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26.03.2015

Kummer extensions, Hilbert’s Theorem 90 and judicious expansion

Filed under: ETH,Exercise,Mathematics @ 17:45

This semester, I am teaching “Algebra II” for the first time. After “Algebra I” which covers standard “Groups, rings and fields”, this follow-up is largely Galois theory. In particular, I have to classify cyclic extensions.

In the simplest case where L/K is a cyclic extension of degree n\geq 1 and K contains all n-th roots of unity (and n is coprime to the characteristic of K), this essentially means proving that if L/K has cyclic Galois group of order n, then there is some b\in L with L=K(b) and b^n=a belongs to K^{\times}.

Indeed, the converse is relatively simple (in the technical sense that I can do it on paper or on the blackboard without having to think about it in advance, by just following the general principles that I remember).

I had however the memory that the second step is trickier, and didn’t remember exactly how it was done. The texts I use (the notes of M. Reid, Lang’s “Algebra” and Chambert-Loir’s delightful “Algèbre corporelle”, or rather its English translation) all give “the formula” for the element b but they do not really motivate it. This is certainly rather quick, but since I can’t remember it, and yet I would like to motivate as much as possible all steps in this construction, I looked at the question a bit more carefully.

As it turns out, a judicious expansion and lengthening of the argument makes it (to me) more memorable and understandable.

The first step (which is standard and motivated by the converse) is to recognize that it is enough to find some element x in L^{\times} such that \sigma(x)=\xi x, where \sigma is a generator of the Galois group G=\mathrm{Gal}(L/K) and \xi is a primitive n-th root of unity in L. This is a statement about the K-linear action of G on L, or in other words about the representation of G on the K-vector space L. So, as usual, the first question is to see what we know about this representation.

And we know quite a bit! Indeed, the normal basis theorem states that L is isomorphic to the left-regular representation of G on the vector space V of K-valued functions \varphi\,:\, G\longrightarrow K, which is given by
(\sigma\cdot \varphi)(\tau)=\varphi(\sigma^{-1}\tau).
(It is more usual to use the group algebra K[G], but both are isomorphic).

The desired equation implies (because G is generated by \sigma) that Kx is a sub-representation of L. In V, we have an explicit decomposition in direct sum
V=\bigoplus_{\chi} K\chi,
where \chi runs over all characters \chi\,:\, G\longrightarrow K (these really run over all characters of G over an algebraic closure of K, because K contains all n-th roots of unity and G has exponent n). So x (if it is to exist) must correspond to some character. The only thing to check now is whether we can find one with the right \sigma eigenvalue.

So we just see what happens (or we remember that it works). For a character \chi\in V such that \chi(\sigma) = \omega, and x\in L^{\times} the element corresponding to \chi under the G-isomorphism L\simeq V, we obtain \sigma(x)=\omega^{-1}x. But by easy character theory (recall that G is cyclic of order n) we can find \chi with \chi(\sigma)=\xi^{-1}, and we are done.

I noticed that Lang hides the formula in Hilbert’s Theorem 90: an element of norm 1 in a cyclic extension, with \sigma a generator of the Galois group, is of the form \sigma(x)/x for some non-zero x; this is applied to the n-th root of unity in L. The proof of Hilbert’s Theorem 90 uses something with the same flavor as the representation theory argument: Artin’s Lemma to the effect that the elements of G are linearly independent as linear maps on L. I haven’t completely elucidated the parallel however.

(P.S. Chambert-Loir’s blog has some recent very interesting posts on elementary Galois theory, which are highly recommended.)

16.03.2015

A parity lemma of A. Irving

Filed under: Exercise,Mathematics @ 12:31

In his recent work on the divisor function in arithmetic progressions to smooth moduli, A. Irving proves the following rather amusing lemma (see Lemma 4.5 in his paper):

Lemma Let p be an odd prime number, let k\geq 1 be an integer and let h=(h_1,\ldots,h_k) be a k-tuple of elements of \mathbf{F}_p. For any subset I of \{1,\ldots, k\}, denote
h_I=\sum_{i\in I}{h_i},
and for any x\in\mathbf{F}_p, let
\nu_h(x)=|\{I\subset \{1,\ldots, k\}\,\mid\, h_I=x\}|
denote the multiplicity of x among the (h_I).
Then if none of the h_i is zero, there exists some x for which \nu_h(x) is odd.

I will explain two proofs of this result, first Irving’s, and then one that I came up with. I’m tempted to guess that there is also a proof using some graph theory, but I didn’t succeed in crafting one yet.

Irving’s proof. This is very elegant. Let \xi be a primitive p-th root of unity. We proceed by contraposition, hence assume that all multiplicities \nu_h(x) are even. Now consider the element
N=\prod_{i=1}^k(1+\xi^{h_i})
of the cyclotomic field K_p=\mathbf{Q}(\xi). By expanding and using the assumption we see that
N=\sum_{x\in\mathbf{F}_p} \nu_h(x)\xi^{x}\in 2\mathbf{Z}[\xi].
In particular, the norm (from K_p to \mathbf{Q}) of N is an even integer, but because p is odd, the norm of 1+\xi^{h_i} is known to be odd for all h_i\not=0. Hence some factor must have h_i=0, as desired.

A second proof. When I heard of Irving’s Lemma, I didn’t have his paper at hand (or internet), so I tried to come up with a proof. Here’s the one I found, which is a bit longer but maybe easier to find by trial and error.

First we note that
\sum_{x\in \mathbf{F}_p} \nu_h(x)=2^k
is even. In particular, since p is odd, there is at least some x with \nu_h(x) even.

Now we argue by induction on k\geq 1. For k=1, the result is immediate: there are two potential sums 0 and h_1, and so if h_1\not=0, there is some odd multiplicity.

Now assume that k\geq 2 and that the result holds for all (k-1)-tuples. Let h be a k-tuple, with no h_i equal to zero, and which has all multiplicities \nu_h(x) even. We wish to derive a contradiction. For this, let j=(h_1,\ldots,h_{k-1}). For any x\in\mathbf{F}_p, we have
\nu_h(x)=\nu_j(x)+\nu_j(x-h_k),
by counting separately those I with sum x which contain k or not.

Now take x such that \nu_j(x) is odd, which exists by induction. Our assumptions imply that \nu_j(x-h_k) is also odd. Then, iterating, we deduce that \nu_j(x-nh_k) is odd for all integers n\geq 0. But the map n\mapsto x-nh_k is surjective onto \mathbf{F}_p, since h_k is non-zero. Hence our assumption would imply that all multiplicities \nu_j(y) are odd, which we have seen is not the case… Hence we have a contradiction.

27.10.2014

0.00023814967230605090687395214144185337601

Filed under: Exercise,Mathematics,Science @ 19:38

Yesterday my younger son was playing dice; the game involved throwing 6 dices simultaneously, and he threw a complete set 1, 2, 3, 4, 5, 6, twice in a row!

Is that a millenial-style coincidence worth cosmic pronouncements? Actually, not that much: since the dices are indistinguishable, the probability of a single throw of this type is

\frac{6!}{6^6}\simeq  0.015432098765432098765432098765432098765,

so about one and a half percent. And for two, assuming independence, we get a probability

\frac{(6!)^2}{6^{12}}\simeq 0.00023814967230605090687395214144185337601,

or a bit more than one chance in five throusand. This is small, but not extraordinarily so.

(The dices are thrown from a cup, so the independence assumption is quite reliable here.)

11.06.2014

Leo’s first theorem

Filed under: Exercise,Mathematics @ 11:46

I learnt the following from my son Léo: the teacher asks to compute 9+9; that’s easy
9\ +\ 9\ =\ 18.
But no! The actual question is to compute 9 times 9! We must correct this! But it’s just as easy without starting from scratch: we turn the “plus” cross a quarter turn on the left-hand side:
9\ \times\ 9
and then switch the digits on the right-hand side:
9\ \times\ 9\ =\ 81.

This is a fun little random fact about integers and decimal expansions, certainly.

But there’s a bit more to it than that: it is in fact independent of the choice of base 10, in the sense that if we pick any other integer b\geq 2, and consider base b expansions, then we also have

(b-1)\ +\ (b-1)\ =\ 2b-2\ =\ b+(b-2)= \underline{1}\,\underline{b-2}

as well as

(b-1)\ \times\ (b-1)\ =\ (b-1)^2=b(b-2)+1= \underline{b-2}\,\underline{1},

(where we underline individual digits in base b expansion.)

At this point it is natural to ask if there are any other Léo-pairs (x,y) to base b, i.e., pairs of digits in base b such that the base b expansions of the sum and the product of x and y are related by switching the two digits (where we always get two digits in the result by viewing a one-digit result z as \underline{0}\, \underline{z}).

It turns out that, whatever the base b, the only such pairs are (b-1,b-1) and the “degenerate” case (0,0).

To see this, there are two cases: either the addition x+y leads to a carry, or not.

If it does, this means that y=b-z where x>z. The sum is then

x+y=b+(x-z)=\underline{1}\,\underline{x-z}.

So this is a Léo-pair if and only if

xy=\underline{x-z}\,\underline{1}.

This equation, in terms of x and z, becomes

x(b-z)=b(x-z)+1,

which holds if and only if z(b-x)=1. Since the factors are integers and non-negative, this is only possible if z=b-x=1, which means x=y=b-1, the solution found by Léo.

Now suppose there is no carry. This means that we have 0\leq x,y\leq b-1 and x+y\leq b-1. Then
x+y=\underline{0}\,\underline{x+y},
and we have a Léo-pair if and only if
xy=\underline{x+y}\,\underline{0},
i.e., if and only if xy=b(x+y).

This is not an uninteresting little equation! For a fixed b (which could now be any non-zero rational), this defines a simple quadratic curve. Without the restrictions on the size of the solution (x,y), there is always a point on this curve, namely
(x_0,y_0)=(2b,2b).
This does not fit our conditions, of course. But we can use it to find all other integral solutions, as usual for quadratic curves. First, any line through (x_0,y_0) intersects the curve in a a second point, which has rational coordinates if the line is also defined by rational coefficients, and conversely.

Doing this, some re-arranging and checking leads to the parameterization

\begin{cases} x=b+k\\  y=b+\frac{b^2}{k}\end{cases}

of the rational solutions to xy=b(x+y), where k is an arbitrary non-zero rational number. In this case, this can also be found more easily by simply writing the equation in the form
0=xy-bx-by=(x-b)(y-b)-b^2\ldots

Now assume that b\geq 2 is an integer, and we want (x,y) to be integers. This holds if and only if k is an integer such that k\mid b^2.

Such solutions certainly exist, but do they satisfy the digit condition? The answer is yes if and only if k=-b, which means x=y=0, giving the expected degenerate pair. Indeed, to have x<b, the parameter k must be a negative divisor of k^2. We write k=-d with d\mid k^2 positive. Then to have non-negative digits, we must have
\begin{cases} x=b-d\geq 0\\ y=b-\frac{b^2}{d}\geq 0\end{cases},
the first one of these inequalities means b\geq d, while the second means that b\leq d

11.05.2014

The discrete spectrum is discrete

Filed under: Exercise,Mathematics @ 19:06

No, this post is not an exercise in tautological reasoning: the point is that the word “discrete” is relatively overloaded. In the theory of automorphic forms, “discrete spectrum” (or “spectre discret”) is the same as “cuspidal spectrum”, and refers to those automorphic representations (of a given group G over a given global field F) which are realized as closed subspaces of the relevant representation space on L^2(G(\mathbf{Q})\backslash G(\mathbf{A}_F)), by opposition with the “continuous spectrum”, whose components may fail to be actual subrepresentations.

However, we can also think of automorphic representations as points in the unitary dual of the relevant adélic group G(\mathbf{A}_F), and this has a natural topology (the Fell topology), for which it then makes sense to ask whether the set of all cuspidal automorphic representations is discrete or not.

The two notions might well be unrelated: for instance, if we take the group \mathrm{SL}_2(\mathbf{R}) and the direct sum over t\in\mathbf{Q} of the principal series with parameter t, we obtain a representation containing only “discrete” spectrum, but parameterized by a “non-discrete” subset of the unitary dual.

It is nevertheless true that the discrete spectrum is discrete in the unitary dual, in the automorphic case. I am sure this is well-known. (Unless the topology on the unitary dual is much weirder than I expect; my expertise in this respect is quite limited, but I remember asking A. Venkatesh who told me that the pathologies of the unitary dual topology are basically irrelevant as far as the automorphic case is concerned). I think this must be well-known, but I don’t remember seeing it mentioned anywhere (hence this post…)

Here is the argument, at least over number fields, and for \mathrm{GL}_n. Let \pi be a given cuspidal representation. The unitary dual topology on the adélic group is, if I understand right, the restricted direct product topology with respect to the unramified spectrum. So a neighborhood of \pi is determined by a finite set S of places and corresponding neighborhoods U_s of \pi_s for s\in S. We want to find such a neighborhood in which \pi is the unique automorphic cuspidal representation.

First, we can fix a neighborhood U_{\infty} of the archimedean (Langlands) parameters that is relatively compact. Next, we note (or claim…) that for a finite place v, the exponent of the conductor is locally constant, so we get neighborhoods U_v of \pi_v, with U_v the unramified spectrum when \pi is unramified at v, such that all automorphic representations in
U=U_{\infty}\times \prod_{v} U_v
have arithmetic conductor equal to that of \pi. With the archimedean condition, it follows that the Iwaniec-Sarnak analytic conductor of cuspidal representations in U is bounded.

However, a basic “height-like” property is that there are only finitely many cuspidal representations with bounded analytic conductor (this is proved by Michel-Venkatesh in Section 2.6.5 of this paper, and a different proof based on spherical codes, as suggested by Venkatesh, is due to Brumley). Thus U almost isolates \pi, in the sense that only finitely many other cuspidal representations can lie in U. Denote by X this set of representations.

Now, the unitary dual is (or should be…) at least minimally separated so that, for any two cuspidal representations \pi_1 and \pi_2, there is an open set which contains \pi_1 and not \pi_2, say V_{\pi_1,\pi_2}. Then
U'=U\cap \bigcap_{\rho\in X-\{\pi\}} V_{\pi,\rho}
is an open neighborhood of \pi which only contains the cuspidal representation \pi

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