Ag.algebraic geometry – automorphisms of the modular curve defined by $ mathbb {Q} $

To let $ p geq $ 3 be a prime. The modular curve $ X (p) $ can be considered as a coherent smooth projective curve over complex numbers. There is a subgroup $ mathrm {PSL} _2 ( mathbb {F} _p) $ within his group of automorphisms (Serre has proved that for $ p geq 7 $ it is the complete group of automorphisms).

For which $ p $ Is there a geometrically coherent smooth projective curve? $ X $ over $ mathbb {Q} $ so that $ X (p) $ is the base change of $ X $ and so that all $ mathrm {PSL} _2 ( mathbb {F} _p) $-Automorphisms of $ X (p) $ are changed by automorphisms of the base $ X $?

dg.differential geometry – Effect of the inverse exponential map on the curvature of a given curve

Suppose you have a curve $ alpha $ in a manifold. You are at one point $ alpha (t) $ this curve. The curvature of $ alpha (t) $ corresponds to the curvature of the curve $ exp ^ {- 1} _ { alpha (t)} ( alpha (s)) $ at the $ s = t $, The previous curve is what you see from the tangent space of $ alpha (t) $,

Imagine, you are moving to another point $ x $, and want to determine how the curvature of $ alpha (t) $ has changed from the point of view of the tan space $ x $,

That's the curvature of $ exp ^ {- 1} _ {x} ( alpha (s)) $ at the $ s = t $? (when $ x $ and $ alpha (t) $ are close enough that we have a local diffeomorphism, for example).

For me it is alright to know the answer if the distributor is a space of constant cross-sectional curvature. I assume that in such a case it depends only on the distance between $ x $ and $ alpha (t) $ and the relative position between the geodesic link $ x $ and $ alpha (t) $ with respect to the hyperplane (in the tangent space of $ alpha (t) $tangential to $ alpha $ at the $ t $,

Calculation Number Theory – Find a CM point with the image in the elliptic curve under modular parametrization

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Certain Integrals – Do you use the theorem of Green for a non-closed curve by adding or subtracting another curve / line?

I understand that Green's theorem can only be applied to simple and closed curves. If you look at these two examples, you may seem to add a line to close the curve and use Green's theorem.

Example A: Let C be the curve from points (2,0) to (0,0) along the circle $ (x-1) ^ 2 + y ^ 2 = 1 $

In this example, I have added the curve L, which is a line y = 0, from (0,0) to (2,0). So now the integral $ int_ {C} vec {F} cdot d vec {r} + int_ {L} vec {F} cdot d vec {r} = int_ {C + L} vec {F } cdot d vec {r} = int int_ {R} -ydA $

In another example, B is: Look at the vector field, $ vec {F} (x, y) = <-x^3y,sqrt{1+y^3}>$ defined for all real x and $ y geq-1 $, Let C be the curve from (-1,1) to (1,1) along the parabola $ y = x ^ 2 $,

To calculate the line integral, I first let R (the area ??) be the region enclosed by the lines $ C-L $where L is the straight line from point (-1,1) to (1,1). That's why: $ int_ {C} vec {F} cdot vec {r} – int_ {L} vec {F} cdot vec {r} = int_ {CL} vec {F} cdot d vec {r} = int int_ {R} x ^ 3dA $

My question is, why do we add in the first example, but subtract in the second example? My first assumption was always adding, as if we were "adding the two lines" to create an enclosed curve.

ag.algebraic geometry – Formal group and formal completion of an elliptic curve

To let $ A $ be a ring, $ S $ the spectrum of $ A $. $ f: E to S $ an elliptic curve.
Then I assume $ f _ * Omega_ {E / A} $ is a free over $ S $. $ has {E} $ (the formal degree along the $ 0 $-Section.) $ cong operatorname {spf} (A ((T))) $,

I intuitively think that this isomorphism means "$ E $ is on site $ 0 $, only the line ",
and that the formal group with $ E $ is the group law of $ E $ around $ 0 $,
(At least over a field.)
So I think that the group schema structure of $ E $ induces the "group formal scheme" (I do not know if such a thing exists or not …) structure $ has {E} $and it is closely related to the formal group of $ E $,
But I can not even get that relationship through a filing.

(I do not know the difficult theory of formal groups and formal schemes.)

Any help is greatly appreciated!

Differential Geometry – How to Define Variations of a Curve on a Smooth Manifold

I'm trying to understand how the calculus of variations works when you set smooth manifolds. The texts I read usually change from Euclidean space to manifolds, as if there is no difference, but I am someone who (at least once) has to do things formally.

The case of Euclidean space: $ $ To let $ I = (a, b) subset mathbb {R} $ and let it go $ q_0: I to mathbb {R} ^ n $ be a curve. We define a deformation from $ q_0 $ with fixed endpoints as another curve $ q: I times (- epsilon, epsilon) to mathbb {R} ^ n $ satisfying:

  • $ q (t, 0) = q_0 (t) $ for all $ t in I $,
  • $ q (a, s) = q_0 (a) $ and $ q (b, s) = q_0 (b) $ for all $ s in (- epsilon, epsilon) $,

The variation of $ q_0 $ in terms of deformation $ q $designated $ Delta q $, is defined by:

$$
Left. Delta q_0 (t) right. = left. frac { partial} { partial s} q (t, s) right | _ {s = 0}
$$

Smooth distributor: The definitions of curves and thus deformations work essentially the same and only replace them $ mathbb {R} ^ n $ from a few varied ones $ Q $, To define the variation, you can switch to local charts and then glue the pieces together carefully.

partition wall $ (a, b) $ as $ {a = t_0, t_1, cdots, t_ {n-1}, t_n = b } $ so the limitation of $ q $ to the subinterval $ (t_i, t_ {i + 1}) $ is contained in the local table $ (U_i, phi_i) $, In each diagram, the previous definition of $ Delta q_0 $ is well defined:

$$
Left. delta ( phi circ q_0) (t) right. = left. frac { partial} { partial s} ( phi circ q) (t, s) right | _ {s = 0}
$$

It can also be shown that these "local" deformations coincide with each other at the intersection of diagrams, so that conceptually there should be no problem in sticking the parts together again. I'm not sure though from where This object lives or why, so I do not feel comfortable going on.

Should the deformation be a different curve $ Q $? A curve continues $ TQ $? Does anyone know a clearer way to do this?

Fourier series, parametric curve of an image using the Fourier transform

I have performed the following actions:
img = Import("https://www.clipartmax.com/png/middle/100-1003682_homer2-homer-simpson-crazy-png.png");
img = Binarize(img~ColorConvert~"Grayscale"~ImageResize~500~Blur~2);
pts = DeleteDuplicates@Cases(Normal@ListContourPlot(Reverse@ImageData(img), Contours -> {0.5}), _Line, -1)((1, 1))

and got a bunch of points. How can I obtain a parametric equation (x (t), y (t)) of these points with Fourier. FourierTrigSeries? I want equations like x(t) = 245.196 + 121.653 Cos(t) + 17.6594 Cos(2 t). y(t) = 347.468 - 202.673 Cos(t) - 26.0902 Cos(2 t) - 12.7999 Cos(3 t) -
6.15289 Cos(4 t) + 4.381 Cos(5 t)
with given accuracy

ag.algebraic geometry – formal completion of an elliptic curve along the $ 0 $ section and formal extension of functions

To let $ S = operatorname {Spec} A $ be an affine scheme, $ f: E to S $ an elliptic curve and $ mathscr {I} $ the ideal bundle of $ 0 $-Section.
(This is invertible because the section defines the effective relative Cartier divisor.)
Accept that $ f_ * Omega_ {E / S}, f _ * mathscr {I} ^ n $ are free over $ mathscr {O} _S $,
($ n = 1, cdots, 6 $)
I want to show $ has {E} cong operatorname {spf} A ((T)) $,
And I do not understand the formal extension of a base $ omega $ from $ f_ * Omega_ {E / S} $ and a base of $ f _ * mathscr {I} ^ n $, along the $ 0 $-Section.

I have tried the following:
Since the $ 0 $Section is a regular immersion, for everyone $ x in S $there are affine openings $ x in V subseteq S $. $ 0 (x) in U subset E $ s.t. $ 0 (V) subseteq U $ and the diagram

$$ require {AMScd}
begin {CD}
S @> {0} >> E \
@VV {1} V @VV {f} V \
S @> {1} >> p
end {CD} $$

corresponds to

$$ require {AMScd}
begin {CD}
C @ <{0} << B \
@A { text {localization by one element}} AA @AAA \
A @ <{1} << A,
end {CD} $$

where the kernel $ I $ from $ B to C $ is generated by $ t in B $a regular element.

I showed that $ has {B} $ (the completion of $ B $ along the kernel $ I $) $ cong C ((t)) $,
And $ Omega_ {B / A} otimes_B has {B} = dt has {B} = dt C ((t)). $
I can show that $ has {E} cong operatorname {Spf} A ((t)) $ locally, and I can expand $ omega $ locally.

How can I extend this process globally?
I have also shown that isomorphism $ has {B} cong C ((t)) $ is compatible with localization.
So I intuitively think that these isomorphisms (at any time) are glued together, and we have $ has {E} cong operatorname {Spf} A ((t)) $,

Any help is greatly appreciated!