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Results tagged with nt.number-theory
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user 404
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
5
votes
Do there exist modern expositions of Klein's Icosahedron?
There is the outrageously expensive "Geometry of the quintic" by Jerry Shurman, which discusses both Klein and Doyle-McMullen approaches (and then some more).
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32
votes
Useless math that became useful
Fast Fourier transform: Originally developed by Gauss in early 19th century. Gauss thought it is unworthy of publication, because there were better computational techniques. It only appeared in his co …
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4
votes
If an abelian variety has an m-torsion point, is the set of all Galois conjugates of the m-t...
No. Take $E=Z(zy^2 = x(x^2+z^2))$, identifying zero with $(0:1:0)$. Then $(-1:0:1)$ is the only real non trivial 2-torsion point of $E(\mathbb{R})$.
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3
votes
Analysis of a quadratic diophantine equation
As Robin and Fedor observed the variety in question is a quartic Del Pezzo surface. There is a nice treatment in Igor Dolgachevs "Topics in classical algebraic geometry I" section 8.5 (including expli …
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3
votes
When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
With 2^2 isogeny it never works:
F. Richelot, De transformatione integralium Abelianorum primi ordinis comentatio. J. reine angew. Math. 16 (1837) 221-341
G. Humbert, Sur la transformation ordinaire …
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2
votes
Modular curves of genus zero and normal forms for elliptic curves
The first thing you'd need in order to define a normal form is unirationality of the moduli space (otherwise you don't even have the correct number of parameters). In dimension 1, this means that you …
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3
votes
Accepted
Choosing a fast computer algebra system that works in characteristic p?
My personal experience is a few years old, but I don't think things have changed much. Sage is (or actually, was) more about ease of use then about performance. The only three CAS's you want to consid …
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6
votes
Geometry Vs Arithmetic of schemes
Look at Dan Abramovich's Birational geometry for number theorists
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9
votes
Etale cohomology -- Why study it?
One of the annoying aspects with sheaf cohomology in algebraic geometry is that - even for curves over the complex numbers - the cohomological dimensions are not what you want them to be, if you are u …
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1
vote
Can we count isogeny classes of abelian varieties?
For any genus, as long as you have non-ismorphic curves, you have non-isomorphic Abelian variety.
For genus 2, you have the Honda-Tate theorem, giving the classes of Abelian varieties (and there is …
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6
votes
Proof of no rational point on Selmer's Curve $3x^3+4y^3+5z^3=0$
The "standard" technique for killing the Hasse priniciple for elliptic curves is to show that the Tate-Shafarevich group has a copy of (Z/mZ)^2 for some m - see chapter X in Silverman's the arithmetic …
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3
votes
Solving polynomial equations when you know in which number field the solutions live
I have to say that this sounds suspiciously close to Matiyasevich's proof of Hilbert's 10th problem (Yuri V. Matiyasevich, Hilbert's Tenth Problem, MIT Press, Cambridge, Massachusetts, 1993). The "yog …
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2
votes
Why are functional equations important?
Sometimes you get very concrete algebro-geometric facts from functional equations:
Example 1: The functional equation relating Weierstrass P (for a lattice L in C) and it's derivative is the equation …
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