Briefly, the Selberg class is a set of functions $F:\C\to\C$ such that $f(s)$ can be written as a Dirichlet series for $\Re(s) > 1$ and that satisfies a form of analytic continuation, a functional equation, a Ramanujan hypothesis bound on coefficients of the Dirichlet series, and an Euler product formula.

Andrew Booker in [1] has extended the Selberg class in a different way, in the notion of an $L$-datum. In this post, we’ll state Booker’s definition of an $L$-datum, state his converse theorem, and explain his corollary that the completed $L$-function of a unitary cuspidal automorphic representation of $\GL_3(\A_\Q)$ has infinitely many zeroes of odd order.
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Posted by Jason Polak on 29. September 2015 · Write a comment · Categories: math

Relative trace formula in the context of automorphic representations is an idea that goes back to Jacquet, and takes into account the distinction of automorphic representations. Distinction is easily defined: if $\pi$ is an automorphic representation of $G(\A_F)$ where $G$ is a reductive algebraic group $G$, $F$ is a global field and $\A_F$ denotes the adeles of $F$, then $\pi$ is called distinguished with respect to a subgroup $H$ if the period integral
\int_{H(F)Z\backslash H(\mathbb{A}_F)} \phi(x)dx
is nonzero for some $\phi$ in the space of $\pi$. Here, $Z$ is a central subgroup of $H(\mathbb{A}_F)$. One may also weight this definition by a character.
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Posted by Jason Polak on 18. September 2015 · Write a comment · Categories: math

Often, in the Sage notebook, if you need a function you can just guess what it might be called. For instance, say I wanted the prime factorisation of 8095897323:

Notice that I use the symbol “>” for the output of a command. But what if I wanted to have the prime factors as a list instead? We could do:

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Posted by Jason Polak on 17. September 2015 · Write a comment · Categories: algebraic-geometry · Tags: , ,

One policy of Aleph Zero Categorical is that any lecture notes posted on the arXiv that I manage to see will be announced and advertised here. Today I saw that Lars Kindler and Kay Rülling have posted their notes entited:

I quote from the introduction for a summary of these notes:

These are the notes accompanying 13 lectures given by the authors at the
Clay Mathematics Institute Summer School 2014 in Madrid. The goal of
this lecture series is to introduce the audience to the theory of $\ell$-adic sheaves
with emphasis on their ramification theory. Ideally, the lectures and these
notes will equip the audience with the necessary background knowledge to
read current literature on the subject, particularly [16], which is the focus of
a second series of lectures at the same summer school. We do not attempt
to give a panoramic exposition of recent research in the subject.

Here, reference 16 refers to the paper “A finiteness theorem for Galois representations of function fields over finite fields (after Deligne)” by Esnault and Kerz. Here is one last quotation that gives the prerequisites for these notes:

The text can roughly be divided into two parts: Sections 2 to 4 deal
with the local theory and only assume a basic knowledge of commutative
algebra, while the following sections are more global in nature and require
some familiarity with algebraic geometry.

The algebraic geometry in these notes is actually quite gentle: they define etale morphism and state basic properties, and the define the etale fundamental group and even prove things about it. Not only that, but these notes have an index!

Posted by Jason Polak on 15. September 2015 · Write a comment · Categories: math

Let $X$ be a measure space and let $T:X\to X$ be a measure-preserving measurable transformation. If $E\subseteq X$ and $x\in E$ then $x$ is said to be recurrent with respect to $E$ and $T$ if there exists a positive integer $n$ such that $T^n(x) \in E$.

Theorem. If $X$ has finite measure then almost every point of $E$ is recurrent.

The proof of this comes down to showing that for all nonnegative $n$ the sets $T^{-n}(F)$ are pairwise disjoint where $F$ is the set of nonrecurrent points of $E$. Since $T$ is measure preserving and $X$ has finite measure, this means that $F$ has measure zero.
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Let $R$ be any commutative ring. The content of a polynomial $f\in R[x]$ is by definition the two-sided ideal in $R$ generated by the coefficients of $f$. If $f,g\in R[x]$, then $c(fg)\subseteq c(f)c(g)$, because each coefficient of $fg$ is a linear combination of elements of $c(f)c(g)$. Sometimes, however, this inclusion is strict. For example, if $k$ is a field of characteristic two, and $R = k[u,v]$ then $f = u + vX$ satisfies $c(f^2)\subset c(f)^2$, where the inclusion is strict. Indeed, $f^2 = u^2 + v^2X^2$ so $c(f^2) = (u^2,v^2)$, whereas $c(f)c(f) = (u^2,v^2,uv)$. A ring $R$ in which $c(fg) = c(f)c(g)$ for all $f,g\in R[x]$ is called Gaussian. We have just seen that $k[u,v]$ is not Gaussian, and in fact, we didn’t even have to specify that $k$ is characteristic two. What about a polynomial ring $k[u]$ over a field $k$ in one variable? Since $k$ is a field, $k[u]$ is a principal ideal domain (PID), and PIDs are always Gaussian. These observations can be clarified by looking at the concept of weak dimension.
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Posted by Jason Polak on 28. August 2015 · Write a comment · Categories: math

The other day we were camping and I had to rinse my Nalgene bottle (post not sponsored by Nalgene).
I only had a fixed amount of water, less than the capacity of the bottle, and I had to decide how to best use the water to minimise the concentration of contaminants. I could pour all the water in the bottle, give it a good shake, and then spill it out. Or, I could divide the rinsing water into several portions. I decided to choose amongst the methods, indexed by $n$, of dividing the rinsing water into $n$ equal portions, and then doing a sequence of $n$ separate rinses. The problem is, how should I choose $n$?
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Posted by Jason Polak on 25. August 2015 · Write a comment · Categories: algebraic-geometry, number-theory · Tags: ,

In the Arthur-Selberg trace formula and other formulas, one encounters so-called ‘orbital integrals’. These integrals might appear forbidding and abstract at first, but actually they are quite concrete objects. In this post we’ll look at an example that should make orbital integrals seem more friendly and approachable. Let $k = \mathbb{F}_q$ be a finite field and let $F = k( (t))$ be the Laurent series field over $k$. We will denote the ring of integers of $F$ by $\mathfrak{o} := k[ [t]]$ and the valuation $v:F^\times\to \mathbb{Z}$ is normalised so that $v(t) = 1$.

Let $G$ be a reductive algebraic group over $\mathfrak{o}$. Orbital integrals are defined with respect to some $\gamma\in G(F)$. Often, $\gamma$ is semisimple, and regular in the sense that the orbit $G\cdot\gamma$ has maximal dimension. One then defines for a compactly supported smooth function $f:G(F)\to \mathbb{C}$ the orbital integral
\Ocl_\gamma(f) = \int_{I_\gamma(F)\backslash G(F)} f(g^{-1}\gamma g) \frac{dg}{dg_\gamma}.
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The flat dimension of an $R$-module $M$ is the infimum over lengths of flat resolutions of $M$, and the weak dimension (or $\mathrm{Tor}$-dimension) of $R$ is the supremum over all possible flat dimensions of modules. Let’s use $\mathrm{w.dim}(R)$ to denote the weak dimension of $R$. As with the global dimension, the weak dimension of $R$ can be computed as the supremum over the set of flat dimensions of the modules $R/I$ for $I$ running over the set of all left-ideals or right-ideals, either is fine!

So, if every ideal is flat, then $\mathrm{w.dim}(R) \leq 1$. What about the converse? If $\mathrm{w.dim}(R) \leq 1$, is it true that every ideal is flat? Let’s make a side remark in that if we replace weak dimension with global dimension, and flat with projective, then the answer follows from Schanuel’s lemma. However, as far as I know there is no Schanuel’s lemma when ‘projective’ is replaced by ‘flat’.

However, we can get away with using part of the proof of Schanuel’s lemma. Before continuing, the reader may wish to check out the statement and proof of Schanuel’s lemma using a double complex spectral sequence.
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Posted by Jason Polak on 07. July 2015 · 1 comment · Categories: math

Reading a math book cover to cover is a rite of passage for some, but the process can be pleasant or a nightmare; it can be rewarding or a mindless verification of theorems. This post is a guide to reading mathematics books, and is aimed for someone who is new at the process such as undergraduate or graduate students.

1. Find a book that suits your style of exploration

Some authors will be closer in line with your way of thinking than others. Unfortunately, it’s often quite difficult to tell which is which. One way that might help is reading through the first few pages of a book to give you a better impression of the author’s style. That, together with looking at the table of contents and introduction ought to be good enough to choose from a list of possible books on a given topic.

For example, I originally starting learning homological algebra through Mac Lane’s book ‘Homology’. After a while, it seemed as though the author was spending too much time on explicit descriptions of different homological constructions and less time on the big picture, or organising principles, though I didn’t know exactly what those were at the time. After about forty pages, I stopped and switched over to Weibel’s ‘An Introduction to Homological Algebra’, and it was much more pleasant and I ended up reading the entire book. This is, by the way, one of the rare times that I’ve actually finished an entire book.
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