nt.number theory - Is there a canonical notion of "mod-l automorphic representation"?

This is in my mind a central open problem.

Here is an explicit example which I believe is still wide open. Serre's conjecture (the Khare-Wintenberger theorem) says that if I have a continuous odd irreducible 2-dimensional mod p representation of the absolute Galois group of the rationals then it should come from a mod p modular form. We have a perfectly good definition of mod p modular forms (sections of the usual line bundles on mod p modular curves).

But what is the "even" analogue of Serre's conjecture?

One problem is that in characteristic zero (i.e., for 2-dimensional continuous complex representations, which must have finite image) these guys are expected to come from algebraic Maass forms, which have, as far as anyone knows, no algebraic definition: they are genuinely analytic objects and it is a complete miracle that their Hecke eigenvalues are algebraic. So there's a big open problem: give an algebraic definition of an algebraic Maass form which doesn't use analytic functions on the upper half plane which are eigenvectors for the Laplacian with eigenvalue 1/4. I have no idea how to do this and I don't think that anyone else has either. The problem is that the forms are not holomorphic (so you can't use GAGA and alg geom) and they're not cohomological (so you can't use group cohomology either), and those are in some sense the only tricks we have (other than Langlands transfer, but I don't know any functorial transfer of these guys which (a) loses no information and (b) gives rise to a form which is cohomological or holomorphic). It might be an interesting PhD problem to check this out in fact. Blasius and Ramakrishnan once tried to go up to an im quad field and then to Sp_4 over Q but the resulting form isn't holomorphic. One might define an algebraic automorphic rep to be "accessible" if it's cohomological or perhaps limit of discrete series (perhaps these are the ones for which there's a chance of proving they're arithmetic), and then try and find examples of algebraic auto reps which should never transfer to an "accessible" rep on any other group.

But even after that, there's another problem, which is that as far as I know one does not expect a continuous even irreducible mod p representation of this Galois group to lift to a de Rham representation in characteristic zero (and indeed perhaps Frank Calegari might be able to give explicit counterexamples to this, after his recent observation that oddness sometimes follows from other assumptions). So even if one could give an algebraic definition of a Maass form, one wouldn't have enough Maass forms---they would all (at least conjecturally) give rise to Galois representations with images all of whose Jordan-Hoelder factors are cyclic or A_5 (by the classification of finite subgroups of PGL_2(C)). So there's another non-trivial sticking point.

In summary---nice question but I don't think that mathematics has a good answer yet (unless you're willing to just cheat and say that an automorphic representation "is" a Galois rep with some properties).

co.combinatorics - Average distance between numbers of the form $2^{a}3^{b}$

The point of this answer is simply to repeat Scott's answer in a more visible place. If he wants to post it himself, I'll delete my post:

Let $a_i$ be the sequence of these integers, sorted into order. Let $a_r leq N < a_{r+1}$. We want to estimate
$$frac{1}{r-1} sum (a_{i+1} - a_i) = (a_r-a_1)/(r-1).$$

As Scott explains,
$$r=(1/2) cdot (log N/log 2 + O(1)) cdot (log N/log 3 + O(1)) = (log N)^2/(2 log 2 log 3) + O(log N) .$$

Also, there is clearly a power of $2$ between $N$ and $N/2$, so $a_r sim N$.

So the average distance is $sim N/(log N)^2$. If you work harder, you can probably tighten up the bounds to show that it is $N (2 log 2 log 3)/(log N)^2 (1+O(1/log N))$

dg.differential geometry - What are CR manifolds like?

CR submanifolds of a complex manifold are defined as submanifolds M⊂X such that TM∩iTM⊂TX has constant rank (i is the imaginary unit). Note that the condition is automatically verified if M has codimension one; for higher codimension this is not true.

An abstract CR manifold is a real manifold M, with a distinguished subbundle HM⊂TM, corresponding to TM∩iTM, endowed with a linear endomorphism J with J²=-Id. The structure is furthermore required to satisfy a so called integrability condition:
For all sections X,Y of HM:

• [X,JY]+[JX,Y] is a section of HM




• ([X,Y]-[JX,JY]) + J([X,JY]+[JX,Y]) = 0

Not every abstract CR manifold can be realized as a CR submanifold.

ct.category theory - What are κappa-categories?

The intuitive explanation is that $kappa$-categories are to first-order functions what cartesian closed categories are to higher-order functions.

This all started with Lambek's work on polynomial categories; the best reference for that is
J. Lambek. Functional completeness of cartesian categories. Annals of Mathematical Logic, 6:251–292, 1973. That paper introduced the choice of the letter $kappa$. A polynomial category is what you get when you take a category with a terminal object, pick an object $X$, and then freely adjoin a new morphism $f:1to X$ and close under composition. This is very much like the ring of polynomials $R[X]$ over a ring $R$ arrived at by adjoining an indeterminate element and closing under the ring operations. Lambek shows that this property can be stated in universal terms -- as the unique category admitting a particular kind of functor from the original category. This is formally more satisfying but not the best route for beginners.

Later, Hasegawa developed this idea much further in M. Hasegawa. Decomposing typed lambda calculus into a couple of categorical programming languages. Lecture Notes in Computer Science, 953, 1995. He showed that just as the $lambda$-calculus can be used as a "syntax" for specifying morphisms in a cartesian closed category, so too can the $kappa$-calculus -- roughly the $lambda$-calculus without first-class functions -- be used as a "syntax" for specifying morphisms. I recommend studying the figure immediately after the first paragraph of section 3 in his paper (very carefully). It conveys both the essence of these categories and their relevance to the study of programming languages.

To wrap up, cartesian closed categories have been an immensely useful tool in understanding programming languages with first-class functions. Unfortunately they can only be used to study languages with the property that for every pair of types $B$ and $C$ there is also a type of functions from $BRightarrow C$ and the ways of getting a $BRightarrow C$ from an $A$ are in one-to-one-correspondence with the ways of getting a $C$ from a (cartesian) pair of an $A$ and a $B$. Lambek-Hasegawa $kappa$-categories are an elegant way to extend these techniques to languages in which this assumption does not hold.

Lastly, as a postscript, both Lambek and Hasegawa assume that the underlying monoidal structure of their categories is Cartesian. Some of the most fascinating results arise when you repeat their constructions in categories which are merely binoidal or premonoidal -- you'd be surprised how few modifications are required.

nt.number theory - Uniformly computable classes of graphs

This question sounds more like a research project than a definite problem. Part of the reason for me to say this is because this question is only interesting if you impose some (ill-defined) qualifications on what an interesting answer can be; and even then, it is not clear how one can answer the question without having the insights into the structure of the integers which you are hoping to find. Having said that, I'll hazard some elementary observations.

Throughout the following, I will refer to the integer which corresponds to a vertex in a graph as its index.

The problem of Gödel numbering

Your question is still about computable predicates. As long as you do so, without imposing further restrictions, there will still be too much room open for Gödel numbering to give solutions to sub-problems.

The predicate V₀ that I described before, while 'elaborate' (in that it would not be so simple in closed form), relied quite heavily on using prime factorization as a means of describing compound data structures --- specifically, in order to construct the indices of vertices to represent a sort of label for the vertex, its adjacency relations to other vertices, and even complete information about the entire graph.

In order to obtain "interesting" representations of graphs just by families of indices and relations on them, we obviously want to reduce reliance on Gödel numbering. We can do this by disallowing Gödel numbering in the indices,* i.e. in the first argument n to the predicate V and the first two arguments to the predicate E. But even if we do this, a very modest amount of Gödel numbering in the third argument μ to E gives away the whole game, as I show below.

^{* Of course, this pre-supposes that determining what constitutes Gödel numbering is a computable problem; I would argue that it isn't even well-defined. We can concievably describe a restriction to how much one can exploit the prime factorization of indices, but this would not address information being carried by the integer in e.g. representation of the index in base 1000, or any arbitrarily esoteric-but-computable representation of the index. Furthermore, even while prohibiting Gödel numbering, we still presumably want/need the index to bear some information; just not potentially arbitrarily complex information. But let's ignore that technical problem and suppose it can be done for the sake of argument.}

A "significantly less complex" universal set of predicates

It suffices to do the following:

• Set $nu$ to be a multiple of the first n primes, for a graph on n vertices.


• Set $V(n,nu) equiv big( nu in n mathbb N ;;&;; n ~text{prime}big)$.


• Set $mu = 2^{e_1} 3^{e_2} cdots p_m^{e_m}$, for a graph on m edges, where each integer e_j is a product of two primes which are the indices of adjacent vertices.


• Set $E(n,m,mu) equiv (exists p~text{prime}):big(mu in p^{nm}mathbb N ;;&;; mu notin p^{nm + 1} mathbb Nbig)$.

The sole explicit instance of "Gödel numbering" in this scheme is in the construction of μ. However, it suffices to admit very simple choices of V and E to represent an arbitrary graph. The number μ isn't even terribly esoteric: the only retriction on μ is that the exponents of its prime factorization are always square-free, and have exactly two prime factors when they are non-zero. This construction generalizes to hypergraphs and graphs with loops trivially.

The key to this construction, of course, is that the prime numbers serve as "easily distinguishable" indices for the vertices, to the extent that any number which is a multiple of two or more primes may be easily interpreted as a set; and then we store multiple sets by storing them in the exponents of a prime factorization of some integer.

I may have fallen afoul of some restriction you have in mind; for instance, it is likely that the set of μ of the form above (either in the explicit construction or the generalization for prime power indices) have small "measure", for many reasonable definitions of measures. From you examples e.g. of cycle graphs, however, I assume the fact that not all μ fall under this construction is not a problem.

Approaches to trees which avoid Gödel numbering

'Universal' predicates which rely on Gödel numbering are obviously boring, in that they indicate only what is possible. After one understands Gödel numbering as a means of representing data structures, they are not in themselves interesting. The challenge is then to see what one can do without using Gödel numbering (which I take in a practical sense to refer to using integers only to denote sequences or collections of other integers).

This is tricky for a graph class such as "arbitrary trees" (or even "arbitrary binary trees"), because there is little structure to deal with; it seems to me that natural representations of them will amount to something like lattices of subsets with added restrictions.

The trick is that for each index, you would like to identify a unique vertex (depending possibly on μ) which can be its "parent" (in the picture of rooted trees); or more generally, however one defines the set of allowed indices, at most one element of the set can be its parent, with exactly one vertex failing to have a parent.

This is where your question becomes ill-defined. Your motivation seems to be to somehow plumb the structure of the integers using graphs; but to do this, we must somehow already have a structure to hand to represent trees. This is not a research problem, so much as it is a very open-ended research programme.

Two potential approaches --- simple ones, and so likely not to be realizable --- occur to me.

1. Once more, we can consider prime factorizations. To avoid the temptation of Gödel numbering, we should not be too picky with the indices; the index-set should admit integers with non-trivial common divisors. A natural choice of edge-predicate is then
   $$ E(n,m,mu) equiv big(exists A,Bbig):Big({A,B} = {n,m} ;;&;; A/B ~text{is the largest prime factor of $A$}Big) ,$$
   that is, where transition from each vertex v to its parent corresponds to division by the largest prime factor of the index of v. A natural choice of predicate V is then to choose indices which are factors of some particular integer.

   
   It is not clear to me how general a class of trees we can get from this; we can easily obtain paths and stars, and some variety of other trees. It is also not clear to me how to get a reasonably varied class of binary trees.




2. We can consider ways of partitioning integers into a sum of smaller integers. Because addition in itself has so "little" structure, any structure would have to come from the allowed set of summands, which has to be given by the integer μ; this comes dangerously close to Gödel numbering. A simple solution is to use this integer to parameterize a family of summands, e.g. powers of μ. This too can lead to Gödel numbering of a different sort; a possible approach is to use ternary expansions of integers to represent the location of a node in a tree by left-right branching from the root, with log₃(n) representing the distance from the root, and the j^th digit representing whether one takes the left or right branch at the j^th level to reach the vertex in question. So there is a real question of what there is that one can do, under the constraint of doing something interesting.

   
   A compromise is to succumb to prime factorization, but to more or less ignore the exponents to avoid the temptation to do Gödel numbering. So, the possible transitions may be taken to be the (maximal) prime-power factors of μ. This still uses μ to denote a set of integers, but not a set with arbitrary elements. To identify a parent of an index, one may take the number which may be reached by subtracting the largest allowable transition.
   One may easily obtain paths (e.g. restrict to odd indices and take μ = 2) and arbitrary star graphs (e.g. restrict indices to 1 and p+1 for p prime, and take μ to be a suitable square-free integer). It's not obvious how to get arbitrary trees.

   
   One can think of generalizing this to allow transitions which are e.g. arbitrary prime-power factors, not just maximal prime-power factors. This can be used to get a wide variety of trees. For instance, we may consider restricting the indices to integers n which are a sum of at most k powers of two, and let the parent of each index be the one which is obtained by subtracting the largest power 2^r < n. This is pretty simple; but also dangerously close to the ternary scheme described above. It is not clear exactly how to delineate the boundary between 'interesting' and 'using Gödel numbering'. (Then again, perhaps this last decomposition isn't interesting anyway --- or perhaps it is merely the concept of 'the structure of an arbitrary tree' which is not especially interesting.)

In closing

Unless your objective is actually to embark on an open-ended research project, you might want to restrict your question a bit more. In particular, you should give some thought to imposing conditions which would prevent "trivial" solutions, especially using numerical tricks to embed the structures you are interested in into the indices and/or parameters.

Unless you somehow refine your question, it is just extremely open-ended. Not to say that it isn't a nice sort of project.

gr.group theory - Finite index normal subgroups of a free group.

Let $G= (mathbb{Z} bigoplus mathbb{Z}) star (mathbb{Z} bigoplus mathbb{Z})$, where $star$ denotes the free product, let F be the commutator subgroup of G, it is free by a theorem of Kurosh. Find a proper normal subgroup of F (other than the trivial one) such that it is of infinite index.

nt.number theory - Writing down minimal Weierstrass equations

Let $E$ be an elliptic curve over $mathbb Q_p$. It is possible that $E$ has bad reduction but then when you see $E$ as a curve over a finite extension $K$ of $mathbb Q_p$, it obtains good reduction. Let $v$ be the valuation defined on $K$ and $R$ its valuation ring. I was interested in checking $E$ has good reduction over $K$ by hand, using the Weierstrass equation. What that amounts to then is writing down the Weierstrass equation $y^2+a_1xy + a_3y = x^3 + a_2x^2+a_4x + a_6$ with the $a_i in R$ and considering changes of coordinates $x=u^2x' + r$ and $y=u^3y' + u^2sx' + t$ for $u,r,s,t in R$ in hopes of finding an equation with $v(Delta')$ minimized, subject to each $a_i'$ being in $R$. There are certain congruence conditions that guarantee minimality of the new equation, e.g. $v(Delta') < 12$, which only depend on the choice of $u$. However, guaranteeing the new equation has coefficients in $R$ requires solving other congruence relations depending on $r,s$ and $t$, e.g. you need $v(a_1+2s)geq v(u)$ (because $a_1' = u^{-1}(a_1+2s)$). The few times I have done this by hand, I have just had to look at the equations and make some choices until something worked out.

My question is whether or not there exists a general method for obtaining a good change of coordinates $u,r,s,t$ and if not, then how do people go about writing down minimal Weierstrass models. I can't imagine there should be general methods for solving the system of non-linear congruences (higher powers of $u,r,s$ and $t$ appear in the other congruences) in the ring $R$, but if there is then I would also be interested in understanding that as well.