18.786: Number Theory II (lecture notes) Taught by Bjorn Poonen Spring 2015, MIT Last updated: May 14, 2015 1 Disclaimer These are my notes from Prof. Poonen's course on number theory, given at MIT in spring 2015. I have made them public in the hope that they might be useful to others, but these are not official notes in any way. In particular, mistakes are my fault; if you find any, please report them to: Eva Belmont ekbelmont at gmail.com 2 Contents 1 February 36 Analysis review: Γ function, Fourier transform, θ function, measure theory; proof of the analytic continuation of ξ (and ζ) using the functional equation for θ 2 February 5 10 More analysis review: Lp spaces, Riesz representation theorem, Haar measures, LCA groups, characters, Pontryagin duality 3 February 12 14 Fourier inversion formula for LCA groups; equivalent definition of local fields; all characters are gotten by shifting a \standard" one, and Fb =∼ F 4 February 19 17 Conductor of a character; self-dual Haar measure of F (under Fourier transform) with standard ; unramified characters are j js; all characters are η · j js; twisted dual; L(χ) and local zeta integral Z(f; χ); statement of meromorphic continuation and functional equation for Z(f; χ) 5 February 24 21 Proof of meromorphic continuation and functional equation of the local zeta integral Z(f; χ) 6 February 26 25 Standard additive character ; Tamagawa measure on A; vol(A=K) = 1; Schwarz- Bruhat functions; Fourier transforms for A $ A and A=K $ K; Poisson summation formula 7 March 3 28 Proof of the Poisson summation formula; proof of Riemann-Roch using Poisson summation formula; norm of an id`ele;id`eleclass characters 8 March 5 32 Meromorphic continuation and functional equation for global zeta integrals Z(f; χ) 9 March 10 36 Local to global information; G-modules; definition(s) of group cohomology 3 10 March 12 41 More equivalent definitions of group cohomology; restriction, inflation, corestriction maps; Shapiro's lemma; homology 11 March 17 44 Tate cohomology Hb ∗ (for a finite group); Tate cohomology is simple for Z=NZ; cup products on H∗ and Hb ∗; definition/examples of profinite groups 12 March 19 48 Supernatural numbers; pro-p-groups and Sylow p-subgroups; discrete G-modules; cohomology of profinite groups; cohomological dimension 13 March 31 52 Nonabelian cohomology (H0 and H1); L=K-twists; Hilbert Theorem 90; Galois ´etale k-algebras 14 April 2 55 Equivalence of categories between k-vector spaces and L-vector spaces with semilinear 1 G-action; H (G; GLr(L)) = f0g (general version of Hilbert theorem 90); examples: K¨ummer theory, Artin-Schreier theory, elliptic curves 17 April 14 59 Galois representations ρ : GK ! GLn(Qp); semisimplifications of representations and Brauer-Nesbitt theorem; Tate twists; geometric Frobenius; image of Frobenius in Gk; Weil group 18 April 16 62 Grothendieck's `-adic monodromy theorem (formula for ρ : W ! GL(V ) on some open subgroup in terms of some nilpotent twisted matrix); Weil-Deligne representations 19 April 23 66 Deligne's classification of `-adic representations of W (they are $ Weil-Deligne representations); attempting to extend representations from W to G 20 April 28 69 More on conditions for representations of W to be representations of G; unipotent and semisimple matrices; an analogue of Jordan decomposition for non-algebraically closed fields; Frobenius-semisimple WD representations 4 21 April 30 73 Characterization of indecomposable Frobenius-semisimple WD representations; admissible representations of GLn(K); statement of local Langlands correspondence 22 May 5 76 Big diagram depicting relationships between various representation groups studied so far; local L-factors for representations of W , and for Weil-Deligne representations 23 May 7 80 24 May 12 84 25 May 14 87 5 Number theory Lecture 1 Lecture 1: February 3 Topics: • Tate's thesis (what led to the Langlands program) • Galois cohomology • Introduction to Galois representation theory (also tied in to the Langlands program) Reference (for Tate): Deitmar Introduction to harmonic analysis. Also, Bjorn will be posting official notes! Before Tate, there was Riemann { the prototype for Tate's thesis is the analytic continuation and functional equation for the Riemann zeta function. For Re s > 1, recall Y X ζ(s) = (1 − p−s)−1 = n−s: n≥1 Theorem 1.1 (Riemann, 1860). (Analytic continuation) The function ζ(s) extends to a meromorphic function on C, which is holomorphic except for a simple pole at 1. (Functional equation) The completed zeta function1 ξ(s) = π−s=2Γ(s=2)ζ(s), where Γ is the gamma function, satisfies ξ(s) = ξ(1 − s). Hecke (1918, 1920) generalized this to ζK (s) and other L-functions. Margaret Matchett (1946) started reinterpreting this in adelic terms, and John Tate (1950) finished this in his Ph.D. thesis. (It's not just a reinterpretation: it gives more information than Hecke's proofs.) Analytic prerequisites. Recall that the Gamma function is Z 1 dt Γ(s) = e−tt−s : 0 t −t × −s × × dt Note that e is a function R ! C and t (as a function of t) takes R>0 ! C and t is a × Haar measure on R>0 (it's translation-invariant on the multiplicative group). (If you do this sort of thing over a finite field you get Gauss sums!) This is convergent for Re s > 0. Proposition 1.2. (1) Γ(s + 1) = sΓ(s) (use integration by parts) (2) Γ(s) extends to a meromorphic function on C with simple poles at 0; −1; −2;::: , and no zeros (3) Γ(n) = (n − 1)! for n 2 Z≥1 1 p R 1 −x2 p (4) Γ 2 = π (by a change of variables this is equivalent to −∞ e dx = π) Now I'll review the Fourier transform. 1think of ξ as ζ with extra factors corresponding to the infinite places 6 Number theory Lecture 1 n Definition 1.3. f : R ! C tends to zero rapidly if for every n ≥ 1, x f(x) ! 0 as jxj ! 1 1 (i.e. jf(x)j = O jxjn ). (r) Definition 1.4. Call f : R ! C a Schwartz function if for every r ≥ 0, f tends to zero rapidly. Write S = S (R) for the set of Schwartz functions. Examples: e−x2 , zero, bump functions (any C1 function with compact support). Definition 1.5. Given f 2 S , define the Fourier transform Z fb = f(x)e−2πixydx: R Because f is a Schwartz function, this converges, and it turns out that fb is also a Schwartz function. 2 Example 1.6. If f(x) = e−πx , then f 2 S and fb = f. You can also define Fourier transforms on L2. Eventually we'll have to generalize all of this from R to compact abelian groups. Theorem 1.7 (Fourier inversion formula). If f 2 S then Z f(x) = feb 2πixydy: R In particular, fbb(x) = f(−x). Theorem 1.8 (Poisson summation formula). If f 2 S , then X X f(n) = fb(n): n2Z n2Z Definition 1.9. For real t > 0, define X 2 θ(t) = e−πn t: n2Z This actually makes sense for any t in the right half plane. You can also define θ(it) = θ(t), defined in the upper half plane; it is a modular form. In general, theta functions are associated to lattices. In this case, the lattice is just Z ⊂ R. Theorem 1.10 (Functional equation of θ). For every real t > 0, 1 − 2 1 θ(t) = t θ t : 7 Number theory Lecture 1 x Proof. If f 2 S and c 6= 0, then the Fourier transform of f c is cfb(cy). Apply −πx2 − 1 −πtx2 this to f(x) = e and c = t 2 to get that the Fourier transform of ft(x) = e is −π 1 y2 − 1 e ( t ) fbt(y) = t 2 . Now apply the Poisson summation formula to ft(x) to get −πn2t X − 1 −πn2· 1 e = t 2 e t : n2Z P −πn2t Note that θ(t) = 1 + 2 n≥1 e , so X 2 θ(t) − 1 e−πn t = : 2 n≥1 P −s=2 s −s Proof of Theorem 1.1. Recall ξ(s) = n π Γ( 2 )n ; start by looking at an indi- vidual summand: 1 s s Z dx − 2 s −s − 2 −s −x s=2 π Γ 2 n = π n e x 0 x 2 dx Make a change of variables x = πn t and recall that x is translation-invariant: 1 Z 2 dt = e−πn tts=2 0 t Now sum over n. As long as you can justify interchanging the sum and integral, Z 1 θ(t) − 1 dt ξ(s) = ts=2 0 2 t Why can you interchange the sum and integral? For s 2 R>1, this is OK because everything is nonnegative and the sum on the left converges. In fact, P R ::: converges absolutely for any complex s: changing the imaginary part does not affect the absolute value jts=2j. If s < 0, you can't expect this to make any sense: if t is close to zero, then the ts=2 part won't converge, and the θ part doesn't help enough. Now I want to replace this expression with something that does make sense for all s. Plan: we have Z 1 θ(t) − 1 dt Z 1 θ(t) − 1 dt ξ(s) = ts=2 + ts=2 1 2 t 0 2 t I(s) θ(t)−1 P −πn2t where I(s) converges for all s 2 C: this is because 2 = n≥1 e , and as t ! 1, the first term (n = 1) dominates.