Opened 13 months ago
Closed 10 months ago
#32258 closed defect (fixed)
Adjustments for the 2adic lseries of elliptic curves
Reported by:  wuthrich  Owned by:  

Priority:  minor  Milestone:  sage9.5 
Component:  elliptic curves  Keywords:  elliptic curves, padic Lseries 
Cc:  loeffler  Merged in:  
Authors:  Chris Wuthrich  Reviewers:  Frédéric Chapoton 
Report Upstream:  N/A  Work issues:  
Branch:  7b7f6ad (Commits, GitHub, GitLab)  Commit:  7b7f6ad24620b744d4decd6dcf648bd479762948 
Dependencies:  Stopgaps: 
Description
This is a minor bug that I am sure to be the first one to notice. Currently one cannot evaluate the (unique) Teichmuller twist of the 2adic Lseries of an elliptic curve. Furthermore the odd powers of the Teichmuller twists are incorrectly normalised. This is harmless except for p=2.
Change History (10)
comment:1 Changed 13 months ago by
comment:2 Changed 13 months ago by
 Branch set to u/wuthrich/ticket_32258
 Commit set to f073844d8b3878b679d7858c03da999564dfca7a
But I couldn't do any testing yet or adjusting doctests.
New commits:
f073844  adjust 2adic lseries for elliptic curves

comment:3 Changed 13 months ago by
I will record here, why the change in the normalisation of the odd powers of the Teichmüller twists are now correct.
Suppose E has good ordinary reduction at p. Let L_p(E) be the padic Lseries of E as sage returns it. If p=2 or p%4 = 1, then the padic Lseries twisted by the (p1)/2th power of the Teichmüller corresponds to the twist by the quadratic character chi by 1 or p respectively. Let L_p(E,chi) to be this padic Lseries returned by taking eta=1
or eta=(p1)/2
respectively. The constant term of L_p(E,chi) is equal to 1/alpha^u * L(E,chi,1) * Gausssum(chi)/ Omega_minus
with u=2 if p=2 and u=1 if p>2. The question is now why is Omega_minus
equal to the smallest positive multiple of i that is in the period lattice of E.
The main conjecture says that L_p(E) and the twisted L_p(E,chi) are the characteristic series of Selmer groups. Greenberg has calculated the constant term even in the case p=2. Suppose E has rank 0 over K = Q(i) or K = Q(sqrt(p) respectively. The main conjecture now predict L_p(E)(0) * L_p(E,chi)(0)
and (11/alpha)^2 * 1/alpha^u * prod c_v(E/K) * Sha(E/K)/ E(K)^2
, where the product runs over all FINITE Tamagawa numbers, have the same padic valuation. The padic BSD of E/K predicts the equality of these two padic numbers.
The following function should therefore return a padic approximation of a power of 4 if the conditions above are satisfied.
def ch(E,p): """ check curve E at p, should return a power of 4. EXAMPLE:: sage: ch(EllipticCurve("443c1"), 13) 1 + O(13^5) sage: ch(EllipticCurve("443c1"), 2) 2^2 + O(2^8) """ if p == 2: u = 2 D = 1 po = 1 k = 8 else: u = 1 D = p po = (p1)//2 k = 4 if p<6 else 3 lp = E.padic_lseries(p,implementation="num") l0 = lp.series(k, prec=3) l1 = lp.series(k, eta = po, prec=3) ct = l0.list()[0] * l1.list()[0] al = lp.alpha() eps = (11/al)**2 * 1/al**u K = QuadraticField(D) EK = E.base_extend(K) Ed = E.quadratic_twist(D) tam = EK.tamagawa_product() sh = E.sha().an() * Ed.sha().an() # that is not SHA/K because of 2torsion. tor = EK.torsion_order() return ct/eps/tam/sh*tor**2
... and I have checked it on quite a number of curves with both rectamgular and nonrectangular period lattice.
comment:4 Changed 13 months ago by
 Commit changed from f073844d8b3878b679d7858c03da999564dfca7a to 56202355f5df69bd015014ade3a7a25732138240
Branch pushed to git repo; I updated commit sha1. New commits:
5620235  trac 32258: add doctest

comment:5 Changed 13 months ago by
 Cc loeffler added
 Status changed from new to needs_review
comment:6 Changed 12 months ago by
 Milestone changed from sage9.4 to sage9.5
comment:7 Changed 11 months ago by
there is a typo in "charachters"
comment:8 Changed 11 months ago by
 Commit changed from 56202355f5df69bd015014ade3a7a25732138240 to 7b7f6ad24620b744d4decd6dcf648bd479762948
comment:9 Changed 11 months ago by
 Reviewers set to Frédéric Chapoton
 Status changed from needs_review to positive_review
I will assume that the math is correct => positive review.
comment:10 Changed 10 months ago by
 Branch changed from u/wuthrich/ticket_32258 to 7b7f6ad24620b744d4decd6dcf648bd479762948
 Resolution set to fixed
 Status changed from positive_review to closed
Expect a correction soon.