Explicit formulas and symmetric function theory

Transcription

1 and symmetric function theory Department of Mathematics University of Michigan Number Theory and Function Fields at the Crossroads University of Exeter January 206

2 Arithmetic functions on Z over short intervals Q: How many primes are there in a random short interval [x, x + H]? Here x [X, 2X ] is random, H = X δ, with δ = (0, ). Theorem (Prime number theorem) X 2X X xnx+h Λ(n) dx H Conjecture (Goldston-Montgomery) X 2X X ( xnx+h Λ(n) H) 2 dx H(log X log H) Conjecture (Good - Churchhouse) X 2X X ( xnx+h ) 2 6 µ(n) dx π H 2

3 Z to F q [T ] One may ask similar questions over F q[t ]. A dictionary: integers Z polynomials F q[t ] positive integers N F q[t ] monic log n deg(f ) n q deg(f ) {n : x n x + H} {g F q[t ] : deg(f g) h} =: I (f ; h) x [X, 2X ] f M n = {f monic : deg(f ) = n} primes irreducible polynomials { deg P, if f = cp k, Λ(n) Λ(f ) := 0, otherwise.

4 F q [T ] analogues for large q Proposition q n g I (f ;h) Λ(g) = q h+ Theorem (Keating-Rudnick) For 0 h n 4, as q Λ(g) q h+ 2 = q h+ (n h 2) + O(q h+/2 ), q n g I (f ;h) Analogous results for e.g µ(f ), Λ j (f ), d k (f ), etc. Theorem (Keating R Roditty-Gershon Rudnick) For 0 h n 4 and h ( /k)n 2, as q ( ) Var f Mn d k (g) = q h+ I k (n; n h 2) + O(q h+/2 ), g I (f ;h)

5 A lattice point count I k (m; N) Theorem I k (m; N) is equal to the count of lattice points x = (x (j) ) Z k2 satisfying each of the relations i 0 x (j) i N for all i, j k 2 x (k) + x (k ) x () = m, and k 3 x A k, where A k is the collection of k k matrices whose entries satisfy the following system of inequalities, x () x (2) x (k) x () 2 x (2) 2 x (k) 2. x () k. x (2) k x (k) k Proposition I k (m; N) = I k (kn m; N) Theorem For m N, (( )) k 2 ( m + k 2 ) I k (m; N) = m = k 2

6 Some related results Theorem (Bary-Soroker, Bender Pollack) For non-zero α M h for h < n, f Mn Λ(f )Λ(f + α) = q n + O(q n /2 ) Fits in a more general picture: Definition (Factorization type) For a squarefree polynomial f = p p k M n, define t f := (deg p,..., deg p k ), deg p deg p 2... Definition (Cycle type) For a permutation σ = σ σ k S n, define t σ := ( σ,..., σ k ), σ σ 2... e.g. f = T (T 2 + ) = p p 2, t f = (2, ) e.g. σ = ()(235)(4), t σ = (3,, ) = 2 3 More general picture: (Andrade Bary-Soroker - Rudnick) As q, Almost all elements of M n are squarefree t f t σ for random f M n and σ S n t f and t f +α are independent for random f M n Note: For f M n squarefree, Λ(f ) = n [t f = (n)] Can be applied to more general factorization functions; µ(f )µ(f + α) due earlier to Carmon-Rudnick.

7 Ideas from the proof for variance in short intervals Use character sums to localize over short intervals Use an explicit formula to relate a character sum to the zeros of an L-function: Tr Θ n χ Λ(f )χ(f ) q n/2 for L(u, χ) det( q /2 uθ χ) = Use an equidistribution theorem (Katz): N ( q /2 uω i ) i= {Θ χ} χ (mod T N+2 ), even {g} g U(N)

8 Factorization functions and explicit formulas Analogous to Λ(f ), other factorization functions are also related to symmetric polynomials in the eigenvalues of Θ χ, with χ (mod T N+2 ): Tr Θ n χ = p n (Θ χ ) := i h n (Θ χ ) := ω i ω in i i n j + +j k =m 0j,...,j k N e j e jk ω n i q n/2 q n/2 q n/2 Λ(f ) χ(f ) µ(f )χ(f ) d k (f )χ(f )

9 Factorization functions and explicit formulas A dictionary for squarefree f : ( ) n e n [deg(f ) = n] h n µ(f )[deg(f ) = n] p n Λ(f )[deg(f ) = n] s λ Q λ (f )?

10 Schur functions: essential facts s λ (ω,..., ω N ) is a symmetric polynomial in ω, indexed by a partition λ = (λ,..., λ l(λ) ) For example s (2,) (ω, ω 2, ω 3) = ω 2 ω 2 + ω ω ω 2 ω 3 + ω ω ω 2 2ω 3 + ω 2ω ω ω 2ω 3. For l(λ) > N, s λ (ω,..., ω N ) = 0. {s λ (ω,..., ω N ) : l(λ) N} is a basis for the space of symmetric polynomials in ω,..., ω N. {s λ (ω,..., ω N ) : l(λ) N} is an orthonormal basis for the space of class functions of U(N): s λ (g)s ν(g) dg = δ λ=ν δ l(λ),l(ν)n U(N) s λ can be expressed in terms of other symmetric functions: e.g. s (,) = h 2 h 2

11 Computations Q λ (f ) for f M 3 and squarefree, λ 3 (,,) (2,) (3) Q (3) - - Q (2,) -2 0 Q (,,) Proposition For λ n, and f squarefree Q λ (f ) = µ(f )X λ (t f )[deg(f ) = n] = ( ) n X λ (t f )[deg(f ) = n] where X λ are characters of the symmetric group. Q λ (f ) for f M 4, λ 4 (,,,) (2,,) (2,2) (3,) (4) Q (4) - - Q (3,) Q (2,2) Q (2,,) Q (,,,)

12 Schur functions of the zeros: Interpretations Proposition For λ n, and χ a Dirichlet character, Heuristic for RMT: s λ (Θ χ) = ( )n q n/2 squarefree {Θ χ} χ (T N+2 ), even {g} U(N) is equivalent to X λ (t f )χ(f ) + O(q /2 ) E χs λ (Θ χ)s ν(θ χ) = δ λ=ν δ l(λ),l(ν)n. But because L(u, χ) has only N non-trivial zeros s λ (Θ χ) = 0, for l(λ) > N. What about E χs λ (Θ χ)s ν(θ χ) = δ λ=ν for l(λ), l(ν) N?

13 Interpretations By harmonic analysis on Dirichlet characters: E χs λ (Θ χ)s ν(θ χ) q n f,g M n squarefree = q n For the diagonal terms: q n X λ X λ X λ (t f )X ν (t g )[deg(f g) n N 2] (tf )X ν (t f ) + q n (tf )X ν (t f ) n! σ S n X λ deg(α)n N 2 α 0 (σ)x ν (σ) = δ λ=ν X λ (t f )X ν (t f +α ). For the off-diagonal terms (by independence of factorization type): ( )( (tf )X ν ) (t q n f +α ) (σ) (σ) = 0 n! n! X λ σ S n X λ σ S n X ν RMT Error terms over α don t add constructively (for l(λ), l(ν) N) (Remark: one must handle λ = n separately, but easily)

14 Interpretations A (very) heuristic takeaway: Subject to the constraint that in the relevant family, each L-function has always a certain number of zeros, factorization information (got from X λ ) is as independent as possible of the metric information got from χ. A rigorous takeaway: Theorem (R.) For a factorization function a : F q[t ] C (that is, a(p e pe k k ) depends only on the data (deg(p ),..., deg(p k ); e,..., e k ), we can write a(f ) = λ â λ X λ (t f ) + b(f ), for b(f ) supported on the squarefuls. We have ( ) Var f Mn a(g) = q h+ g I (f ;h) l(λ )n h 2 λ n â λ 2 + O(q h+/2 ).

15 Thanks: Thanks!