I always use PARI when I need to do computations and I am a big fan of this little program. I believe that it is possible to do in PARI everything you can do with such big programs as Maple and Mathematica. Well… almost everything. Here I’d like to present some tricks to do things in PARI that seem impossible from first sight, or just convenient hints. Readers are very welcome to publish their own tricks in comments. This way we may create something like a library of tricks.

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Faithful readers of Vivastgasse 7 may have noticed that we haven’t posted anything in a while now- the reasons are numerous, but it mostly boils down to the fact that two of us (Anton and I) are in the middle of resolving several quasi-bureaucratic/ quasi-academic things: Anton is right now in Ukraine, finishing up his thesis and busy planning his move to Paris in a couple of months for his first postdoc. I’m busy planning a move as well- off to Durham next month for five months. So both of us are sort of ‘out-of-commission’ right now. However, I do intend to post something soon (maybe this weekend?) either on schemes over the mysterious “field with one element” or better yet, an unpacking of our guest-blogger Sniggy Mahanta’s post on conformal field theories. (Thanks goes to AJ Tolland for pointing out some gross inaccuracies in that post!)

But here is the main reason for this (non-mathematical) post- to vent!

Readers should be warned that the author is not an expert of CFT and, in fact, not even a novice in physics. What follows should be taken with a hefty pinch of salt.

Given any Riemann surface $X$ (as a target manifold) one is able to associate to it an $SCFT(X)$ or a super conformal field theory. The word super can just be construed as a $\mathbb{Z}_2$-grading of the theory. This is a simplistic version as normally one should also take into account several other parameters like a B-field and so on. Within an SCFT there is a topological sector called a TQFT (topological quantum field theory) which is insensitive to the metric on the target space. There are axiomatic descriptions of this theory due to Atiyah and Segal and in its latest version possibly due to Costello. Roughly an $n$-dimensional TQFT is a functor satisfying a lot of axioms from $n$-manifolds with labelled boundaries (incoming and outgoing) to symmetric monoidal DG (differential graded) categories with some twisting.

It seems when people talk about modular forms they tend to forget that they are very related to families of elliptic curves. Here I want to explain some simple way to understand the connection. We will consider modular forms for the full modular group $SL(2, \mathbf Z)$.

So consider the simplest family of elliptic curves, the Weierstrass family:

$y^2=x^3+ax+b.$

Recently, the norm-residue homomorphism has been the subject of intense discussions in the K-theoretic community following the proof of the Bloch-Kato conjecture by Voevodsky, Suslin and Rost (see Rost’s lecture at this year’s Arbeitstagung.) The goal of this post is to explain the norm-residue homomorphism in a down to earth ring-theoretic language.

(Via Anton (my fellow-blogger) and his wife Masha- their seven year old daughter Ivanka’s notes from Hirzebruch’s opening lecture at the 50th Arbeitstagung here a couple of weeks ago)

Recently I asked Faltings for some references for a self-study of Arakelov theory beyond Lang’s Arakelov theory.

He suggested the following (all revolving around the arithmetic Riemann-Roch theorem):

1. Papers by Gillet and Soule on arithmetic intersection theory

2. Papers by Bismut, Gillet and Soule on determinant of cohomology of an arithmetic variety (towards Riemann-Roch for this determinant.)

Recently I was trying to understand what’s behind the mysterious condition

$\sum_i [z_i]\wedge[1-z_i] = 0\in \Lambda^2 \mathbb C^\times$

for elements $z_i\in \mathbb C$ ($z_i\neq 0, 1$) to define an element $\sum_i[z_i]$ in the Bloch group. It appears that the condition naturally appears if one studies hyperbolic $3$-manifolds.

Of late, a lot of people have been asking me about Tannakian categories, so I decided to post this short blurb (mainly to check the Latex-related features of wordpress and also to provide some bird’s-eye view of why we should care about Tannakian categories.)

To begin with:

Definition
A neutral Tannakian category $\mathcal{T}$ over a base field $k$ is a rigid abelian tensor category with an $k$-linear exact faithful tensor $\omega: \mathcal{T} \longrightarrow \textrm{Vect}_k$ ($\rm{Vect}_k$ being the category of vector spaces over $k$) which one calls the fiber functor.