It is not at all clear to me what your (first attached) paper is claiming to do. The "Conclusion" section seems unconnected to the rest. Sage already has things like this: sage: E = EllipticCurve([0,0,0,12,34]); E Elliptic Curve defined by y^2 = x^3 + 12*x + 34 over Rational Field sage: xp = E.weierstrass_p() sage: yp = xp.derivative()/2 sage: xp z^-2 - 12/5*z^2 - 34/7*z^4 + 48/25*z^6 + 1224/385*z^8 + 88052/79625*z^10 - 3264/1925*z^12 - 17082432/14889875*z^14 + 29443592/116491375*z^16 + 3036110016/4094715625*z^18 + O(z^20) sage: yp -z^-3 - 12/5*z - 68/7*z^3 + 144/25*z^5 + 4896/385*z^7 + 88052/15925*z^9 - 19584/1925*z^11 - 17082432/2127125*z^13 + 235548736/116491375*z^15 + 27324990144/4094715625*z^17 + O(z^19)
sage: E.defining_polynomial()([xp,yp,1]) O(z^16) and sage: G = E.formal_group() sage: G.group_law() t1 + t2 - 24*t1^4*t2 - 48*t1^3*t2^2 - 48*t1^2*t2^3 - 24*t1*t2^4 - 102*t1^6*t2 - 306*t1^5*t2^2 - 510*t1^4*t2^3 - 510*t1^3*t2^4 - 306*t1^2*t2^5 - 102*t1*t2^6 - 288*t1^8*t2 + 1152*t1^6*t2^3 + 2304*t1^5*t2^4 + 2304*t1^4*t2^5 + 1152*t1^3*t2^6 - 288*t1*t2^8 + O(t1, t2)^10 John Cremona On 24 February 2016 at 08:23, Padma Ramanathan <[email protected]> wrote: > We are attaching two preprints where we have connected the Weierstrass p - > function to the modular parametrization which is the heart of the Shimura - > Taniyama - Weil conjecture. It is based on well known papers in various sub > branches of the theory of elliptic curves. We have shown that a corollary of > certain papers is the possibility of connecting Bernoulli - Hurwitz numbers > to the L-function in analogy with the Bernoulli numbers and their > connection to the Riemann zeta -function. This is first developed in a paper > by Hurwitz where elliptic functions and circular functions are developed > analogously. The most general paper (to the best of our knowledge) is by > Clarke where he connects generalized Bernoulli numbers to the L-function. > His idea stated simply is the formal exponential is the generating function > of the generalized Bernoulli numbers B_n and the formal logarithm is the > generating function of the a_n which appear in the L-function. You may > wonder why he did not think of STW. The answer is they were all trying to > generalize the von Staudt theorem. > > In the second paper we use well known results including the formal logarithm > to construct explicitly the modular parametrization. > > It would be nice to have a package attached to SAGE which implements these > ideas. > > -- > You received this message because you are subscribed to the Google Groups > "sage-nt" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To post to this group, send email to [email protected]. > Visit this group at https://groups.google.com/group/sage-nt. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send an email to [email protected]. Visit this group at https://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
