Martin Orr's Blog

Matlab rant: Elementwise operations on arrays

Posted by Martin Orr on Saturday, 17 March 2012 at 17:27

I have to teach some simple use of Matlab to my engineering students. I don't much like the Matlab language, but then I am unhappy with pretty much every programming language I have ever used except C (which is great for some purposes but they do not include numerical work). Today I am going to rant about the first bit of code in our Matlab course (and probably the first non-trivial bit of code in many Matlab courses and tutorials).

My complaint concerns elementwise operations on arrays (i.e. applying the same operation to every element of an array). These are very important in Matlab, and sometimes you can write code that looks like it is acting on scalars and it just works elementwise on arrays. If scalar code would always just work elementwise on arrays then I might be happy with this (but I worry that if you had such a language, how would you distinguish operations that act on an array as a unit, not just elementwise?).

But sometimes you need to alter things slightly to work elementwise on arrays. I worry that for beginning programmers, the fact that elementwise array operations look so similar to scalar operations makes it difficult to understand when these alterations are required - it would be better to consistently require a syntax meaning "perform this operation elementwise."

no comments Tags matlab, programming, teaching Read more...

The Faltings height and normed modules

Posted by Martin Orr on Saturday, 31 December 2011 at 15:31

In this post I shall give the definition of the Faltings height of an abelian variety over any number field. Last time we did this over only, and we used two properties of : the integers are a PID and there is only one archimedean place. To do things more generally, we will introduce the technology of normed modules and their degrees.

2 comments Tags abelian-varieties, alg-geom, faltings, maths, number-theory Read more...

The Faltings height of an abelian variety over the rationals

Posted by Martin Orr on Thursday, 17 November 2011 at 15:58

The Faltings height is a real number attached to an abelian variety (defined over a number field), which is at the centre of Faltings' proof of Finiteness Theorem I. In this post all I will do is define the Faltings height of an abelian variety over , as already this requires a lot of preliminaries on cotangent and canonical sheaves of schemes. Further complications arise over other base fields, which I will discuss next time.

For an abelian variety over , the Faltings height is the (logarithm of the) volume of as a complex manifold with respect to a particular volume form, chosen using the -structure of . The preliminaries are needed in order to choose the volume form.

Faltings' proof of Finiteness I proceeds by showing that for any fixed number field, there are finitely many abelian varieties of bounded Faltings height. This is done by showing that the Faltings height is not far away from the classical height of a point representing the abelian variety in the moduli space . Then he shows that the Faltings height is bounded within an isogeny class. Both of these parts are difficult.

5 comments Tags abelian-varieties, alg-geom, faltings, maths, number-theory Read more...

Siegel's theorem for curves of genus 0

Posted by Martin Orr on Friday, 28 October 2011 at 12:35

Last time we proved Siegel's theorem on the finiteness of integer points on affine curves of genus at least 1. The theorem applies also to curves of genus 0 with at least 3 points at infinity. I shall give a simple proof that deduces this from the higher genus case, then another proof using Baker's theorem from transcendental number theory which gives an effective bound on the heights of the points.

Theorem. Let be a number field and a finite set of places of . Let be an affine -curve of genus 0 such that there are at least 3 -points in the projective closure of which are not in . Then has finitely many -integer points.

The condition that there should be at least 3 points at infinity is necessary: the affine line is a genus 0 curve with 1 point at infinity and infinitely many integer points, and the curve for a non-square positive integer has 2 points at infinity and infinitely many integer points.

2 comments Tags maths, number-theory Read more...

Shafarevich and Siegel's theorems

Posted by Martin Orr on Friday, 07 October 2011 at 09:00

In this post I will prove the Shafarevich conjecture for elliptic curves (also called Shafarevich's theorem). The proof is by reducing it to the finiteness of the number of solutions of a certain Diophantine equation, and then applying Siegel's theorem on integral points on curves.

Shafarevich's Theorem. Let be a number field and a finite set of places of . Then there are only finitely many isomorphism classes of elliptic curves over with good reduction outside .

Siegel's Theorem. Let be a number field and a finite set of places of . An absolutely irreducible affine curve over of genus at least has only finitely many -integral points.

Since the reduction of Shafarevich's theorem to Siegel's theorem is short, and Siegel's theorem is of independent interest, most of the post will be about Siegel's theorem.

2 comments Tags alg-geom, faltings, maths, number-theory Read more...