Kazuya Kato

The great arithmetician Kazuya Kato visited twice over the last few weeks, so I thought I’d use the occasion to recommend some writings. An undergraduate level textbook on number theory is

Number Theory I: Fermat’s Dream

published by the American Mathematical Society. It is short and covers fairly standard material, but contains many unusual insights. A research article that represents quite well Kato’s vision of number theory is

Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. Part I

available online if your institution subscribes to Springer. The background required for a genuine reading of the paper is rather extensive, but even without it, you can enjoy the introduction, the first few sections, and the closing remarks.

A main theme of the work is that the somewhat mysterious p-adic zeta functions and L-functions are the objects with direct relevance to the important problems of arithmetic geometry, while the usual complex functions are a sort of intermediary. It took me a long time to come to terms with this view, especially since I still don’t understand these functions (actually elements of some non-commutative algebra of measures) at all well, but it is eventually an essential component of my own thoughts about Diophantine geometry.

MK

Rank of Jordan canonical form

Hi sir, I was practicing on the 2006 paper and I was wondering what the “rank” meant for Jordan canonical forms. I’d appreciate your help.

Cheers
Xiao

Reply:

Rank is a notion that we use for any linear map, and is defined as the dimension of its image. If the map is given by a matrix, it is also the dimension of the column space. If two matrices represent the same linear map, that is, are similar, then they have the same rank. In particular, the rank of the Jordan canonical form is the same as the rank of the original matrix. One important fact to review is the formula relating the rank and nullity of a linear map.

MK

Fundamental groups and Diophantine geometry

A few weeks ago, I gave a colloquium lecture at Leeds university and subsequently wrote up an exposition based on it. It’s still not entirely `popular,’ but may give a somewhat better sense than my previous remarks of at least a few ideas.

Let me know if there are some points on which you would like elaboration.

MK

Algebraic number theory update

On the course web page for algebraic number theory, I have now added a (*) next to each item directly relevant to the exams. Make sure you are familiar with the material therein. In particular, there is a new note on some elementary irrationality and a revised practical summary. The problems that have been recently added to the summary are marked with the label (N).

MK

Jordan normal form

dear Professor,

I have a few questions regarding jordan canonical form, id be grateful if you could point me in the right direction…..im struggling to understand how you can work out Jordan form from the jordan basis…..e.g. if i have

Cha(X)= (x-2)^3
Ma(X)=(x-2)^2

and a Jordan basis of

B1={[1,0,3],[2,-3,0]}
B2={(1,0,0]}

how do i no what the JF is???

I no its…

2
2
2

[along the diagonal]

but what goes in the gaps, ie the 1’s

Reply:

Check the Jordan basis carefully. But you don’t need it to calculate the JNF in this case. The minimal polynomial tells you that the size of the largest Jordan block is 2×2. Think about that a bit.

MK