attachment:lp_teichmuller.sage of days22/wuthrich

   1 # code to compute the series representation
   2 
   3 E = EllipticCurve('11a1')
   4 
   5 n=10; prec=10
   6 Lserie = E.padic_lseries(5)
   7 alp = Lserie.alpha(prec)
   8 MSP = E.modular_symbol()
   9 MSM = E.modular_symbol(sign=-1)
  10 
  11 p = Lserie._p
  12 bounds = Lserie._prec_bounds(n,prec)
  13 padic_prec = max(bounds[1:]) + 5
  14 res_series_prec=min(p**(n-1),prec)
  15 
  16 K = QQ
  17 gamma = K(1+p)
  18 R = PowerSeriesRing(K,'T',res_series_prec)
  19 T = R(R.gen(),res_series_prec)
  20 L = R(0) #going to be the L-series eventually
  21 one_plus_T_factor = R(1)
  22 gamma_power = K(1)
  23 teich = Lserie.teichmuller(padic_prec)
  24 pt = 2 # the power of Teichmuller
  25 p_power = p**(n-1)
  26 
  27 for j in range(p_power):
  28    s = K(0)
  29    for a in range(1,p):
  30        b = teich[a]*gamma_power
  31        if pt & 2 == 0:
  32            s += teich[a]^pt * ( (1/alp)^n * MSP((b)/(p^n) ) -
  33 (1/alp)^(n+1) * MSP((b)/(p^(n-1))) ) # mu(b + p^nZ_p)
  34        else:
  35            s += teich[a]^pt * ( (1/alp)^n * MSM((b)/(p^n) ) -
  36 (1/alp)^(n+1) * MSM((b)/(p^(n-1))) )
  37    L += s * one_plus_T_factor
  38    one_plus_T_factor *= (1+T)
  39    gamma_power *= gamma
  40 
  41 print 'L before p-adic approx is %s' % L
  42 
  43 K = Qp(p,padic_prec,print_mode='series')
  44 R = PowerSeriesRing(K,'T',res_series_prec)
  45 Lprime = R(L,res_series_prec)
  46 print 'Lprime is %s' % Lprime
  47 aj = Lprime.list()
  48 if len(aj) > 0:
  49       aj = [aj[0].add_bigoh(padic_prec-2)] +
  50 [aj[j].add_bigoh(bounds[j]) for j in range(1,len(aj))]
  51 L = R(aj,res_series_prec )
  52 if pt & 2 == 0:
  53    L = L / Lserie._E.real_components()
  54 else:
  55    L = L
  56 
  57 print 'The 5-adic L-function of 11A1 twisted by the (pt)-th power of
  58 the Teichmuller character is %s ' % L