differential equations – How do you find the Inverse of Elliptic ─░ntegral of Second Kind when modulus is large

So I tried to take the inverse of EllipticE when modulus is large, in Mathematica, but the solution gives wrong answer.

InverseSeries(Series(EllipticE(x, -k), {x, 0, 12}, {k, Infinity, 1}),y) = InverseFunction(y,k)

For example, I tried EllipticE(0.5,-9.9) = 0.656 where x:0.5 , k:-9.9, y:0.656

But InverseFunction(y,k) is not equal to 0.5. Am I not correctly taking the inverse of the function?
I need a general form of an equation for the inverse of EllipticE. Polynomial approximation is also fine. The approximation should definitely work around when x-->0 and k-->-infinity.
So for the above example, the approximation function result should yield to 0.5 when y=0.656 and k=-9.9. I need to code this function in MCU, so I need an analytical approximation.

Caner

ag.algebraic geometry – Why is this “the first elliptic curve in nature”?

The LMFDB describes the elliptic curve 11a3 (or 11.a3) as “The first elliptic curve in nature”. It has minimal Weierstra├č equation
$$
y^2 + y = x^3 – x^2.
$$

My guess is that there is some problem in Diophantus’ Arithmetica, or perhaps some other ancient geometry problem, that is equivalent to finding a rational point on this curve. What might it be?

elliptic curves – why $ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p $

I have read an article that says: $ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p $

Where $ E $ is an elliptic curve over $ mathbb {Q} _p $ and $ E_1 ( mathbb {Q} _p) = {P in E ( mathbb {Q} _p): tilde {P} = tilde {O} } $.

The author says the proof is in "Silverith Arithmetic of Elliptic Curves" on page 191, but it says here:

If $ E $ is an elliptic curve over $ mathbb {Q} _p $ and $ has {E} $ is the formal group, then:

$$ E_1 ( mathbb {Q} _p) approx hat {E} (p mathbb {Z} _p) $$

So I don't know a good reference for proving $ E_1 ( mathbb {Q} _p) approx mathbb {Z} _p $.

elliptic curves – degree of morphisms and isogenies

$ renewal command { J} { mathrm {Jac}} renewal command { F} { mathbb {F}} $
I read this paper from B. Gross and there is something I do not understand on p. 22. 945. Here is the context: Fix a prime number $ p equiv 3 pmod 4 $and define the (hyper) elliptic curves via $ F_p $ given by the (affine) equations
$$ X_1: y ^ 2 = x ^ p – x, quad X_2: y ^ 2 = x ^ {p + 1} -1, quad
E_1: y ^ 2 = x ^ 3-x, quad E_2: y ^ 2 = x ^ 4-1. $$

I checked (using the Tate isogenicity theorem) that there is non-zero isogenicity $ alpha: J (X_1) to J (X_2) $ between the jacobian varieties (actually both are isogenic to $ E_1 ^ {(p-1) / 2} $ about $ F_p $) and there is non-zero isogenicity $ beta: E_1 to E_2 $.

There is a morphism $ f_2: X_2 to E_2, (x, y) mapsto (x ^ {(p + 1) / 4}, y) $ that's right now $ (p + 1) / $ 4. Then it is said that this is why we get a morphism $ f_1: X_1 to E_1 $ Degree $ (p + 1) / $ 4, but I don't understand why / how.

I know that $ f_1 $ induces a morphism $ phi_2: J (X_2) to E_2 $We get a morphism $ beta circ phi_2 circ alpha ^ { vee}: J (X_1) to E_1 $hence a morphism $ f_1: X_1 to E_1 $, but I think it has degrees at least the degree of $ f_2 $.
Maybe there is a clever way to compose $ phi_2 $ with other isogenies to maintain the equality of degrees?

Generally with a non-constant morphism $ f_2: X_2 to E_2 $It may not be possible to get a morphism $ f_1: X_1 to E_1 $ to the same extent as $ f_2 $: Just take $ X_2 = E_2 = X_1, f_2 = mathrm {id} $ and $ E_1 $ an elliptic curve is isogenic but not isomorphic $ E_2 $.
I am probably missing something easy, but I would rather ask for clarification.

bip 32 hd wallets – Elliptic Curve Point at Infinity

What exactly is the "point in infinity"?

It is a point added to the points on the curve. Together they form a group. It has the following characteristics:

  • (x, y) + (x, -y) = infinite
  • (x, y) + infinity = (x, y)
  • infinite + (x, y) = (x y)

Is there more than one "point in infinity"?

Only one.

How can I tell if my x and y generated by EC are the "point of infinity"?

The point at infinity is not on the curve, so it has no x or y coordinates. It is displayed whenever a point is added to its own negation (to which the normal addition rule has no answer).

Is there a way to calculate the "point of infinity"?

It is simply "the point in infinity", there is nothing to calculate.

ag.algebraic geometry – the extended upper half-plane $ mathcal {H} ^ * $ as stable elliptic curves

Write $ mathcal {M} (1) $ (and its compactification $ overline { mathcal {M}} (1) $) for the module space (stable) elliptic curves. Display both as a stack $ text {To} _ text {et} $, analytical spaces with the etale topology.

Consider the universal family of elliptic curves and stable elliptic curves and give a pullback diagram
$ requires {AMScd} $
begin {CD}
mathcal {H} @ >>> overline { mathcal {H}} \
@V V V @VV V \
mathcal {M} (1) @ >>> overline { mathcal {M}} (1)
end {CD}

what exhibits $ mathcal {M} (1) simeq ( mathcal {H} / text {SL} _2 ( mathbf {Z})) $ as a global quotient of analytical space $ mathcal {H} $. They are also called quotient stacks $ overline { mathcal {H}} / Gamma (N) = X (N) $ to the $ N> 2 $.

On the other side is the extended upper half level $ mathcal {H} ^ * = mathcal {H} cup mathbf {P} ^ 1 ( mathbf {Q}) $, that is a Hausdorff topological space on which $ text {SL} _2 ( mathbf {Z}) $ looks really discontinuous and $ mathcal {H} ^ * / Gamma (N) simeq X (N) $ as topological spaces.

question: is $ overline { mathcal {H}} $ an analytical space? What is their relationship with $ mathcal {H} ^ * $ (what i think is Not an analytical space, only its quotients are according to congruence subgroups)?

arithmetic geometry – elliptic curves and their neron model

To let $ E $ be an elliptic curve over $ mathbb {Q} $, For a prime number $ p $, To let $ mathcal {E} _p $ designate his neron model over $ mathbb {Z} _p $, Let also $ Phi_p (E) $ denote the component group of $ mathcal {E} _p $,

The structure of $ Phi_p (E) $ is known and I want to study it if $ E $ added multiplicative reduction $ p $, First, if $ E $ split multiplicative reduction at $ p $, then it is known that $ Phi_p (E) simeq mathbb {Z} / {n mathbb {Z}} $, Where $ n $ is the (normalized) $ p $-adic assessment of the discriminant of $ E $, (I admit this fact.) Next, when $ E $ has an undivided multiplicative reduction $ p $, then $ Phi_p (E) simeq mathbb {Z} / {m mathbb {Z}} $, Where $ m = 1 $ or $ 2 $ so that $ m equiv n pmod 2 $,

Here's my question: Suppose that $ E $ has an undivided multiplicative reduction $ p $, As above, we use the same notation ($ m $. $ n $. $ Phi_p (E) $ Etc).
We know the following.

  1. There is an unbranched quadratic extension $ L / { mathbb {Q} _p} $ so that $ E / { mathcal {O} _L} $ divided multiplicative reduction.

  2. The Neron model does not change when the base changes.

If the two above are correct, then the component group $ Phi_p (E) $ can be calculated using the neron model of $ E / L $, Where $ E $ divided multiplicative reduction. Since the $ p $-adic rating does not change under the unchanged extension, the component group of the Neron model of $ E / L $ is isomorphic to $ mathbb {Z} / {n mathbb {Z}} $, Consequently, $ Phi_p (E) $ is isomorphic too $ mathbb {Z} / {n mathbb {Z}} $what is wrong if $ n> 2 $,

I don't know where my argument fails. Please correct me!

Geometry – problem with equating the coefficient of the elliptic curve

To let, $ E: = y ^ 2 = x ^ 3 + Ax + B $ an elliptic curve, 2 points $ P, -Q $ on $ E $ so that $ 2P = -Q $, we can write

$$ y ^ 2-x ^ 3 + Ax + B = (x – e_1) (x – e_2) ^ 2 = 0 $$
Where, $ e_1 = x (Q), e_2 = x (P) $ a double root. So,
$ y = m (x-x (Q)) + y (Q) $, Here $ y $ is the line that goes thoroughly $ P, Q $, and $ m $ is the slope.

$$ (x – e_1) (x ^ 2 – 2xe_2 + e_2 ^ 2) $$
$$ = (x ^ 3 – 2x ^ 2e_2 + xe_2 ^ 2) – (x ^ 2e_1 – 2xe_1e_2 + e_1e_2 ^ 2) $$
$$ = x ^ 3 – 2x ^ 2e_2 + xe_2 ^ 2-x ^ 2e_1 + 2xe_1e_2-e_1e_2 ^ 2 $$
$$ = x ^ 3 + x ^ 2 (-1) (2e_2 + e_1) + x (e_2 ^ 2 + 2e_1e_2) + (- 1) e_1e_2 ^ 2 $$

Of $ y ^ 2-x ^ 3 + Ax + B = 0 $ we get coefficients of $ x ^ 2 $ is $ (- 1) m ^ 2 $, We find the equality coefficient, $ m ^ 2 = (2e_2 + e_1) $,

The problem is, if I simplify both sides and then equate the coefficient, I get the correct coefficient of $ x ^ 2 $ but not for $ x $ (if plug-in value of $ P, Q $, the coefficients of $ x ^ 2 $ just doesn't satisfy $ x $).

What's the problem?

Cryptography – private key for generating public keys bitcoin vb.net reports unknown error – secipt256k1 elliptic curve

I try to use secp256k1 eliptic curve to get that public key of private key but it gives me one unknown error error

 Private Sub Button13_Click(sender As Object, e As EventArgs) Handles Button13.Click
    Dim private_key = "68040878110175628235481263019639686"
    'public key should be Nr6MbFUfMovKCX4vd5YpQnRYsN4rq6pNPEEBKmicAEwwuYLpJrt5LsRvfvR2G8pJ5rMchEMWDYJ7rdYGY7PjxHEa
    Dim public_key As String


    Using eliptic As New ECDsaCng()
        eliptic.HashAlgorithm = CngAlgorithm.ECDsaP256

        Dim data() As Byte = Encoding.UTF8.GetBytes(private_key)
        Dim key As Byte() = eliptic.SignData(data)
        public_key = key.ToString
    End Using

    TextBox15.Text = public_key
End Sub