n where n is divisible by 3). */ function ValPrimeTo(f,p) if f eq 0 then return 99999999999; end if; v, pos := Min([Valuation(Coefficient(f,i)) : i in [1..Degree(f)] | (i mod p ne 0) and Valuation(Coefficient(f,i)) ne 0]); return v; end function; function IsCongruentToW(f) return ValPrimeTo(f-w,3) ge 1; end function; function pDeprive(f,p) R<q> := Parent(f); for i in [1..Ceiling(Degree(f)/3)] do f -:= Coefficient(f,3*i)*q^(3*i); end for; return f; end function; function Level33Search(k1, k2) R9 := PowerSeriesRing(Integers(9)); RZ := PowerSeriesRing(Integers()); nearby_forms := [R9|]; for k in [w : w in [k1..k2] | IsEven(w)] do "k = ",k; v := ZpRationalNewforms(11,k,3,prec) cat ZpRationalNewforms(33,k,3,prec); "v = ", v; for f in v do if IsCongruentToW(f) then g := pDeprive(R9!(RZ!f),3); Append(~nearby_forms, g); "g = ",g; end if; end for; end for; return nearby_forms; end function;