Constructible sets, flat maps are open and generic freeness.

I will cover a set of results that are connected and illustrate some interesting techniques about working with finiteness conditions (finitely generated as a ring, module etc) in algebraic geometry over a field.

Initially, I will prove Chevalley’s theorem that the image of a constructible set is constructible. This will involve proving Grothendieck’s generic freeness and Noetherian induction. Next, I will give a couple of applications of Chevalley’s theorem – I will prove that flat maps are open and provide a different proof of the Nullstellensatz.

Chevalley’s Theorem on Constructible sets:

Given a scheme X, a (topological) subspace is constructible if it the union of sets of the form U \cap V for U,V open and closed respectively. That is, it is a finite union of locally closed subspaces.

Theorem 1 (Chevalley): Suppose f:X \to Y is a finite type morphism between Noetherian schemes. Then the image of a constructible set under the morphism is constructible.

To prove this, we will use Grothendieck’s generic freeness lemma:

Theorem 2 (Grothendieck’s generic freeness): Suppose \phi: B \to A is a finite type morphism of rings where B is a Noetherian integral domain. Let M be a finite module over A. Then, there is a f \in B such that M_f is a free B_f module.

This theorem can obviously be generalized to the case of schemes and coherent sheaves with appropriate finiteness hypothesis. The proof of this theorem proceeds by an application of spreading out:

Proof: For an exact sequence of modules 0 \to M \to N \to P \to 0, if M,P are generically free in the above sense, then clearly so is N. Therefore, by using a sufficiently fine filtration of the given module M, we can reduce to the case where M = A/\mathfrak p for a prime ideal \mathfrak p \subset A (using the theory of associated primes). In short, we can assume that M= A and A is an integral domain.

The induction will be on the transcendence degree of A over B. If A is a finite extension, then the torsion of A has a closed locus of support and we pick a f that kills the torsion. In general, consider A\otimes_B K(B) where K(B) is the fraction field of B.

This is a finite type algebra over K(B) and by Noether Normalization (see here), we can find a factorization of the form K(B) \to K(B)[t_1,\dots,t_n] \to A\otimes_B K(B) where B[t_1,\dots,t_n] \to A is finite. Now, it is not necessary that this is true integrally but after inverting by all the relevant denominators, we can find a f\in B such that we have a factorization B_f \to B_f[t_1,\dots, t_n] \to A_f as before.

Moreover, since A is generically finite over B[t_1,\dots,t_n], we can find some integral generators (this time as a module) such that we have an exact sequence

0 \to B[t_1,\dots,t_n]^m \to A \to D

where the simple factors of D have lower transcendence degree than n. By applying the inductive hypothesis, we are done.

\Box

To prove Chevalley’s theorem, it clearly suffices to prove that f(X) is constructible. This follows using noetherian induction from generic freeness:

Proof: By generic freeness, we can find an open set U over the base such that the structure sheaf over the inverse image is free. In particular, either this open set is entirely contained in f(X) or entirely disjoint from it. Either way, it suffices to show that f(X)\cap (Y-U) is constructible but we are then immediately done by Noetherian induction.

\Box

We can remove Noetherian assumptions in the standard way by relaxing to finitely presented and expressing our rings as the base of change of a finitely presented map over Z[x_1,\dots,x_n] which is Noetherian.

Let us now use Chevalley’s theorem to prove that flat maps of finite type are open. Initially, let us assume that our schemes are Noetherian. Then the proof proceeds through the following steps:

  1. Show that for constructible sets, being open is equivalent to containing all generalizations.
  2. Show that the above reduces, in the case of local rings, to the map being surjective.
  3. Show that for a map B \to A between local rings and a finite module M over A, being faithfully flat over B is equivalent to being flat and non-zero.
  4. Putting the pieces together, this shows that flat local maps are always surjective and hence flat maps of finite type are always open since they take open sets to constructible sets by Chevalley.

I will only prove 3 here:

Theorem 3: Support (B,\mathfrak m) \to (A,\mathfrak n) is a map of local rings and M is a finite type module over A. Then, if it is flat (over B) and non zero, it is faithfully flat. That is, for any non zero B-module N, N\otimes_B M \neq 0.

Proof: Suppose N were finitely generated. Then notice that N\otimes_B M is a finitely generated A module (generated by generators for N and M). Hence, by Nakayama, it suffices to show that (N\otimes (B/\mathfrak m))\otimes_{B/\mathfrak m}(N\otimes A/\mathfrak n) \neq 0. But this is the tensor product of two non zero vector spaces (by Nakayama) and hence is non zero.

Now, in the general case, we write N as a filtered colimit of it’s finitely generated submodules. Since M is flat, the tensor product commutes with the colimit and moreover preserves injective maps. Since each object in the colimit is non zero and the maps in the colimit are injective, the resulting colimit is non-zero too.

In particular, the key example to key in mind is N = \kappa(\mathfrak p) for \mathfrak p a prime ideal of B. This case also suffices to show surjectivity.

\Box

Finally, let us prove the Nullstellensatz using these ideas. Recall a version of the nullstellensatz:

Theorem 4: (Nullstellensatz) Suppose k is an algebraically closed field and k \to L is a finite type map (as k-algebras) such that L is also a field. Then L = k.

In particular, this implies that for a finite type scheme over a field, closed points are exactly those with residue field a finite extension of the base field since base change to the algebraic closure does not affect being an integral extension.

This last statement immediately implies that for a map f: X\to Y between finite type schemes over a field, the image of a closed point is closed. Moreover, this lets us conclude that the closed points are dense in a finite type scheme over a field since distinguished opens are still finite type.

Proof: By assumption, there is a factorization k \to k[t_1,\dots,t_n] \to L where the second map is surjective. In particular, we have finite type maps g_i: k[t_i]\to L. By Chevalley, the image of a closed point has to be constructible but it is easy to show that the only constructible points on k[t] are the classical closed points by the division algorithm and using that k is algebraically closed.

Therefore, the map k[t_i] \to L factors through k[t_i] \to k \to L and since this is true for all i and k[t_1,\dots,t_n] \to L is surjective, it implies that k = L.

\Box

 

A crucial point in the above proof is that we can explicitly classify all ideals in k[t]. We can do a similar thing for Z[t] and this lets us prove that maximal ideals for finite type algebras over Z correspond exactly to having a finite residue field.

Advertisements

Irreducibility of Galois representations attached to Elliptic Curves

Epistemic status: I have not checked this carefully for errors, it is entirely possible there are mistakes in this.

 

We will be mostly be concerned with Elliptic curves over number fields E/K. However, let us start with a general lemma about Elliptic curves with endomorphism ring \mathbb Z.

Lemma 1: Let E be an elliptic curve over a field with endomorphism ring exactly \mathbb Z. Then for elliptic curves E',E'' with isogenies f',f'': E',E'' \to E with non isomorphic cyclic kernels, then E',E'' are not isomorphic.

Proof: Say the kernels have degree n',n'' and suppose E',E'' are isomorphic. Consider the isogeny g: E \to E' \to E'' \to E where the central arrow is the supposed isomorphism and the last arrow is the dual of f''. This is an endomorphism of E and therefore, by assumption is of the form [n] (multiplication by n).

Moreover, the kernel of g contains the kernel of f' which is a cyclic subgroup. Therefore, \mathbb Z/n'\mathbb Z is contained in \mathbb Z/a\mathbb Z. That is, n'|a. Moreover, the quotient by the kernel of f contains the cyclic subgroup \mathbb Z/n''\mathbb Z. By a similar argument, this shows that n''|a.

That is to say, n'n'' = a^2 and n',n''|a and therefore n'=n'' contrary to assumption.

\Box

This might seem like quite a specialized lemma but it has surprising utility. For instance, this lemma features implicitly in showing that the various definitions of a supersingular elliptic curve are equivalent. We will use it together with Shafarevich’s theorem to prove the irreducibility of the Galois representation on the Tate module.

Theorem 1[Shafarevich’s Theorem]: If E is an elliptic curve over a number field K and S is a finite set of finite places of K, then there are only finitely many isomorphism classes of Elliptic curves with good reduction away from S.

This has the important corollary:

Corollary 1: There are only finitely many isomorphism classes of Elliptic curves isogenous to E/K.

I will not prove either theorem here but the proofs are not too hard. The idea is to bound the number of Elliptic curves that can occur by considering the Weierstrass equation and showing that there are only finitely many options that can occur for any given discriminant (by Siegel’s theorem). The number of discriminants that can occur is further bounded since they are units in \mathscr O_{K,S} determined upto a 12th power.

We can use the lemma and corollary to prove the irreducibility of Galois representations. Let V_l be the rational Tate module of E and E[l] the l- torsion of E. Both of these are G_K = \mathrm{Gal}(\overline K/K) modules.

Theorem 2: For curves E/K with no Complex multiplication, V_l is irreducible for all l and E[l] is irreducible for almost all l.

Proof: If E[l] is reducible, then it contains a cyclic submodule X_l defined over K. Then, E/X_l are elliptic curves defined over K, isogenous to E with cyclic kernels of order l and therefore by the lemma are pairwise non isomorphic. However, by the corollary, there can only be finitely many such isomorphism classes and therefore, finitely many l such that E[l] is reducible.

Similarly, suppose V_l is reducible. Then it contains a cyclic submodule Y. Since the integral Tate module T_l is also Galois-invariant, we can define X = Y_l \cap T_l defined over K. Consider X_n = X/l^nX. These are cyclic subgroups of E[l^n]. As such, we can define E_n = E/X_n to be curves over K isogenous to E with cyclic kernels.

As before, by the lemma they are pairwise distinct but by Shafarevich’s theorem this is impossible. In this case, we can also avoid Shafarevich’s theorem by the following argument:

By the lemma, we know that there E_n \cong E_m for some m>n. Note that X_n \subset X_m in this case. Therefore, there is an isogeny E/X_n \to E/X_m with cyclic kernel but this is also an endomorphism since these curves were assumed to be isomorphic. Since E/X_n does not have CM by assumption, this is an impossibility (endomorphisms cannot have cyclic kernels in this case).

\Box

This theorem is quite interesting but also quite far from the best known. A few remarks follow:

Remark 1:  The above theorem is false for curves with complex multiplication. In that case, suppose K is a field that contains the field of endomorphisms L of E. If p = \pi_1\pi_2 is a prime that splits in L, then E[p] is reducible. This is because there exists an isogeny [\pi_1] defined over K with kernel of size N(\pi_1) = p.

Remark 2:  Serre proved a much stronger version of this where he showed that, under the hypothesis of the theorem, the image of G_K in the E[l] is in fact isomorphic to the entire group GL_2(\mathbb F_p) for almost all l. This is proved in his famous 1972 paper “Proprites galoisiennes des points d’ordre fini des courbes elliptiques”. This is of course much stronger than simple irreducibility.

In fact, we can beef up irreducibility to absolute irreducibility easily enough:

First, note that there is an element (called complex conjugation) in G_K that has order 2 and determinant -1 by the Weil pairing. Hence, it has eigenvalues 1,-1. In particular, we can find a rational basis of eigenvectors for complex conjugation. Since the action on E[l] is by the reduction of the action on V_l, the same is true for the characteristic l reduction.

This is to say, we have an element that has two eigenvectors with eigenvalues 1,-1 defined over the base field. This is sufficient to prove absolute irreducibility:

Corollary 3: Let V be a two dimensional irreducible representation of a group G over the field K. Suppose that there is an element g with two eigenvectors v,w with eigenvalues 1,-1 respectively. Then, the representation is absolutely irreducible.

Proof: Let L/K be a field extension such that V\otimes_K L is reducible. Then, there is an invariant subspace generated by a vector of the form \alpha v + \beta w. In particular, g fixes it which implies that \alpha or \beta is 0.

However, this immediately implies that the vector space spanned by v or w in V is in fact invariant under G and hence V is irreducible.

\Box

Noether Normalization, Spreading out and the Nullstellensatz .

Hilbert’s Nullstellensatz plays a central role in algebraic geometry. It can be seen as the fundamental link between the modern theory of schemes and the classical theory of algebraic varieties over fields. Since this is one of the first results a novice in algebraic geometry learns and is often proved very algebraically, one often does not gain a good understanding of the proof till much later.

I will present three proofs of the Nullstellensatz found in the literature from a geometric perspective. This will highlight the role of the “spreading out and specializing” common to the proofs that might not be obvious from an algebraic presentation. The last proof is a very short, self contained demonstration of the techniques. Along the way, we will also see a geometric proof of Noether Normalization.

The proof of Hilbert’s lemma is usually broken up into the following two steps: 1) Prove the weak Nullstellensatz and 2) Derive the strong Nullstellensatz using the Rabinowitsch or other means. I will be focusing solely on the first step in this post. Nothing in this is original except for the presentation.

Continue reading

Weak Mordell-Weil and various approaches to it

I will show a few different approaches to the Weak Mordell-Weil theorem and how they are all really the same proof. The various proofs use the Hermite-Minkowski theorem, Class Field Theory and the Dirichlet’s Unit theorem/finiteness of Class group.

 

Let E/K be an elliptic curve over a number field. The Mordell-Weil theorem says that the group of rational points E(K) is finitely generated. This is usually proven in two steps:

  1. Weak Mordell-Weil Theorem: We prove for some n that E(K)/nE(K) is finite.
  2. Theory of heights: We define the notion of a height of a point on E (roughly, how many bits of information one would need to store the point). Using this and (1), the completion of the proof is quite formal.

I will focus here on the weak Mordell-Weil theorem and in particular, an approach to it using the Hermite-Minkowski theorem. This approach will apply without very little change to the case of Abelian varieties and the general technique seems to be applicable in great generality.

The idea of the proof is as follows:

  1. Reduce the case of general K to supposing that K contains the n-torsion using the Kummer sequence.
  2. Use the Kummer Pairing to reduce to showing that the inverse image of E(K) under the multiplication by n [n] map generates a finite extension of K.
  3. Show that there is a smooth, proper model of E/K over an open subset of \mathcal O_K and hence the inverse image of [n] generates an extension etale over R, not just K.
  4. Apply Hermite-Minkowski.

 

 

Continue reading

Weil pairing and Galois descent

There is an interesting way in which Weil pairing on Abelian Varieties is nothing more than Galois descent for a particular Galois extension. I have not seen this connection used before in the literature but I have also not seen a lot of literature…

Let A/k be an abelian variety over a field k of characteristic p. Suppose m is an integer coprime to p and let [m]: A \to A denote the multiplication by m map with kernel A[m].

Recall that there is a dual abelian variety A^\vee representing the Picard functor for A. In particular, A^\vee[m] is the group of line bundles \mathcal L on A such that [m]^*\mathcal L is trivial. A and A^\vee are finite abelian groups and the Weil pairing is a perfect pairing of the form:

\langle-,-\rangle:  A[m] \times A^\vee[m] \to \mu_m.

It’s definition goes as follows: Let D be a divisor corresponding to \mathcal L and let g(x) be a rational function on A such that \mathrm{div}\ g(x) = [m]^*D. Then, for a \in A[m], we define \langle a,\mathcal L\rangle = g(x+a)/g(x).

If t_a denotes translation by a, then g(x+a) is the pull back of g(x) along t_a. Since [m]\circ t_a = [m], g(x+a) and g(x) have the same divisor and so g(x+a)/g(x) is regular everywhere on A and hence constant by properness of A. A little more work shows that this rational functions is in \mu_m. We will see below that this is an immediate conclusion of our alternate viewpoint.

This is the standard picture. However, there is also this alternate way of looking at things:

Consider the etale Galois extension [m]: A \to A. It’s galois group is canonically identified with A[m]. Moreover, A^\vee[m] is precisely the set of line bundles on A trivialized by this etale cover. In other words, it is the set of Galois twists of \mathcal O_A (or even for any line bundle). Since descent is effective for this extension, this will immediately imply that:

A^\vee[m] \cong H^1(\mathrm{Gal}([m]), \mathcal O_A^\times) = H^1(A([m]), k^\times) = \mathrm{Hom}\ (A[m],k^\times).

The final equality is because A[m] acts trivially on \mathcal O_A^\times = k^\times. We also see immediately that \mathrm{Hom}\ (A[m],k^\times) = \mathrm{Hom}\ (A[m],\mu_m) since A[m] is m-torison as an Abelian group. Therefore, this gives us a pairing:

A[m]\times A^\vee[m] \to \mu_m.

Explicitly, the map goes as follows: Given a line bundle \mathcal L, we pick an isomorphism g: O_A \to [m]^*\mathcal L. Then, the corresponding 1-cocycle for a \in A[m] is defined by a \to (t_a^*g)^{-1}g \in \mathrm{Aut}(\mathcal O_A). It is easily seen that this is the same explicit construction as in the standard viewpoint.

Conclusion:

I find the Galois descent viewpoint conceptually satisfying. The standard treatments of the Weil pairing can seem arbitrary and it not clear why such a pairing should exist or be useful. On the other hand, [m]:A \to A is a perfectly natural Galois extension connected to A and \mathcal O_A torsors are clearly an interesting thing to consider.

Some notes of mine

I wrote these notes a while back for my own reference. I don’t know how useful they will be to others but they are my attempt to understand the subjects they talk about. They have no references to where I got the material since they are only intended for private use (despite my publishing them here…).

1. Modular forms over finite characteristic:

These notes contain material from Serre’s article on modular forms mod-p and at the very end, a little bit about modular forms over the p-adics as in Serre’s paper culminating in a definition of the Kubota-Leopoldt p-adic zeta function using this theory.

After a brief summary of classical modular forms, the results should be more or less self contained. The most interesting section (in my opinion…) is Section 4 where I briefly discuss Katz’ perspective on modular forms and use it to show that the Hasse invariant is a modular form and compute it’s q-expansion using the Tate curve. This section is also scarce on details and I understood this stuff by reading Prof. Emerton’s wonderful expository article here.

The article is here: Modular_Forms_mod_P.

 

2. Complex Multiplication

This is my attempt to streamline and summarize the main results of complex multiplication as I see them. This is short (about 5-6 pages) and the final section is my answer to the question here.

The article is here: Complex_Multiplication_notes.

 

3. Growth of Class number in Z_p extensions

A summary of the relevant chapter in Washington’s books. Nothing new here.

The article is: Growth_of_class_groups_in_Z_p_extensions.

 

 

 

Formal Summation and Dirichlet L-functions

Recall the classical Riemann zeta function:

\zeta(s) = \sum_{n\geq 1}\frac{1}{n^s}

and the Dirichlet L-functions for a character \chi: \mathbb Z \to \mathbb C:

S(s,\chi) = \sum_{n\geq 1}\frac{\chi(n)}{n^s}.

defined for \Re(s) > 1.  These functions can be analytically continued to the entire complex plane (except a pole at s=1 in the case of \zeta(s)). In particular, the values at non positive integers carry great arithmetic significance and enjoy many properties.

For instance, L(s,\chi) is an algebraic integer and in fact equal to B_{k+1,\chi}/(k+1) where B_{k,\chi} are the generalized Bernoulli numbers. Moreover, these values satisfy p-adic congruences and integrality properties such as the Kummer congruence (and generalizations to the L-functions).

The standard proof of these proceeds by showing that L(s,\chi) satisfies a functional equation that relates L(1-s,\chi) to L(s,\overline{\chi}) and then computing the values L(n,\chi) for n \geq 1 integral using analytic techniques (such as Fourier analysis).

Proving the p-adic properties is then by working directly with the definition of the generalized Bernoulli numbers instead of the L-functions. However, the L-functions are clearly the fundamental object here and it would be nice to have a way to directly work with the values at negative integers (without using the functional equation).

One might be tempted to extend the series definition to the negative integers and say:

\zeta(-k)  "="  1^k + 2^k + 3^k + \dots

For instance, consider the following (bogus) computation:

\zeta(0)\frac{t^0}{0!} = 1^0\frac{t^0}{0!} + 2^0\frac{t^0}{0!} + \dots

\zeta(-1)\frac{t^1}{1!} = 1^1\frac{t^1}{1!} + 2^0\frac{t^1}{1!} + \dots

\zeta(-2)\frac{t^2}{2!} = 1^2\frac{t^2}{2!} + 2^2\frac{t^2}{2!} + \dots

. . .

and let us “sum” the columns first:

\sum_{k\geq 0}\zeta(-k)\frac{t^{k}}{k!} = e^{t} + e^{2t} + \dots = \frac{e^t}{1 + e^t}

which, remarkably enough, is the right generating function for \zeta(-k)! We have exchanged the summation over two divergent summations and ended up with the right answer.

In fact, it is possible to rigorously justify this procedure of divergent summation and moreover, one can use it to prove a lot of arithmetic properties of these values rather easily (like the Kummer congruence). I learnt the basic method from some lecture notes of Prof. Akshay Venkatesh here: Section 3, Analytic Class Number formula and L-functions.

I then discovered that one could use these techniques to compute the explicit values (along the outline above) and prove some more stuff. I wrote this up in an article (that also explains the basic technique and should be (almost) self contained) here: Divergent_series_summation.