Are ζ-functions able to solve Diophantine equations?

Are ζ-functions able to solve Diophantine equations? An introduction to (non-commutative) Iwasawa theory Mathematical Institute University of Heidelberg CMS Winter 2007 Meeting

Leibniz (1673) L-functions The analytic class number formula 1 1 3 + 1 5 1 7 + 1 9 1 11 + = π 4 (already known to GREGORY and MADHAVA)

L-functions The analytic class number formula Special values of L-functions

L-functions The analytic class number formula N 1, (Z/NZ) units of ring Z/NZ. Dirichlet Character (modulo N) : χ : (Z/NZ) C extends to N χ(n) := { χ(n mod N), (n, N) = 1; 0, otherwise.

L-functions The analytic class number formula Dirichlet L-function w.r.t. χ : satisfies: - Euler product L(s, χ) = n=1 χ(n), s ɛ C, R(s) > 1. ns L(s, χ) = p 1 1 χ(p)p s, - meromorphic continuation to C, - functional equation.

Examples L-functions The analytic class number formula χ 1 : Riemann ζ-function 1 ζ(s) = n s = p n=1 1 1 p s, χ 1 : (Z/4Z) = {1, 3} C, χ 1 (1) = 1, χ 1 (3) = 1 L(1, χ 1 ) = 1 1 3 + 1 5 1 7 + 1 9 1 11 + = π 4

Examples L-functions The analytic class number formula χ 1 : Riemann ζ-function 1 ζ(s) = n s = p n=1 1 1 p s, χ 1 : (Z/4Z) = {1, 3} C, χ 1 (1) = 1, χ 1 (3) = 1 L(1, χ 1 ) = 1 1 3 + 1 5 1 7 + 1 9 1 11 + = π 4

L-functions The analytic class number formula

L-functions The analytic class number formula Conjectures of Catalan and Fermat p, q prime numbers Catalan(1844), Theorem(MIHǍILESCU, 2002): x p y q = 1, has unique solution in Z with x, y > 0. 3 2 2 3 = 1 Fermat(1665), Theorem(WILES et al., 1994): x p + y p = z p, p > 2, has no solution in Z with xyz 0.

L-functions The analytic class number formula Conjectures of Catalan and Fermat p, q prime numbers Catalan(1844), Theorem(MIHǍILESCU, 2002): x p y q = 1, has unique solution in Z with x, y > 0. 3 2 2 3 = 1 Fermat(1665), Theorem(WILES et al., 1994): x p + y p = z p, p > 2, has no solution in Z with xyz 0.

L-functions The analytic class number formula Factorisation over larger ring of integers ζ m primitive mth root of unity Z[ζ m ] the ring of integers of Q(ζ m ), e.g. for m = 4 with i 2 = 1 we have in Z[i] = {a + bi a, b ɛ Z} : x 3 y 2 = 1 x 3 = (y + i)(y i) and for m = p n we have in Z[ζ p n] : x pn +y pn = (x + y)(x + ζ p ny)(x + ζ 2 p ny)... (x + ζpn 1 p n y) = z pn.

The strategy L-functions The analytic class number formula Hope: Use unique prime factorisation to conclude a contradiction from the assumption that the Catalan or Fermat equation has a non-trivial solution. Problem: In general, Z[ζ m ] is not a unique factorisation domain (UFD), e.g. Z[ζ 23 ]!

The strategy L-functions The analytic class number formula Hope: Use unique prime factorisation to conclude a contradiction from the assumption that the Catalan or Fermat equation has a non-trivial solution. Problem: In general, Z[ζ m ] is not a unique factorisation domain (UFD), e.g. Z[ζ 23 ]!

Ideals L-functions The analytic class number formula Kummer: Replace numbers by ideal numbers : For ideals(=z[ζ m ]-submodules) 0 a Z[ζ m ] we have unique factorisation into prime ideals P i 0 : a = n i=1 P n i i Principal ideals: (a) = Z[ζ m ]a

Ideal class group L-functions The analytic class number formula Cl(Q(ζ m )) : = { ideals of Z[ζ m ]}/{ principal ideals of Z[ζ m ]} = Pic(Z[ζ m ]) Fundamental Theorem of algebraic number theory: #Cl(Q(ζ m )) < and h Q(ζm) := #Cl(Q(ζ m )) = 1 Z[ζ m ] is a UFD Nevertheless, Cl(Q(ζ m )) is difficult to determine, too many relations!

The L-function solves the problem L-functions The analytic class number formula How can we compute h Q(i)? It is a mystery that knows the answer! L(s, χ 1 )

The L-function solves the problem L-functions The analytic class number formula How can we compute h Q(i)? It is a mystery that knows the answer! L(s, χ 1 )

The cyclotomic character L-functions The analytic class number formula Gauß: G(Q(ζ N )/Q) κ N = (Z/NZ) with g(ζ N ) = ζ κ N(g) N N = 4 : for all g ɛ G(Q(ζ N )/Q) χ 1 is character of Galois group G(Q(i)/Q) L(s, χ 1 ) (analytic) invariant of Q(i).

L-functions The analytic class number formula Analytic class number formula for imaginary quadratic number fields: h Q(i) = #µ(q(i)) N L(1, χ 1 ) 2π = 4 2 2π L(1, χ 1) = 4 π L(1, χ 1) = 1 (by Leibniz formula) Z[i] is a UFD.

L-functions The analytic class number formula Analytic class number formula for imaginary quadratic number fields: h Q(i) = #µ(q(i)) N L(1, χ 1 ) 2π = 4 2 2π L(1, χ 1) = 4 π L(1, χ 1) = 1 (by Leibniz formula) Z[i] is a UFD.

L-functions The analytic class number formula Analytic class number formula for imaginary quadratic number fields: h Q(i) = #µ(q(i)) N L(1, χ 1 ) 2π = 4 2 2π L(1, χ 1) = 4 π L(1, χ 1) = 1 (by Leibniz formula) Z[i] is a UFD.

L-functions The analytic class number formula A special case of the Catalan equation Since L(s, χ 1 ) knows the arithmetic of Q(i), it is able to solve our problem: Claim: x 3 y 2 = 1 has no solution in Z. In the decomposition x 3 = (y + i)(y i) the factors (y + i) and (y i) are coprime (easy!) y + i = (a + bi) 3 for some a, b ɛ Z taking Re( ) and Im( ) gives: y = 0, contradiction!

Regular primes L-functions The analytic class number formula Similarly ζ(s) knows for which p Cl(Q(ζ p ))(p) = 1 holds! Then the Fermat equation does not have any non-trivial solution. But 37 h Cl(Q(ζ37 ))! Iwasawa: Cl(Q(ζ p n))(p) =? for n 1.

Number fields versus function fields Q F l (X) = K (P 1 F l ) K /Q number field K (C)/F l (X) function field C P n F l smooth, projective curve, i.e. K (C)/F l (X) finite extension

Counting points on C N r := #C(F l r ) cardinality of F l r -rational points φ : C C x i xi l Frobenius-automorphism Lefschetz-Trace-Formula N r = #{Fix points of C(F l ) under φ r } = 2 ( 1) n Tr ( φ r H n (C) ) n=0

ζ-function of C, WEIL (1948) ζ C (s) : = x ɛ C 1, s ɛ C, R(s) > 1, 1 (#k(x)) s ( t r ) = exp N r, t = l s r = r=1 2 det(1 φt H n (C)) ( 1)n+1 n=0 = det(1 φt Pic0 (C) ) (1 t)(1 lt) ɛ Q(t)

Riemann hypothesis for C ζ C is a rational function in t and has poles at: s = 0, s = 1 zeroes at certain s = α satisfying R(α) = 1 2. Can the Riemann ζ-function also be expressed as rational function?

p-adic ζ-functions Main Conjecture

Tower of number fields p-adic ζ-functions Main Conjecture Studying the class number formula in a whole tower of number fields simultaneously: Q F 1... F n F n+1... F := n 0 F n. with F n := Q(ζ p n), 1 n, F Z p = lim n Z/p n Z Q p := Quot(Z p ), κ : G := G(F /Q) = Z p, F n G g(ζ p n) = ζ κ(g) p n for all g ɛ G, n 0 Q

p-adic ζ-functions Main Conjecture Z p = Z/(p 1)Z Z p, i.e. G = Γ with = G(F 1 /Q) = Z/(p 1)Z and Γ = G(F /F 1 ) = < γ > = Z p. Iwasawa-Algebra Λ(G) := lim Z p [G/G ] = Z p [ ][[T ]] G G open with T := γ 1.

p-adic functions p-adic ζ-functions Main Conjecture Maximal ring of quotients of Λ(G) : Q(G) = p 1 i=1 Q(Z p [[T ]]). Z = (Z 1 (T ),..., Z p 1 (T )) ɛ Q(G) are functions on Z p : for n ɛ N Z (n) := Z i(n) (κ(γ) n 1) ɛ Q p { }, i(n) n mod (p 1)

Analytic p-adic ζ-function p-adic ζ-functions Main Conjecture KUBOTA, LEOPOLDT and IWASAWA: ζ p ɛ Q(G) such that for k < 0 ζ p (k) = (1 p k )ζ(k), i.e. ζ p interpolates - up to the Euler-factor at p - the Riemann ζ-function p-adically.

p-adic ζ-functions Main Conjecture Ideal class group over F IWASAWA: #Cl(F n )(p) = p nrk Zp (X)+const where X := X(F ) = lim n Cl(F n )(p) with G-action, Z p (1) := lim n µ p n with G-action, X Zp Q p, Q p (1) := Z p (1) Zp Q p finite-dimensional Q p -vector spaces with operation by γ.

Iwasawa Main Conjecture p-adic ζ-functions Main Conjecture MAZUR and WILES (1986): ζ p det(1 γt X Zp Q p ) det(1 γt Q p (1)) mod Λ(G) i det(1 γt H i ) ( 1)i+1 mod Λ(G) analytic algebraic p-adic ζ-function Trace formula

The analogy p-adic ζ-functions Main Conjecture function field number field F l = F l (µ) F = Q(µ(p)) φ γ C ζ C G m ζ p Pic 0 (C) X = Pic(F )

Non-commutative Iwasawa Theory

From G m to arbitrary representations up to now: coefficients in cohomology: Z p (1) G m, µ(p) tower of number fields: F = Q(µ(p))

Generalisations ρ : G Q GL(V ) (continuous) representation with V = Q n p and Galois-stable lattice T = Z n p. coefficients in cohomology: T ρ tower of fields: K = Q ker(ρ) Example: E elliptic curve over Q, T = T p E = lim E[p n ] = Z 2 p, V := T Zp Q p. n

p-adic Lie extensions F K such that G := G(K /Q) GL n (Z p ) is a p-adic Lie group with subgroup H such that Γ := G/H = Z p H F Γ Q G K

Philosophy Attach to (ρ, V ) analytic p-adic L-function L(V, K ) with interpolation property L(V, K ) L(1, V χ) for χ : G GL n (Z p ) with finite image. algebraic p-adic L-function F(V, K ). Problem: Λ(G) in general not commutative! Non-commutative determinants?

Non-commutative Iwasawa Main Conjecture COATES, FUKAYA, KATO, SUJATHA, V.: There exists a canonical localisation Λ(G) S of Λ(G), such that F(V, K ) exists in K 1 (Λ(G) S ). Also L(V, K ) should live in this K -group. Main Conjecture: L(V, K ) F(V, K ) mod K 1 (Λ(G)).

Non-commutative characteristic polynomials M Λ(G)-module, which is finitely generated Λ(H)-module BURNS, SCHNEIDER, V.: Λ(G) S Λ(H) M 1 γ Λ(G) S Λ(H) M = induces det(1 γt M) ɛ K 1 (Λ(G) S ). Main Conjecture over K : Trace formula" in K 1 (Λ(G) S ) mod K 1 (Λ(G)): L(K, Z p (1)) det(1 γt H (K, Z p (1)))

New Congruences If G = Z p Z p and coefficients: Z p (1) KATO: K 1 (Λ(G)) χ i O i [[T ]] K 1 (Λ(G) S ) χ i Quot(O i [[T ]]) L(K /Q) (L p (χ i, F )) i Existence of L(K /Q) congruences between L p (χ i, F ) Main Conjecture /K Main Conjecture /F for all χ i Similar results by RITTER, WEISS for finite H.

A theorem for totally real fields F totally real, F cyc K totally real, G = Z p Z p MAHESH KAKDE, a student of Coates, recently announced: Theorem (Kakde) Assume Leopoldt s conjecture for F. Then the non-commutative Main Conjecture for the Tate motive (i.e. for = G m ) holds over K /F.