@MatheinBoulomenos That is how you would show that homotopy classes of maps correspond to bundles indeed, using the defining property of EG and some obstruction theory
Consider
$$ y = \int_1^{\infty} \frac{li(x)^2 (x - 1)}{x^4} dx $$
Is there a closed form for y ?
It appears that a good approximation is $ 10 \cdot Ci (\frac{56}{19})$.
—-
If there is No closed form would it help to allow the function
$$ t(x) = \int_1^x li(t)^2 dt $$
??
@LeakyNun So you mean that saying "the completion of the algebraic closure of the completion at a nonarchimedean valuation of the localization at the zero ideal of the initial object in the category of modules over the free ablian group of rank 1 with the unique ring structure is $\Bbb C_p$" is difficult to comprehend?
@MikeMiller I see. I wonder if there's some analogy that can be made between topological obstruction theory and $\operatorname{Ext}$ groups in homological algebra, since they also give you obstructions for the existence of sections
@loch if you're interested, consider $f = (1+X)^3-1 \in \Bbb Z_3[X]$. Then, its Lubin-Tate group is $\Bbb G_m = X+Y+XY = (1+X)(1+Y)-1$. Then, since $[x+y]_f = [x]_f +_f [y]_f = (1+[x]_f)(1+[y]_f)-1$, you get $[n]_f = (1+X)^n-1$ for all $n \in \Bbb Z_3$, including things like $n=\frac12 \in \Bbb Z_3$
hi @KasmirKhaan
@loch Then, $f_m = (1+X)^{3^m}-1$ and its roots are quite nice
and then there is a canonical isomorphism $\operatorname{Gal}(\Bbb Q_3(f_m=0)/\Bbb Q_3) \cong \Bbb Z_3/3^m\Bbb Z_3$
@TedShifrin the reason I asked this is because there seems to be an exact sequence $0 \to \Bbb C^\times \to \mathcal{M}^*(X) \to \operatorname{Div}_0(X) \to \operatorname{Jac}(X) \to 0$
And this looks a lot like the exact sequence $ 1 \to (\mathcal{O}_K)^\times \to K^\times \to \operatorname{I}(K) \to \operatorname{Gal}(L/K) \to 1$ where $L$ is the Hilbert class field of the number field $K$ and $I(K)$ is the group of fractional ideals of $K$
I have no idea, @Mathein. It's a fundamental construction in (complex) algebraic geometry to look at the Jacobian ... and there are all sorts of higher-dimensional analogues.
@TedShifrin Artin reciprocity tells us that the class group is isomorphic to the Galois group of the maximally unramified abelian extension (that's just the non-obvious part of the exact sequence above), so geometrically speaking, we have a strong relation between line bundles and unramified abelian coverings. Is there something like that for Riemann surfaces?
Coverings sound like you should have flat line bundles.
Representations of $\pi_1$.
So those would be the line bundles with chern class $0$. So those are the line bundles corresponding to divisors of degree $0$. So it seems to connect.
That's usually called $\text{Pic}^0$ (the Picard variety). But for curves, yeah, it ties into the Jacobian.
can we say something about the fundamental group of the Jacobian (or the Picard variety)? (which should correspond to the Galois group of the Hilbert class field)
that's one step in going from a modular form to a Galois representation
@TedShifrin so if we have the full modular group, then $\Gamma(1) \backslash \Bbb H$ classifies complex tori. What if we take $\Gamma(N) \backslash \Bbb H$? Does this also classify something?
@TedShifrin so what if we take $H \subset \Gamma(1)$ such that $\Gamma(1)/N$ is the monster group, then does $H \backslash \Bbb{H}$ classify something?
Some weird facts: For each prime $p$, look at the normalizer of $N$ of $\Gamma_0(p)$, then the compactified quotient of the upper half plane by that has genus 0 iff p divides the order of the monster group
@TedShifrin yes, I have to find $d\in\mathbb R^2$ such that satisfies the limit. However I don't see it geometrically. I'v learned about feasible and attainable directions, and I know how the 'look like' geometrically.'
Have you shaded in the region $y\ge -x^3$. Draw the tangent line to $y=-x^3$ at the origin. Why do you get all directions at that line and above, and nothing else?
I dunno. It will be covered in precisely the same location someone talks about characteristic classes
I think it unwise to keep a list of things you have to learn (and the corresponding references) and better to just go wild on something that sounds interesting, and go to other references when you get stuck
you have a short exact sequence $1 \to \operatorname{Gal}(\widehat{K}_f^m/\widehat{K}) \to W(\widehat{K}_f^m/K) \to W(\widehat{K}/K) \to 1$ and another SES $1 \to \mathcal{O}^\times / (1+\mathfrak{p}^k) \to K^\times/(1+\mathfrak{p}^k) \to K^\times/ \mathcal{O}^\times \to 1$ and you have comptabile maps between those sequences where you know that the maps on the left and on the right are isomorphisms
Let I and J be bounded intervals (I \subset J), let f : I -> R be a Riemann-Stiltjes integrable function on I and let F: J -> R be the function F(x) := f(x) if (x \in I) and
I want to say "$h(x)$ is continuous, and hence by algebaric limit theorem, $h(2^nx)$ is continuous" in mathematical logic symbols. I am not able to figure out how to write $h(x)$ is continuous. The expression template I have is $(h(x):continuous)\rightarrow((h(2^nx):continuous)$
are all BOUNDED continuous functions on the OPEN interval (like (3,5)) Riemann-Stieltjes integrable?
I know why this can possibly break because in my proof there is one property used from the Riemann integral which breaks down in the Riemann-Stiltjess case HOWEVER I did not use this property I used its version for the piece-constant fucntions fro which it does not break down!
Consider
$$ y = \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^4} dx, $$
where $\operatorname{li}(x)$ is the logarithmic integral. Is there a closed form for y ?
It appears that a good approximation is $ 10 \cdot \operatorname{Ci}\bigl( \frac{56}{19}\bigr)$, where $\operatorname{Ci}(x)...
You'll find that the modular curve $X_0(N) = \Gamma_0(N) \backslash \mathbb{H}$ classifies elliptic curves with a cyclic subgroup of order $N$, $X_1(N) = \Gamma_1(N) \backslash \mathbb{H}$ classifies elliptic curves with an $N$-torsion point, and $X(N) = \Gamma(N) \backslash \mathbb{H}$ classifies elliptic curves $E$ with a basis $P,Q$ for $E[N]$ such that their Weil pairing is equal to $-1$ (although I'm not very sure how to view the Weil pairing).
Actually it's just Theorem 5.2.5 in https://wstein.org/books/ribet-stein/main.pdf.
Using this language, Mazur's theorem on the classification of torsion groups for elliptic curves over $\mathbb{Q}$ becomes a question of asking the existence of $\mathbb{Q}$-rational points on these curves (for various $N$) (although I'm not sure how this language is related to the proof)
Consider
$$ y = \int_1^{\infty} \frac{\operatorname{li}(x)^2 (x - 1)}{x^4} dx, $$
where $\operatorname{li}(x)$ is the logarithmic integral. Is there a closed form for y ?
It appears that a good approximation is $ 10 \cdot \operatorname{Ci}\bigl( \frac{56}{19}\bigr)$, where $\operatorname{Ci}(x)...
Hello, does anyone have a reference for the fact that the stereographic projection (in higher dimensions) maps circles that don't go through the north pole N to circles and maps circles that go through N to straight lines ? I already know the proof in lower dimension but I cannot generalize the proof.
In mathematics the existence of a mathematical object is often proved by contradiction without showing how to construct the object.
I’m wondering, does the existence of the object imply that it is at least possible to construct the object? Or are there mathematical object that do exist but are i...
and then once you made your question more specific, I suspect the answer would be to read about constructivism and the axiom of choice and all that
an example would be the fact that every nonzero commutative ring has a maximal ideal, but this is proved using Zorn’s lemma, so this is not constructive
So if I interpret you correctly, there are examples of existence proofs, which are impossible to give by a constructive proof. In other words, there are objects that mathematically exist which are impossible to construct using a constructive proof?
I didnt study the axiom of choice yet, I did study the axiom of infinity. In some way I guess that this infinite set, that exists by axiom, is also impossible to construct. If not, we wouldnt need the axiom, but could give a constructive proof.
@Kasper if you do use it to define $\mathbb N$, then it's natural (badum psht) to start at $0$, because why start at $\varnothing^+$ when you can start at $\varnothing$?
it's just axiomatizing the idea, but I think everyone agrees that the existence of countable things implies the idea that you can assign things a number of how many of them there are
[Deus Ex Machina Lemma] Let $K$ be a local field ($\mathcal O$, $\mathfrak p$, $\Bbb F_q$) and $L$ be a complete unramified extension of $K$. Let $\varphi \in \operatorname{Gal}(L/K)$ such that $\varphi$ restricts to the arithmetic Frobenius $\pmod {\mathfrak p}$. Suppose $\pi$ is a uniformiser of $L$, and $f \in \mathcal O_L[[X]]$ satisfies $f \equiv \pi X \pmod {X^2}$ and $f \equiv X^q \pmod {\mathfrak p}$. Let $(\pi', f')$ be another such pair. Then, for any $\theta_1 \cdots \theta_t \in \mathcal O_L$ such that $\varphi(\theta_i)/\theta_i = \pi'/\pi$, there is a unique $F \in \mathcal O…
> Which leads me to another remark. Historians of science often seem to write about their subject as if scientific progress were a necessary sociological development. But as far as I can see it is largely driven by enthusiasm, and characterized largely by randomness.
I solved Langley’s Adventitious Angles in a quick fashion. Check out this video - youtu.be/Uuqqe9ewNnA . Please tell me why this happened to be right. By the way, I know the video is bad.
And, finally, the book is not concerned with any problems belonging to the so-called logic and methodology of empirical sciences. I must say that I am inclined to doubt whether any special "logic of empirical sciences", as opposed to logic in general, or, to the "logic of deductive sciences", exists at all
I thought more about number fields and Riemann surfaces and I figured out how you can recover a compact Riemann surface from its field of meromorphic functions, but only as a set. I don't understand how you get the topology or the complex structure, though. I have a vague idea, but I don't know if it works
If the construction I have in mind actually works, that should be the inverse of the functor from compact Riemann surfaces to finitely generated field extensions of $\Bbb C$ of transcendence degree $1$
technically when you're blowing up a sheaf of ideals (read this as closed subscheme) then the blow up is taking the relative proj of the rees algebra of the ideal.
but it's not that bad once you unravel definitions, and does give you the classical description in the classical case (i.e. blowing up a point in a plane etc.) :p
@MatheinBoulomenos Only theoretically. I don't think I'll be able to follow through your construction. You should write it out in the garbological room as a general essay though, so other people who are more interested can read it
@MatheinBoulomenos the algebro-geometric side of the story is that curves over $k$ are equivalent to transcendental extensions of $k$ of degree $1$, and probably you want to take the analyticification of the curves to make them riemann surfaces?
I'm googling to see if people explicitly discuss the use of zorn's lemma here
i found another statement of the proof that uses zorns lemma, and another statement of your proof that does not
maybe it's a matter of convention whether the empty set is ordered
"Note that in lots of places the empty set is not a partially ordered set. If you are not going to deal with ordinals, then it is sometimes better to just go with it. "
go with what -_-
but a convention shouldn't determine whether the whole proof works so I dont believe that's the difference Jech's proof and Folland's Proof
Consider
$$ y = \int_1^{\infty} \frac{li(x)^2 (x - 1)}{x^4} dx $$
Is there a closed form for y ?
It appears that a good approximation is $ 10 \cdot Ci (\frac{56}{19})$.
—-
If there is No closed form would it help to allow the function
$$ t(x) = \int_1^x li(t)^2 dt $$
??
If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
