## #32258 closed defect (fixed)

Reported by: Owned by: wuthrich minor sage-9.5 elliptic curves elliptic curves, p-adic L-series loeffler Chris Wuthrich Frédéric Chapoton N/A 7b7f6ad 7b7f6ad24620b744d4decd6dcf648bd479762948

### 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 2-adic L-series of an elliptic curve. Furthermore the odd powers of the Teichmuller twists are incorrectly normalised. This is harmless except for p=2.

### comment:1 Changed 13 months ago by wuthrich

Expect a correction soon.

### comment:2 Changed 13 months ago by wuthrich

• Authors set to Chris Wuthrich
• 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 2-adic lseries for elliptic curves`

### comment:3 Changed 13 months ago by wuthrich

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 p-adic L-series of E as sage returns it. If p=2 or p%4 = 1, then the p-adic L-series twisted by the (p-1)/2-th 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 p-adic L-series returned by taking `eta=1` or `eta=(p-1)/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 `(1-1/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 p-adic valuation. The p-adic BSD of E/K predicts the equality of these two p-adic numbers.

The following function should therefore return a p-adic 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 = (p-1)//2
k = 4 if p<6 else 3

l0 = lp.series(k, prec=3)
l1 = lp.series(k, eta = po, prec=3)
ct = l0.list() * l1.list()

al = lp.alpha()
eps = (1-1/al)**2 * 1/al**u

EK = E.base_extend(K)
tam = EK.tamagawa_product()
sh = E.sha().an() * Ed.sha().an() # that is not SHA/K because of 2-torsion.
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 non-rectangular period lattice.

### comment:4 Changed 13 months ago by git

• 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 wuthrich

• Status changed from new to needs_review

### comment:6 Changed 12 months ago by mkoeppe

• Milestone changed from sage-9.4 to sage-9.5

### comment:7 Changed 11 months ago by chapoton

there is a typo in "charachters"

### comment:8 Changed 11 months ago by git

Branch pushed to git repo; I updated commit sha1. New commits:

 ​b859fd6 `Merge branch 'develop' of git://github.com/sagemath/sage into twoadiclseries` ​7b7f6ad `trac 32258: typos`

### comment:9 Changed 11 months ago by chapoton

• 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 vbraun

• Branch changed from u/wuthrich/ticket_32258 to 7b7f6ad24620b744d4decd6dcf648bd479762948
• Resolution set to fixed
• Status changed from positive_review to closed
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