THE ANALYTIC CLASS NUMBER FORMULA AND L-FUNCTIONS AKSHAY VENKATESH NOTES BY TONY FENG CONTENTS 1. Overview 3 2. L-functions: analytic properties 5 2.1. Analytic continuation 5 2.2. Epstein ζ functions 7 2.3. The Dedekind zeta function 10 2.4. Generalizations 14 3. L-functions: arithmetic properties 18 3.1. Rationality 18 3.2. p-adic continuation. 18 3.3. Abstract sequence spaces 19 3.4. Application to rationality 21 4. p-adic L-functions 25 4.1. Analyticity for real quadratic fields 25 4.2. Totally real fields 30 5. Hurwitz zeta functions 35 5.1. Interlude on analysis 35 5.2. Hurwitz zeta functions 36 5.3. Explicit evaluations 38 6. Artin L-functions 41 6.1. Motivation 41 6.2. Artin’s conjecture 42 6.3. The conductor-discriminant formula 44 6.4. Analytic properties 46 6.5. Positive characteristic speculation 49 7. Stark’s conjectures 51 7.1. The class number formula 51 7.2. Aside: how to compute L(s,ρ) 52 7.3. Stark’s conjectures 54 7.4. Compatibility with class number formula 58 7.5. Imaginary quadratic fields 61 8. Class numbers of cyclotomic fields 67 1 Math 263C 2015 8.1. Reformulating Stark’s conjecture 67 8.2. Stickelberger’s Theorem 69 8.3. Herbrand’s Theorem 76 9. Converse to Herbrand’s Theorem 79 9.1. Ribet’s proof 79 9.2. Cyclotomic units 80 9.3. Converse to Herbrand 88 9.4. Euler Systems 89 2 Math 263C 2015 1. OVERVIEW Let K be a number field. Then we have an associated Dedekind ζ func- tion X s ζK (s ) = (Nm I )− I K ⊂O generalizing the Riemann zeta function (which is the special case K = Q), and possessing the following basic properties: The series defining ζK (s ) is convergent for Reps > 1. (This is easy; • it follows from the observation that there are very few ideals of a given norm.) ζK (s ) has an Euler product factorization: • Y 1 ζ s . K ( ) = 1 Nmp s p prime ( )− − ζK (s ) has a meromorphic continuation to s C, with the only • pole being a simple pole at s = 1. 2 We have a class number formula • hR r1 r2 Ress =1 ζK (s ) = 2 (2π) (1) w pdisc K Here R is the regulator, which can be thought of as the “volume” of K∗ , h is the class number, w is the number of roots of unity, andOr1 and r2 are the number of real and complex places of K . It satisfies a functional equation. An elegant way to phrase this is • that s =2 r1 r2 (disc K ) ΓR(s ) ΓC(s ) ζK (s ) is symmetric under s 1 s. $ − Here s s =2 s 2 and s 2 2 s s . ΓR( ) = π− Γ( = ) ΓC( ) = ( π)− Γ( ) We will begin by discussing the proofs of these properties, following the classical method of Hecke. If K is Galois over Q, then Y ζK (s ) = L(s,ρ) ρ irred. rep’n. of Gal(K =Q) where L(s,ρ) is the Artin L-function attach to ρ. Each L(s,ρ) has (at least conjecturally) analogous properties to those mentioned above. Example 1.0.1. If K = Q(i ), then X 1 ζ s K ( ) = m 2 n 2 s n+mi ( + ) 3 Math 263C 2015 and the corresponding factorization into Artin L-functions is s s s ζK (s ) = ζ(s ) (1− 3− + 5− ...). |{z}| − {z − } ρ=triv ρ=sign The main point of the course is to discuss the following question: since ζK factors, there should be a corresponding factorization of the class num- ber formula (1): so the right hand side should be expressible as hR Y h ρ R ρ “ ( ) ( )”. w = w ρ (ρ) Even to formulate precisely what this factorization should be is a little tricky. Example 1.0.2. Even for the case K = Q(i ) (when the Galois group has size 2, and there are two representations), what goes on is subtle because of issues at 2. The problem is analogous to the following general issue: if A is a finite abelian group with an action of Z=2, then one can define “eigenspaces” A+,A . The map A+ A A fails to be an isomorphism − − in the presence of 2-torsion, which× appears! in both “eigenspaces.” This idea motivates many things in number theory, like Stark’s conjec- ture and the Main Conjecture of Iwasawa theory. 4 Math 263C 2015 2. L-FUNCTIONS: ANALYTIC PROPERTIES We’ll now prove some of the basic properties of ζK , following Hecke. 2.1. Analytic continuation. We first digress briefly about what it means to have analytic continuation (we use this blanket term even when re- ferring to what might technically be called meromorphic continuation). When one says that the standard ζ function X 1 ζ(s ) = s n n has “analytic continuation,” it usually comes with connotations of com- plex analysis. For us, complex analysis is not important; what’s important is that there is a way to “make sense” of the expression for any s . Example 2.1.1. How might we make sense of P 1 ? Observe that at least pn P 1 for Reps > 1, n s can be approximated by an integral. Therefore, we consider N Z N X1 1 X 1 d x = lim . n s N n s x s n=1 !1 n=1 − 0 N 1 s − You can prove that this has a limit for Reps > 0. The integral is 1 s , so − N X 1 N 1 s lim − N n s 1 s n 1 − !1 = − exists for Reps > 0. It equals ζ(s ) for Reps > 1, and we can “make sense” of ζ(s ) by simply setting it to be the above expression for all Reps > 0. You can play similar games to extend it further “by hand.” Example 2.1.2. Let K = Q(i ). Then X 1 1 X 1 . m 2 n 2 s = 4 m 2 n 2 s m+ni ( + ) (m,n)=(0,0) ( + ) 6 We can interpret this as the sum of a particular binary quadratic form 2 over the lattice Z C. We can generalize this by asking about the sum of any (say definite) quadratic⊂ form Q over a lattice Λ C. ⊂ ¨ positive definite « binary quadratic forms ax 2 bx y cy 2 =SL2(Z) o / H=SL2(Z) + + O disc=1 n lattices Λ C o of covolume⊂ 1 up to rotation 5 Math 263C 2015 TONY: [discriminant should be 1 to be positive-definite, eh?] The horizontal♠♠♠ map sends Q to the root of Q−(z ,1) lying in H. The vertical map sends z 1,z =Im(z )1=2, with the associated quadratic form Λ (choos- ing a positively7! h i oriented basis). We actually show that there is a kind of functional equation for a gen- eral sum of the form X 2s z − , z =Q(z ). z Λ 0 jj jj jj jj 2 −{ g This is (for fixed s ) a real-analytic Eisenstein series on SL2(Z) H. More n precisely, we’ll show that if Λ R is a lattice of covolume 1, n ⊂ X s ΓR(s ) v − v Λ jj jj 2 has an analytic continuation with simple poles at s = 0 and s = n, and is symmetric under s n s . Then we’ll recover the results for ζQ(i ) by taking Λ = Z[i ] C. However,7! − it’s not so obvious what to do for real qua- dratic fields like⊂ . ζQ(p2) Example 2.1.3. Just to convince you that this is not a mysterious object, let’s write it in a more concrete way. X 1 ζ ( 2)(s ) = . Q p (a 2 2b 2)s a+bp2 Z[p2]=units 2 − The unit group 2 is generated by 1, 2 1. Let 2 1; then is Z[p ]∗ p α = p α positive under one embedding and negative− − under the other.− Therefore, any β = a +bp2 can be made totally positive by multiplying by a unit. If 2 β = a +bp2 is totally positive, then by multiplying by something in (α )Z we can assume that β 1 < α4. β ≤ You can check that in terms of a and b, this is equivalent to b > 0,3b 2a. This gives a fundamental domain in 2 for 2 2 . So the≤ zeta R Z[p ]=Z[p ]∗ 2 2 1 function is the sum over a,b in some cone in of 2 2 s . This ( ) Z R (a 2b ) ⊂ − 2 is again a sum of a quadratic form over lattice points in a cone in R , but the quadratic form is indefinite. However, the cone is away from the indefinite locus, so it doesn’t “see” the indefiniteness. This idea is due to Shintani, and we’ll discuss it again later. The cones are not simplicial, i.e. the integer points on the interior are not an inte- ger linear combination of those on the boundary. Shintani works with simplicial subdivisions to resolve this, but it isn’t important for us. 6 Math 263C 2015 For any arbitrary imaginary quadratic field K , ζK (s ) is a sum of sums of the form P z 2s as ranges over the ideal classes of K . Hecke re- z Λ 0 − Λ alized that there2 −{ isg j aj way to extend this to real quadratic fields as well. For imaginary K , we get a finite subset of SL2(Z) H associated to K (namely, the points corresponding to the lattices of then ideal classes).