Computing zeta functions and L-functions Lecture 1 Andrew V. Sutherland June 24, 2019 CMI-HIMR Summer School in Computational Number Theory Zeta functions of curves and their function fields Recall that the zeta function of a nice curve X=Fq is defined by 0 1 X Nr Z (T ) := exp T r ; X @ r A r≥1 where Nr := #X(Fqr ). Equivalently, if we put K := Fq(X) then X n Y e −ce ZX (T ) = ZK (T ) := bnT = (1 − T ) ; n≥1 e≥1 where bn counts effective divisors of degree n, and ce counts prime divisors (places of K, equivalently, closed points of X) of degree e. Indeed, we have X Nr = #X(Fqr ) = ece; ejr X X X 1 X Nr log Z (T ) = − c log(1 − T e) = c T de = T r: X e e d r e≥1 e≥1 d≥1 r≥1 Key properties of the zeta function of a curve From the Weil conjectures for curves (and abelian varieties), we have LX (T ) 1. ZX (T ) = (1−T )(1−qT ) with LX 2 Z[T ] of degree 2g. g 2g g−1 2g−1 g+1 g 2. LX (T ) = q T + q a1T + ··· + qag−1T + ag T g−1 1 + a1T + ··· + ag−1T + Q2g 1=2 3. LX (T ) = i=1(1 − αiT ) with jαij = q ; r P2g r Q2g r 4. #X(Fqr ) = q + 1 − 1 αi and # Jac(X)(Fqr ) = i=1(1 − αi ). It follows that a1; : : : ; ag determine N1;:::;Ng and conversely, and that both determine #X(Fqr ) and # Jac(X)(Fqr ) for all r ≥ 1. We also have the bounds ja j ≤ 2gqi=2 (which are not tight in general). p i i p Setting all αi to q, and then to − q, yields the Hasse-Weil bounds p 2g p 2g ( q − 1) ≤ # Jac(Fq) ≤ ( q + 1) spanning an interval of width 4gqg−1=2 + O(qg−3=2). The L-function of a curve Now let X be a nice curve of genus g over a number field K. The L-function of X is defined the Euler product X −s Y −s −1 L(X; s) = L(Jac(X); s) := ann := Lp(N(p) ) : n≥1 p where p varies over the primes of K (prime ideals of OK ) and N(p) := #Fp is the cardinality of the residue field Fp := OK =p. For primes p of good reduction for X we have Lp(T ) := LXp (T ), where Xp denotes the reduction of X to the residue field Fp. In every case Lp 2 Z[T ] has degree at most 2g. Thus the an are integers and L(X; s) is an arithmetic L-function of degree 2g with analytic 1 normalization Lan(X; s + 2 ) . It can happen that X has bad reduction at p but Jac(X) does not; from the L-function perspective, these are good primes. The Selberg class with polynomial Euler factors poly P −s The Selberg class S consists of Dirichlet series L(s) = n≥1 ann : 1. L(s) has an analytic continuation that is holomorphic at s 6= 1; s Qr 2. For some γ(s) = Q i=1 Γ(λis + µi) and ", the completed L-function Λ(s) := γ(s)L(s) satisfies the functional equation Λ(s) = "Λ(1 − s¯); Pr where Q > 0, λi > 0, Re(µi) ≥ 0, j"j = 1. Define deg L := 2 i λi. 3. a1 = 1 and an = O(n ) for all > 0 (Ramanujan conjecture). Q −s −1 4. L(s) = p Lp(p ) for some Lp 2 Z[T ] with deg Lp ≤ deg L (has an Euler product). 1 The Dirichlet series Lan(s; X) := L(X; s + 2 ) satisfies (3) and (4), and conjecturally lies in Spoly; for g = 1 and K totally real this is known. Strong multiplicity one Theorem (Kaczorowski-Perelli 2001) P −s P −s poly If A(s) = n≥1 ann and B(s) = n≥1 bnn lie in S and ap = bp for all but finitely many primes p, then A(s) = B(s). Corollary poly If Lan(s; X) lies in S then it is completely determined by any choice of all but finitely many coefficients ap. poly Henceforth we assume that Lan(s; X) 2 S . s g Let ΓC(s) = 2(2π) Γ(s) and define Λ(X; s) := ΓC(s) L(X; s). Then Λ(X; s) = "N 1−sΛ(X; 2 − s): where the analytic root number " = ±1 and the analytic conductor N 2 Z≥1 are determined by the ap (let us take these as definitions). Testing the functional equation g R 1 s−1 Let G(x) be the inverse Mellin transform of ΓC(s) = 0 G(x)x dx, and define 1 X S(x) := a G(n=x); x n R 1 −s so that Λ(X; s) = 0 S(x)x dx, and for all x > 0 we have S(x) = "S(N=x): The function G(x) decays rapidly, and for sufficiently large c0 we have 1 X S(x) ≈ S (x) := a G(n=x); 0 x n n≤c0x with an explicit bound on the error jS(x) − S0(x)j. Effective strong multiplicity one Fix a finite set of small primes S (e.g. S = f2g) and an integer M that we know is a multiple of the conductor N (e.g. M = ∆(X)). There is a finite set of possibilities for " = ±1, NjM, and the Euler factors Lp 2 Z[T ] for p 2 S (the coefficients of Lp(T ) are bounded). p Suppose we can compute an for n ≤ c1 M whenever p - n for p 2 S. p We now compute δ(x) := jS0(x) − "S0(N=x)j with x = c1 N for every possible choice of ", N, and Lp(T ) for p 2 S. If all but one choice makes δ(x) larger than our explicit error bound, we know the correct choice. 1 For a suitable choice of c1 this is guaranteed to happen. One can explicitly determine a set of O(N ) candidate values of c1, one of which is guaranteed to work; in practice the first one usually works. 1Subject to our assumptions; if it does not happen then we have found an explicit counterexample to the conjectured Langlands correspondence. Conductor bounds The formula of Brumer and Kramer gives explicit bounds on the p-adic valuation of the algebraic conductor N of Jac(X): vp(N) ≤ 2g + pd + (p − 1)λp(d); 2g P i P i where d = b p−1 c and λp(d) = idip , with d = dip , 0 ≤ di < p. g p = 2 p = 3 p = 5 p = 7 p > 7 1 8 5 2 2 2 2 20 10 9 4 4 3 28 21 11 13 6 For g ≤ 2 these bounds are tight (see www.lmfdb.org for examples). For hyperelliptic curves N divides ∆(X). What about other curves? Algorithms to compute zeta functions Given X=Q of genus g, we want to compute Lp(T ) for all good p ≤ B. complexity per prime (ignoring factors of O(log log p)) algorithm g = 1 g = 2 g = 3 point enumeration p log p p2 log p p3(log p)2 group computation p1=4 log pp 3=4 log p p(log p)2 p-adic cohomology p1=2(log p)2 p1=2(log p)2 p1=2(log p)2 CRT (Schoof-Pila) (log p)5 (log p)8 (log p)14 average poly-time (log p)4 (log p)4 (log p)4 P −s 2 3 For L(X; s) = ann , we only need ap2 for p ≤ B, and ap3 for p ≤ B. For 1 < r ≤ g we r can compute all apr with p ≤ B in time O(B log B) using naive point counting. The bottom line: it all comes down to computing ap's at good primes, equivalently, computing #X(Fp) = p + 1 − ap (aka counting points). The divisor group of a curve (function field) Recall that we have a (contravariant) equivalence of categories fnice curves X=Fqg ! ffunction fields K=Fqg ; ∗ which sends X to Fq(X) and morphisms ': X ! Y to field embeddings ' : Fq(Y ) ! Fq(X) defined by f 7! f ◦ '. We have a bijection between closed points P of X (GFq -orbits of X(Fq)) and places P of K (equivalence classes of absolute values of K). The divisor group Div(X) = Div(K) is the free abelian group on closed points (places) P . Each D 2 Div(X) has the form X D = nP P: P × P For f 2 K we define div(f) := P vP (f)P , and let Princ(X) denote the subgroup fdiv(f): f 2 K×g [ f0g of principal divisors. The divisor class group P Define the homomorphism deg: Div(X) ! Z by D 7! P nP deg(P ), where deg(P ) = #P = [κ(P ): Fq]; note that Princ(X) ⊆ ker deg. Div(X) 0 We now define Pic(X) := Princ(X) , and the divisor class group Pic (X) as the kernel of the degree map Pic(X) ! Z, yielding the exact sequence 0 0 −! Pic (X) −! Pic(X) −! Z −! 0: Provided that X has a rational point we have a functorial isomorphism 0 0 2 Pic (X) ' Jac(X)(Fq), meaning Pic (XL) ' Jac(X)(L) for all L=Fq. We shall henceforth assume X(Fq) contains a rational point O. W now define the Abel-Jacobi map X ! Pic0(X) P 7! [O − P ] When X is an elliptic curve this map is an isomorphism.