Frobenius Elements, the Chebotarev Density Theorem, and Recirocity Dylan Yott July 30, 204 Motivation Recall Dirichlet s theorem from elementary number theory. Theorem.. For a, m) =, there are infinitely many rimes a mod m). The oint of this exosition is to resent a theorem which generalizes the above result and has many alications that will hel us later in the seminar. Then, as always, we will rerove quadratic recirocity. Suose L/K is a Galois) extension of number fields with Galois grou G. Suose P is a rime of L lying over a rime of K. Set O L /P = F P and O K / = F. Then recall the following exact sequence: IP/) DP/) GalF P /F ) Now we remind the reader of these objects and mas. We ve defined DP/) = {σ G σp) = P)}. Given an element σ DP/), this gives an automorhism of O L fixing P, which is an automorhism of F P, and fixes F, giving an element of GalF P /F ). It is a standard fact from algebraic number theory that this ma is surjective. Then we ve chosen to define IP/) to be the kernel of this surjection. It turns out that this is equivalent to defining IP/) = {σ DP/) σα) α mod P), α O L }. More imortantly for us, when is unramified in L, we have IP/) =. This gives an isomorhism between DP/) and GalF P /F ). It is well known that GalF P /F ) is cyclic of order fp/), and in fact has a canonical generator given by x x N), where N) = F := q. Lifting this canonical generator along our isomorhism to DP/) gives an element which we call F rob P, which is characterized by the roerty that α O L, we have F rob P α) α q mod O L ). Since we re interested in using these Frobenius elements to understand extensions of K in terms of the arithmetic of K, it would be a bit disaointing if these Frobenius elements deended too much on our choice of P/.
Exercise.2. Show that if P and P are two rimes over, let σ G be such that σp) = P. Show F rob P = σ F rob P σ. By the last exercise, when L/K is abelian, there is no deendence at all on the choice of P, and we can make sense of F rob. In fact, the decomosition grous DP/) also only deend on, which is similarly not hard to show. 2 Examles Consider Qi)/Q. Here, the only ramified rime is 2. Otherwise, slits mod 4) and is inert 3 mod 4). Since D) is trivial in the slit and ramified case and is equal to G otherwise, it is clear what the Frobenius elements are. If we write G = {±} then F rob = when mod 4) and F rob = otherwise. It is very useful to write F rob = There is a more honest and erhas more enlightening way to see F rob = ). ) that uses the roerty satisfied by Frobenius. Let > 2 be rime, and suose we want to know which element σ G acts like raising to the th ower modulo. That is, which element is a lift of Frobenius? Well: a + bi) = a + bi ) Since i =, we see that when mod 4), then σ = is the right automorhism, and conversely for the other case. Exercise 2.. Generalize the above to Q d), d squarefree. Now consider Qζ n )/Q. Then it is well known that the rimes that ramify are recisely those dividing n. Another well known fact is that G can be canonically identified with Z/nZ), with an element m being given by the automorhism determined by ζ n ζ m n. So for, n) =, we want to comute F rob. It is straightforward to check that the automorhism which lifts Frobenius in this case is Z/nZ). Cyclotomic fields are very nice in the sense that their ramification and Frobenius elements are very exlicit, which will be very handy in what follows. 3 Quadratic Recirocity We can now give a very concetual and clean roof of quadratic recirocity. ) Theorem 3.. For, q distinct odd rimes, we have q q ) = ) q 2 2. ) Proof. It is straightforward to check that this is equivalent to q = q ), where ) =. Next, using either ramification theory and basic Galois theory, or using the theory of Gauss sums, we have the following tower of fields and Frobenius elements. 2
Qζ ) F rob q = q Q ) F rob q = q ) Q If we know that F rob q Z/Z) restricts to the Frobenius element in {±, then we would be done, because q restricts to q ). Showing that Frobenius elements restrict correctly is not difficult since Frobenius elements were defined canonically, and as such behave in a functorial way. 4 Chebotarev Density Theorem We can rehrase Dirichlet s theorem about rimes in arithmetic rogressions in terms of Frobenius elements in an obvious way. a mod m) F rob = a Z/mZ Thus, roving Dirichlet s theorem comes down to understanding the distribution of Frobenius elements. As such it is natural to study the distribution of Frobenius elements for arbitrary abelian extensions, and hoefully obtain results similar to Dirichlet s theorem. This is recisely what the Chebotarev density theorem tells us. Theorem 4.. For L/K an abelian extension of number fields, F rob are equally distributed in G = GalL/K). Remark 4.2. The term equally distributed has a meaning which we will not make recise. We have created a machine that given arithmetic data of K rimes), roduces elements of a galois grou G = GalL/K). Now we re going to ut these Frobenius elements together to construct the Artin ma. Suose now for simlicity of exoisiton that all the rimes of K are unramified in L, and that L/K is abelian as usual. Then we have the following: Art L/K I K : GalL/K) This ma is defined by Art L/K ) = F rob and extended to I K by multilicativity. By the Chebotarev density theorem, this ma is surjective, and as such I K /KerArt L/K ) = GalL/K). This is interesting because the left hand side is urely arithmetic data of K 3
and the right is information about the extensions of K. This suggests that we might be able to understand extensions of K by studying subgrous ossible kernels of the Artin mas) of I K. This will be covered in later lectures where we discuss Artin recirocity. Before we finish we resent a theorem. Theorem 4.3. For K a number field, there exists L, the maximal unramified abelian extension of K with KerArt L/K ) = P K, the subgrou of rincial ideals. L is called the Hilbert class field of K. Let K be a number field, and L its Hilbert class field. Then CO K ) = GalL/K). Thus, the rime ideals of K are equally distributed in the ideal classes. 5 Proof of Dirichlet s Theorem Rather than rove the Chebotarev density theorem, we rove Dirichlet s theorem, since the general roof of the Chebotarev density theorem ends u reducing to Dirichlet s theorem in some sense, and the ideas resent in Dirichlet s theorem already demonstrate the general theory very well. Let : Z/mZ) C be a character which we extend to Z by 0. Define an L-function as follows. Ls, ) = n) n s n For m = 0, and the trivial character we obtain Ls, chi) = ζs), the Riemann zeta function. For m > 0 and trivial, we get a slightly deficient Riemann zeta function ζs) m s. Now we resent a general fact about L-functions without roof: Lemma 5.. Let Ls, ) = a n n s, and suose A n = n k= a k is bounded as a function of n. Then Ls, ) converges for Res) > 0. The L-functions as in the above lemma arise when is a non-trivial character. Whenever is trivial, we only get convergence for Res) >. Now we use the Euler roduct for these L-functions to maniulate them into a more useful form before we 4
roceed with the roof. Ls, ) = logls, )) = = = = ) s log ) s ) n ) s ) n n ns + n q ) s + βs, ) ) n n ns The imortant thing is that βs, ) converges for Res) > 2. Now we roceed with the roof of Dirichlet s theorem. Theorem 5.2. For a, m) =, there are infinitely many rimes a mod m). Proof. Consider the following sum over the characters of Z/mZ) : a) logls, )) = ) a) ) s + βs, ) ) = s a ) + a) βs, ) 2) = φm) s + β s, ) 3) a mod m) In the above, β s, ) converges at s =. To deduce the theorem we need to know that L, ) 0 whenever is nontrivial. Assuming this fact for the meantime, then since Ls, 0 ) diverges for 0 the trivial character, we see that the LHS of ) must diverge at s =, which says that there must be infinitely many rimes a mod m), and we are done. There are various ways of roving L, ) 0 for 0, and this fact certainly constitutes the main analytic meat of the roof. There are ways of doing this using certain techniques from analysis, as well as one using the analytic class number formula. 5