From Nuclear Physics to Number Theory

How the Manhattan Project Helps us Understand Primes and Elliptic Curves

Steven J. Miller

Brown University

University of Connecticut, March 24^{th}, 2005 http://www.math.brown.edu/∼sjmiller

Acknowledgments

Random Matrices (with Peter Sarnak)

• Rebecca Lehman

• Yi-Kai Liu

• Chris Hammond

• Inna Zakharevich

Elliptic Curves (with Eduardo Dueñez)

• Adam O’Brien

• Aaron Lint

Fundamental Problem: Spacing Between Events

General Formulation: Studying system, observe values at t_{1}, t_{2}, t_{3}, . . ..

Question: what rules govern the spacings between the t_{i}?

Fundamental Problem: Spacing Between Events

General Formulation: Studying system, observe values at t_{1}, t_{2}, t_{3}, . . .. Question: what rules govern the spacings between the t_{i}? Examples:

• Spacings between Primes.

• Spacings between Energy Levels of Nuclei.

• Spacings between Eigenvalues of Matrices.

• Spacings between Zeros of Functions.

Goals of the Talk

• Determine correct scale to study spacings.

• See similar behavior in different systems.

• Discuss tools / techniques needed to prove the results.

• Predictive power of Random Matrix Theory: suggests answers for questions in Number Theory.

PART I

NORMALIZED SPACINGS

Example: Fractional Parts

• For α ∉ ℚ, set x_{n} = n^{2}α mod 1.

• Order x_{1}, …, x_{N}: 0 ≤ y_{1} ≤ ⋯ ≤ y_{N} ≤ 1.

Example: Fractional Parts

• For α ∉ ℚ, set x_{n} = n^{2}α mod 1.

• Order x_{1}, …, x_{N}: 0 ≤ y_{1} ≤ ⋯ ≤ y_{N} ≤ 1.

• Expect spacings between adjacent y’s of size \frac{1}{N}.

Example: Fractional Parts

• For α ∉ ℚ, set x_{n} = n^{2}α mod 1.

• Order x_{1}, …, x_{N}: 0 ≤ y_{1} ≤ ⋯ ≤ y_{N} ≤ 1.

• Expect spacings between adjacent y’s of size \frac{1}{N}.

• Should study \frac{y_{n+1}−y_{n}}{1/N}.

• Poissonian behavior for most α: behave like N numbers chosen uniformly in [0, 1].

Example: Primes

• π(x) = #\{p : p prime, p ≤ x\} ∼ \frac{x}{log x}

Example: Primes

• π(x) = #\{p : p prime, p ≤ x\} ∼ \frac{x}{log x}

• Average spacing between primes at most x is \frac{x}{π(x)} ∼ log x.

• If p_{n}, p_{n+1} ∼ x, study \frac{p_{n+1}}{log p_{n+1}} − \frac{p_{n}}{log p_{n}}.

Example: Primes

• π(x) = #\{p : p prime, p ≤ x\} ∼ \frac{x}{log x}

• Average spacing between primes at most x is \frac{x}{π(x)} ∼ log x.

• If p_{n}, p_{n+1} ∼ x, study \frac{p_{n+1}}{log p_{n+1}} − \frac{p_{n}}{log p_{n}}.

• One reason why twin primes are so hard: \frac{2}{log x} → 0.

Example: Primes

• π(x) = #\{p : p prime, p ≤ x\} ∼ \frac{x}{log x}

• Average spacing between primes at most x is \frac{x}{π(x)} ∼ log x.

• If p_{n}, p_{n+1} ∼ x, study \frac{p_{n+1}}{log p_{n+1}} − \frac{p_{n}}{log p_{n}}.

• One reason why twin primes are so hard: \frac{2}{log x} → 0.

Aside: (Nicely 1994) Twin primes led to finding the Pentium bug!

∑_{{p \atop p,p+2 prime}} (\frac{1}{p} + \frac{1}{p + 2}) ≈ 1.90216058.

Poisson Process: Spacings between Primes

Observation: See similar behavior in spacings between normalized fractional parts n^{2}α mod 1 and normalized primes.

Poisson Process: Spacings between Primes

Observation: See similar behavior in spacings between normalized fractional parts n^{2}α mod 1 and normalized primes.

Study spacings between normalized adjacent primes in I = [10^{15}, 10^{15} + 2 · 10^{7}].

Numerically observe and conjecture that as |I| → ∞

\frac{# \{p_{i} ∈ I : \frac{p_{i+1}}{log p_{i+1}} − \frac{p_{i}}{log p_{i}} ∈ [α, β]\}}{#\{p_{i} ∈ I\}} ⟶ ∫_{α}^{β} e^{−x}dx.

15

PART II

RANDOM MATRIX THEORY

Origins of Random Matrix Theory

Classical Mechanics: 3-Body Problem Intractable.

Origins of Random Matrix Theory

Classical Mechanics: 3-Body Problem Intractable. Heavy nuclei like Uranium (200+ protons / neutrons) even worse!

Origins of Random Matrix Theory

Classical Mechanics: 3-Body Problem Intractable. Heavy nuclei like Uranium (200+ protons / neutrons) even worse! Get some info by shooting high-energy neutrons into nucleus, see what comes out.

Origins of Random Matrix Theory

Classical Mechanics: 3-Body Problem Intractable. Heavy nuclei like Uranium (200+ protons / neutrons) even worse! Get some info by shooting high-energy neutrons into nucleus, see what comes out. Fundamental Equation: Hψ_{n} = E_{n}ψ_{n}

H : matrix, entries depend on system E_{n} : energy levels ψ_{n} : energy eigenfunctions

Origins of Random Matrix Theory (continued)

Statistical Mechanics: for each configuration, calculate quantity (say pressure).

Origins of Random Matrix Theory (continued)

Statistical Mechanics: for each configuration, calculate quantity (say pressure). Average over all configurations – most configurations close to system average.

Origins of Random Matrix Theory (continued)

Statistical Mechanics: for each configuration, calculate quantity (say pressure). Average over all configurations – most configurations close to system average. Nuclear physics: choose matrix at random, calculate eigenvalues, average over matrices.

Origins of Random Matrix Theory (continued)

Statistical Mechanics: for each configuration, calculate quantity (say pressure). Average over all configurations – most configurations close to system average. Nuclear physics: choose matrix at random, calculate eigenvalues, average over matrices. Look at: Real Symmetric (A^{T} = A), Complex Hermitian (\overline{A}^{T} = A), Classical Compact groups (unitary, orthogonal).

Random Matrix Ensembles

Real Symmetric Matrices:

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & ⋯ & a_{1N} \\ a_{21} & a_{22} & a_{23} & ⋯ & a_{2N} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{N1} & a_{N2} & a_{N3} & ⋯ & a_{NN} \end{matrix} ) = A^{T} , a_{ij} = a_{ji}, λ_{i} ∈ ℝ.

Random Matrix Ensembles

Real Symmetric Matrices:

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & ⋯ & a_{1N} \\ a_{21} & a_{22} & a_{23} & ⋯ & a_{2N} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{N1} & a_{N2} & a_{N3} & ⋯ & a_{NN} \end{matrix} ) = A^{T} , a_{ij} = a_{ji}, λ_{i} ∈ ℝ.

Define

Prob(A : a_{ij} ∈ [α_{ij}, β_{ij}]) = ∏_{1≤i≤j≤N} ∫_{α_{ij}}^{β_{ij}} p(x_{ij})dx_{ij}.

Want to understand eigenvalues of randomly chosen A.

MAIN TOOL: Eigenvalue Trace Lemma

Trace(A) = a_{11} + a_{22} + ⋯ + a_{NN}.

Eigenvalue Trace Lemma: Trace(A^{k}) = ∑_{i=1}^{N} λ_{i}(A)^{k}.

MAIN TOOL: Eigenvalue Trace Lemma

Trace(A) = a_{11} + a_{22} + ⋯ + a_{NN}.

Eigenvalue Trace Lemma: Trace(A^{k}) = ∑_{i=1}^{N} λ_{i}(A)^{k}.

• Will give correct normalization for zeros;

• Allows us to pass from knowledge of matrix entries to knowledge of eigenvalues.

Correct Scale for Eigenvalues of Real Symmetric Matrices Entries chosen from Mean 0, Variance 1 Density

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

By the Central Limit Theorem:

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2} ∼ N^{2}

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

By the Central Limit Theorem:

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2} ∼ N^{2}

∑_{i=1}^{N} λ_{i}(A)^{2} ∼ N^{2}

Trace(A^{2}) = ∑_{i=1}^{N} λ_{i}(A)^{2}.

By the Central Limit Theorem:

Trace(A^{2}) = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}a_{ji} = ∑_{i=1}^{N} ∑_{j=1}^{N} a_{ij}^{2} ∼ N^{2}

∑_{i=1}^{N} λ_{i}(A)^{2} ∼ N^{2}

Gives NAverage(λ_{i}(A)^{2}) ∼ N^{2} or Average(λ_{i}(A)) ∼ \sqrt{N}.

Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}.

Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}. For each N × N matrix A, attach a probability measure:

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}}) .

Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}. For each N × N matrix A, attach a probability measure:

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}}) .

Equivalently,

∫_{α}^{β} μ_{A,N}(x)dx = \frac{# \{λ_{i} : \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [α, β]\}}{N}.

Eigenvalue Distribution

δ(x − x_{0}) is a unit point mass at x_{0}. For each N × N matrix A, attach a probability measure:

μ_{A,N}(x) = \frac{1}{N} ∑_{i=1}^{N} δ (x − \frac{λ_{i}(A)}{2\sqrt{N}}) .

Equivalently,

∫_{α}^{β} μ_{A,N}(x)dx = \frac{# \{λ_{i} : \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [α, β]\}}{N}.

k^{th} Moment of μ_{A,N} = \frac{1}{N} ∑_{i=1}^{N} \frac{λ_{i}(A)^{k}}{(2\sqrt{N})^{k}} = \frac{Trace(A^{k})}{2^{k}N^{\frac{k}{2}+1}}.

Wigner’s Semi-Circle Law

N × N real symmetric matrices, upper triangular entries independently chosen from a fixed probability density p on ℝ.

∫_{α}^{β} μ_{A,N}(x)dx = \frac{# \{λ_{i} : \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [α, β]\}}{N}.

∫_{α}^{β} μ_{A,N}(x)dx = \frac{# \{λ_{i} : \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [α, β]\}}{N}.

THEOREM: Wigner’s Semi-Circle Law: Assume p has mean 0, variance 1, other moments finite. As N → ∞, almost all A have μ_{A,N} close to the Semi-Circle density

S(x) = \begin{cases}\sqrt[\frac{2}{π}]{1 − x^{2}} & if |x| ≤ 1 \\ 0 & otherwise. \end{cases}

∫_{α}^{β} μ_{A,N}(x)dx = \frac{# \{λ_{i} : \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [α, β]\}}{N}.

THEOREM: Wigner’s Semi-Circle Law: Assume p has mean 0, variance 1, other moments finite. Then as N → ∞ almost all A have μ_{A,N} close to the Semi-Circle density

S(x) = \begin{cases}\sqrt[\frac{2}{π}]{1 − x^{2}} & if |x| ≤ 1 \\ 0 & otherwise. \end{cases}

Technical: As N → ∞ with probability one the Kolmogorov-Smirnov discrepency between μ_{A,N} and S tends to zero.

Disc(μ_{A,N}, S) = sup_{x} |∫_{−∞}^{x} μ_{A,N}(t)dt − ∫_{−∞}^{x} S(t)dt|

Proof of Wigner’s Semi-Circle Law

1. Eigenvalue Trace Lemma (Trace(A^{k}) = ∑_{i} λ_{i}(A)^{k}) converts sums over eigenvalues to sums over entries of A.

2. Expected value of k^{th}-moment of μ_{A,N}(x) is

∫_{−∞}^{∞} ⋯ ∫_{−∞}^{∞} \frac{Trace(A^{k})}{2^{k}N^{\frac{k}{2}+1}} ∏_{i≤j} p(a_{ij})da_{ij}.

3. Show the expected value of k^{th}-moment of μ_{A,N}(x) equals the k^{th}- moment of the Semi-Circle.

Random Matrix Theory: Semi-Circle Experiments 1. Choose a probability distribution p on ℝ.

2. Choose a large N (size of matrices).

3. Look at a large number of N × N real symmetric matrices, each matrix has its upper triangular entries independently chosen from p.

4. Calculate the normalized eigenvalues \frac{λ_{i}(A)}{2\sqrt{N}}, investigate their properties.

Random Matrix Theory: Semi-Circle Law

500 Matrices: Gaussian 400 × 400 p(x) = \frac{1}{\sqrt{2π}} e^{−x^{2}/2} Random Matrix Theory: Semi-Circle Law Cauchy Distribution: p(x) = \frac{1}{π(1+x^{2})}, x ∈ ℝ

Infinite Variance

Gaussian Orthogonal Ensemble (GOE) Conjecture

Let GOE(x) ≈ \frac{π}{2} xe^{−\frac{π}{4}x^{2}}.

Gaussian Orthogonal Ensemble (GOE) Conjecture Let GOE(x) ≈ \frac{π}{2} xe^{−\frac{π}{4}x^{2}}. Let p be a probability distribution on ℝ with mean zero, variance one and finite higher moments. GOE Conjecture: As N → ∞, for most A the probability density of the spacing between consecutive normalized eigenvalues converges to GOE(x):

\frac{# \{i : \frac{λ_{i+1}(A)}{2\sqrt{N}} − \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [α, β]\}}{N} ⟶ ∫_{α}^{β} GOE(x)dx.

Gaussian Orthogonal Ensemble (GOE) Conjecture Let GOE(x) ≈ \frac{π}{2} xe^{−\frac{π}{4}x^{2}}. Let p be a probability distribution on ℝ with mean zero, variance one and finite higher moments. GOE Conjecture: As N → ∞, for most A the probability density of the spacing between consecutive normalized eigenvalues converges to GOE(x):

\frac{# \{i : \frac{λ_{i+1}(A)}{2\sqrt{N}} − \frac{λ_{i}(A)}{2\sqrt{N}} ∈ [α, β]\}}{N} ⟶ ∫_{α}^{β} GOE(x)dx.

Only proved for p a Gaussian with mean 0 and variance 1. What about numerical data in other cases?

Uniform Distribution: p(x) = \frac{1}{2} for |x| ≤ 1

5000: 300 × 300 uniform on [−1, 1] Cauchy Distribution: p(x) = \frac{1}{π(1+x^{2})}, x ∈ ℝ

5000: 100 × 100 Cauchy

5000: 300 × 300 Cauchy

PART III NUMBER THEORY

IDEA:

Zeros of Random Matrices Provide a Good Model for Zeros of Number Theoretic Functions

Riemann Zeta Function

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Geometric Series (and Extending Functions): If |u| < 1,

∑_{k=0}^{∞} u^{k} = 1 + u + u^{2} + u^{3} + ⋯ = \frac{1}{1 − u}

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Geometric Series (and Extending Functions): If |u| < 1,

∑_{k=0}^{∞} u^{k} = 1 + u + u^{2} + u^{3} + ⋯ = \frac{1}{1 − u}

Unique Factorization: n = p_{1}^{r_{1}} ⋯ p_{m}^{r_{m}}.

∏_{p} (1 − \frac{1}{p^{s}})^{−1} = [1 + \frac{1}{2^{s}} + (\frac{1}{2^{s}})^{2} + ⋯] [1 + \frac{1}{3^{s}} + (\frac{1}{3^{s}})^{2} + ⋯] ⋯

= ∑_{n} \frac{1}{n^{s}}.

Riemann Zeta Function (continued):

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Properties of ζ(s) and Primes: Proofs that π(x) → ∞.

Riemann Zeta Function (continued):

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Properties of ζ(s) and Primes: Proofs that π(x) → ∞.

• lim_{s→1^{+}} ζ(s) = ∞.

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Properties of ζ(s) and Primes: Proofs that π(x) → ∞.

• lim_{s→1^{+}} ζ(s) = ∞.

• ζ(2) = \frac{π^{2}}{6}.

Riemann Zeta Function (continued):

ζ(s) = ∑_{n} \frac{1}{n^{s}} = ∏_{p} (1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1
π(x) = #{p : p is prime, p ≤ x}

Properties of ζ(s) and Primes: Proofs that π(x) → ∞.

• lim_{s→1^{+}} ζ(s) = ∞.

• ζ(2) = \frac{π^{2}}{6}.

• ζ(1 + it) ≠ 0 for t ∈ ℝ implies π(x) ∼ \frac{x}{log x}.

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Functional Equation:

ξ(s) = π^{−\frac{s}{2}}Γ (\frac{s}{2}) ζ(s) = ξ(1 − s).

Riemann Zeta Function (continued):

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Functional Equation:

ξ(s) = π^{−\frac{s}{2}}Γ (\frac{s}{2}) ζ(s) = ξ(1 − s).

Riemann Hypothesis:

• All zeros have Re(s) = \frac{1}{2}; can write zeros as \frac{1}{2} + iγ, γ ∈ ℝ.

• Number of zeros with 0 ≤ γ ≤ T is about T log T.

Riemann Zeta Function (continued):

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = ∏_{p prime}(1 − \frac{1}{p^{s}})^{−1} , Re(s) > 1.

Functional Equation:

ξ(s) = π^{−\frac{s}{2}}Γ (\frac{s}{2}) ζ(s) = ξ(1 − s).

Riemann Hypothesis:

• All zeros have Re(s) = \frac{1}{2}; can write zeros as \frac{1}{2} + iγ, γ ∈ ℝ.

• Number of zeros with 0 ≤ γ ≤ T is about T log T. Observation:

• Spacings between normalized zeros appear same as between normalized eigenvalues of Complex Hermitian matrices (\overline{A}^{T} = A).

Normalized Zeros of Riemann Zeta Function Zeros \frac{1}{2} + iγ, γ ∈ ℝ

Know #{γ : 0 ≤ γ ≤ T } is about T log T.

Normalized Zeros of Riemann Zeta Function Zeros \frac{1}{2} + iγ, γ ∈ ℝ

Know #{γ : 0 ≤ γ ≤ T } is about T log T. Average spacing of zeros with γ ∼ T is \frac{T}{T log T} = \frac{1}{log T}. Contrast with primes: Average spacing of primes p ∼ x is \frac{x}{x/ log x} = log x.

Normalized Zeros of Riemann Zeta Function Zeros \frac{1}{2} + iγ, γ ∈ ℝ

Know #{γ : 0 ≤ γ ≤ T } is about T log T. Average spacing of zeros with γ ∼ T is \frac{T}{T log T} = \frac{1}{log T}. Contrast with primes: Average spacing of primes p ∼ x is \frac{x}{x/ log x} = log x. Normalized zeros: study γ_{n+1} log γ_{n+1} − γ_{n} log γ_{n}. Contrast with primes, where we studied \frac{p_{n+1}}{log p_{n+1}} − \frac{p_{n}}{log p_{n}}.

Zeros of ζ(s) vs. GUE(x):

70 million spacings between adjacent normalized zeros of ζ(s), starting at the 10^{20}^{th} zero (from Odlyzko)

General L-Functions

• Euler Product:

L(s) := ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} = ∏_{p} \frac{1}{L_{p}(p^{−s})}, Re(s) ≫ 0, L_{p}(x) = polynomial.

General L-Functions

• Euler Product:

L(s) := ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} = ∏_{p} \frac{1}{L_{p}(p^{−s})}, Re(s) ≫ 0, L_{p}(x) = polynomial.

• Functional Equation:

Λ(s) := (Γ − Factors) · L(s)
Λ(s) = \frac{ε(s)}{C^{s}}\overline{Λ(1 − \overline{s})}, C > 0 is the Conductor

General L-Functions

• Euler Product:

L(s) := ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} = ∏_{p} \frac{1}{L_{p}(p^{−s})}, Re(s) ≫ 0, L_{p}(x) = polynomial.

• Functional Equation:

Λ(s) := (Γ − Factors) · L(s)
Λ(s) = \frac{ε(s)}{C^{s}}\overline{Λ(1 − \overline{s})}, C > 0 is called the Conductor

• Riemann Hypothesis: All zeros have Re(s) = \frac{1}{2}; can write zeros as \frac{1}{2} + iγ, γ ∈ ℝ.

General L-Functions

• Euler Product:

L(s) := ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}} = ∏_{p} \frac{1}{L_{p}(p^{−s})}, Re(s) ≫ 0, L_{p}(x) = polynomial.

• Functional Equation:

Λ(s) := (Γ − Factors) · L(s)
Λ(s) = \frac{ε(s)}{C^{s}}\overline{Λ(1 − \overline{s})}, C > 0 is called the Conductor

• Riemann Hypothesis: All zeros have Re(s) = \frac{1}{2}; can write zeros as \frac{1}{2} + iγ, γ ∈ ℝ.

• Number of Zeros:

Number of zeros with γ ∼ T is like T log T Zeros near s = \frac{1}{2} have γ ∼ \frac{1}{log C}

(Heuristic): Knowledge of Zeros

P (x) polynomial, zeros r_{1}, …, r_{n}. Then

P (x) = (x − r_{1})(x − r_{2}) ⋯ (x − r_{n})
= x^{n} + a_{n−1}(r_{1}, …, r_{n})x^{n−1} + ⋯ + a_{0}(r_{1}, …, r_{n})

where

a_{n−1}(r_{1}, …, r_{n}) = −(r_{1} + ⋯ + r_{n})
⋮
a_{0}(r_{1}, …, r_{n}) = r_{1}r_{2} ⋯ r_{n}.

Knowledge of zeros gives info on coefficients of P (x).

Explicit Formula / Contour Integration: Analogue of Eigenvalue Trace Lemma

−\frac{ζ′(s)}{ζ(s)} = −\frac{d}{ds} log ζ(s)
= \frac{d}{ds} ∑_{p} log (1 − p^{−s})
= ∑_{p} \frac{log p · p^{−s}}{1 − p^{−s}}
= ∑_{p} \frac{log p}{p^{s}} + Good(s).

Contour Integration:

∫ −\frac{ζ′(s)}{ζ(s)}\frac{x^{s}}{s}ds vs ∑_{p} log p ∫ (\frac{x}{p})^{s} \frac{ds}{s}.

Knowledge of zeros gives info on the L-function coefficients.

Interesting L-Functions

What makes an L-Function interesting?

Interesting L-Functions

What makes an L-Function interesting?

• Coefficients a_{n} of arithmetic significance.

• Look for L-Functions with multiple zeros: • Conjectured that all zeros are simple except for deep reasons;

• Do multiple zeros attract or repel nearby zeros?

Interesting L-Functions

What makes an L-Function interesting?

• Coefficients a_{n} of arithmetic significance.

• Look for L-Functions with multiple zeros: • Conjectured that all zeros are simple except for deep reasons;

• Do multiple zeros attract or repel nearby zeros?

Will see L-Functions of Elliptic Curves are interesting.

Interesting L-Functions

What makes an L-Function interesting?

• Coefficients a_{n} of arithmetic significance.

• Look for L-Functions with multiple zeros: • Conjectured that all zeros are simple except for deep reasons;

• Do multiple zeros attract or repel nearby zeros?

Will see L-Functions of Elliptic Curves are interesting.

• Many have multiple zeros at s = \frac{1}{2}.

• Can investigate if these zeros attract or repel.

Elliptic Curves: E: y^{2} = x^{3} + Ax + B

74

Elliptic Curves: Group of Rational Solutions E(ℚ)

Studying E: y^{2} = x^{3} + Ax + B Mordell-Weil Theorem: Rational solutions: E(ℚ) = ℤ^{r} ⨁ Finite Group.

Elliptic Curves: Group of Rational Solutions E(ℚ)

Studying E: y^{2} = x^{3} + Ax + B Mordell-Weil Theorem: Rational solutions: E(ℚ) = ℤ^{r} ⨁ Finite Group. Question: how does r depend on E?

Elliptic Curves: Group of Rational Solutions E(ℚ)

Studying E: y^{2} = x^{3} + Ax + B Mordell-Weil Theorem: Rational solutions: E(ℚ) = ℤ^{r} ⨁ Finite Group. Question: how does r depend on E? Attach an L-Function to E: As ζ(s) gives us information on primes, expect L-Function gives us information on E.

Elliptic Curves: Group of Rational Solutions E(ℚ)

Studying E: y^{2} = x^{3} + Ax + B Mordell-Weil Theorem: Rational solutions: E(ℚ) = ℤ^{r} ⨁ Finite Group. Question: how does r depend on E? Attach an L-Function to E: As ζ(s) gives us information on primes, expect L-Function gives us information on E. Review: Legendre Symbol: (\frac{0}{p}) = 0 and

(\frac{a}{p}) = \begin{cases}1 & if x^{2} ≡ a mod p has two solutions \\ −1 & if x^{2} ≡ a mod p has no solutions. \end{cases}

Note 1 + (\frac{a}{p}) is the number of solutions to x^{2} ≡ a mod p.

L-Function of an Elliptic Curve E : y^{2} = x^{3} + Ax + B Let N_{p} be the number of solutions mod p:

N_{p} := ∑_{x mod p} [1 + (\frac{x^{3} + Ax + B}{p})] = p + ∑_{x mod p} (\frac{x^{3} + Ax + B}{p})

L-Function of an Elliptic Curve E : y^{2} = x^{3} + Ax + B Let N_{p} be the number of solutions mod p:

N_{p} := ∑_{x mod p} [1 + (\frac{x^{3} + Ax + B}{p})] = p + ∑_{x mod p} (\frac{x^{3} + Ax + B}{p})

Local data: a_{p} = p − N_{p}. Use to build the L-function:

L(E, s) := ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}.

L-Function of an Elliptic Curve E : y^{2} = x^{3} + Ax + B Let N_{p} be the number of solutions mod p:

N_{p} := ∑_{x mod p} [1 + (\frac{x^{3} + Ax + B}{p})] = p + ∑_{x mod p} (\frac{x^{3} + Ax + B}{p})

Local data: a_{p} = p − N_{p}. Use to build the L-function:

L(E, s) := ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}.

Local to Global: {a_{p}}_{p prime} ⟷ E(ℚ) = ℤ^{r} ⨁ Finite Group.

L-Function of an Elliptic Curve E : y^{2} = x^{3} + Ax + B

Let N_{p} be the number of solutions mod p:

N_{p} := ∑_{x mod p} [1 + (\frac{x^{3} + Ax + B}{p})] = p + ∑_{x mod p} (\frac{x^{3} + Ax + B}{p})

Local data: a_{p} = p − N_{p}. Use to build the L-function:

L(E, s) := ∑_{n=1}^{∞} \frac{a_{n}}{n^{s}}.

Local to Global: {a_{p}}_{p prime} ⟷ E(ℚ) = ℤ^{r} ⨁ Finite Group. Birch and Swinnerton-Dyer Conjecture: Geometric rank r equals number of zeros of L(E, s) at s = \frac{1}{2}. Possibility of repulsion / attraction from zeros at s = \frac{1}{2}!

Families of Elliptic Curves:

E : y^{2} = x^{3} + A(T)x + B(T), A(T), B(T) ∈ ℤ[T].

E : y^{2} = x^{3} + A(T)x + B(T), A(T), B(T) ∈ ℤ[T].

Have a FAMILY of L-Functions:

• t ∈ ℤ gives an elliptic curve E_{t} with conductor C_{t}.

• t ∈ ℤ gives a family of L-functions L(E_{t}, s).

• C_{t} is typically growing polynomially in t.

E : y^{2} = x^{3} + A(T)x + B(T), A(T), B(T) ∈ ℤ[T].

Have a FAMILY of L-Functions:

• t ∈ ℤ gives an elliptic curve E_{t} with conductor C_{t}.

• t ∈ ℤ gives a family of L-functions L(E_{t}, s).

• C_{t} is typically growing polynomially in t.

Similar to RMT where we had many N × N matrices • Do E_{t} from the same family have similar properties?

• N → ∞ becomes C_{t} → ∞.

Families of Elliptic Curves

Mordell-Weil Theorem for Families: E : y^{2} = x^{3} + A(T)x + B(T), A(T), B(T) ∈ ℤ[T].

Families of Elliptic Curves

Mordell-Weil Theorem for Families:

E : y^{2} = x^{3} + A(T)x + B(T), A(T), B(T) ∈ ℤ[T].

• Group of Rational Function Solutions: P (T) = (x(T), y(T)). E(ℚ(T)) = ℤ^{r(E)} ⨁ Finite Group.

Families of Elliptic Curves

Mordell-Weil Theorem for Families:

E : y^{2} = x^{3} + A(T)x + B(T), A(T), B(T) ∈ ℤ[T].

• Group of Rational Function Solutions:

P (T) = (x(T), y(T)).

E(ℚ(T)) = ℤ^{r(E)} ⨁ Finite Group.

• Specialization Theorem: For all t ∈ ℤ sufficiently large: r(E_{t}) ≥ r(E).

Families of Elliptic Curves

Mordell-Weil Theorem for Families:

E : y^{2} = x^{3} + A(T)x + B(T), A(T), B(T) ∈ ℤ[T].

• Group of Rational Function Solutions:

P (T) = (x(T), y(T)).

E(ℚ(T)) = ℤ^{r(E)} ⨁ Finite Group.

• Specialization Theorem: For all t ∈ ℤ sufficiently large: r(E_{t}) ≥ r(E). Questions:

• How does r(E_{t}) vary in the family?

• How do the zeros of L(s, E_{t}) vary in the family?

Example of a Family of Elliptic Curves

Consider E : y^{2} = x^{3} − T^{2}x + T^{4}.

• Can find some rational solutions by inspection: • P_{1}(T) = (T, T^{2})

• P_{2}(T) = (T^{2}, T^{3})

Consider E : y^{2} = x^{3} − T^{2}x + T^{4}.

• Can find some rational solutions by inspection: • P_{1}(T) = (T, T^{2})

• P_{2}(T) = (T^{2}, T^{3})

• These points generate infinite part of E(ℚ(T)).

• This means the rank of E(ℚ(T)) is 2.

Consider E : y^{2} = x^{3} − T^{2}x + T^{4}.

• Can find some rational solutions by inspection: • P_{1}(T) = (T, T^{2})

• P_{2}(T) = (T^{2}, T^{3})

• These points generate infinite part of E(ℚ(T)).

• This means the rank of E(ℚ(T)) is 2.

• What is r(E_{t}) for t ∈ ℤ Consider E : y^{2} = x^{3} − T^{2}x + T^{4}.

• Can find some rational solutions by inspection: • P_{1}(T) = (T, T^{2})

• P_{2}(T) = (T^{2}, T^{3})

• These points generate infinite part of E(ℚ(T)).

• This means the rank of E(ℚ(T)) is 2.

• What is r(E_{t}) for t ∈ ℤ • r(E_{1}) = 1. Not surprising as P_{1}(1) = P_{2}(1)!

Consider E : y^{2} = x^{3} − T^{2}x + T^{4}.

• Can find some rational solutions by inspection: • P_{1}(T) = (T, T^{2})

• P_{2}(T) = (T^{2}, T^{3})

• These points generate infinite part of E(ℚ(T)).

• This means the rank of E(ℚ(T)) is 2.

• What is r(E_{t}) for t ∈ ℤ • r(E_{1}) = 1. Not surprising as P_{1}(1) = P_{2}(1)!

• r(E_{2}) = 2.

Consider E : y^{2} = x^{3} − T^{2}x + T^{4}.

• Can find some rational solutions by inspection: • P_{1}(T) = (T, T^{2})

• P_{2}(T) = (T^{2}, T^{3})

• These points generate infinite part of E(ℚ(T)).

• This means the rank of E(ℚ(T)) is 2.

• What is r(E_{t}) for t ∈ ℤ • r(E_{1}) = 1. Not surprising as P_{1}(1) = P_{2}(1)!

• r(E_{2}) = 2.

• r(E_{42}) = 4!

L-Functions and Random Matrix Theory

• Spacings between normalized zeros of all “nice” L-functions look like spacings between normalized eigenvalues of complex Hermitian matrices.

L-Functions and Random Matrix Theory

• Spacings between normalized zeros of all “nice” L-functions look like spacings between normalized eigenvalues of complex Hermitian matrices.

• (GUE Model) Limit as N → ∞ of N × N complex Hermitian matrices with upper triangular entries independently chosen from the standard normal.

• Zeros with γ ∼ T well-modelled by N × N matrices with N ∼ log T.

L-Functions and Random Matrix Theory

• Spacings between normalized zeros of all “nice” L-functions look like spacings between normalized eigenvalues of complex Hermitian matrices.

• (GUE Model) Limit as N → ∞ of N × N complex Hermitian matrices with upper triangular entries independently chosen from the standard normal.

• Zeros with γ ∼ T well-modelled by N × N matrices with N ∼ log T.

• For a while, seemed this was the only set of matrices relevant to number theory.

L-Functions and Random Matrix Theory

• GUE Model cannot be enough for all of Number Theory:

L-Functions and Random Matrix Theory

• GUE Model cannot be enough for all of Number Theory: • Appears to be good for describing behavior of zeros far from s = \frac{1}{2}.

• Insensitive to finitely many zeros.

L-Functions and Random Matrix Theory

• GUE Model cannot be enough for all of Number Theory: • Appears to be good for describing behavior of zeros far from s = \frac{1}{2}.

• Insensitive to finitely many zeros.

• Often zeros near s = \frac{1}{2} are interesting:

L-Functions and Random Matrix Theory

• GUE Model cannot be enough for all of Number Theory: • Appears to be good for describing behavior of zeros far from s = \frac{1}{2}.

• Insensitive to finitely many zeros.

• Often zeros near s = \frac{1}{2} are interesting:

• Different types of L-functions have different behavior of zeros near s = \frac{1}{2}.

• Possibility of multiple zeros at s = \frac{1}{2} and attraction or repulsion.

• Need a theory for these low zeros.

Measures of Spacings: 1-Level Density and Families

Let ϕ be an even Schwartz function whose Fourier Transform is compactly supported. Let L(s) be an L-function with zeros \frac{1}{2} + iγ (γ ∈ ℝ) and conductor C.

Let ϕ be an even Schwartz function whose Fourier Transform is compactly supported. Let L(s) be an L-function with zeros \frac{1}{2} + iγ (γ ∈ ℝ) and conductor C. Define the 1-level density by

∑_{n} ϕ (γ_{n} log C)

Let ϕ be an even Schwartz function whose Fourier Transform is compactly supported. Let L(s) be an L-function with zeros \frac{1}{2} + iγ (γ ∈ ℝ) and conductor C. Define the 1-level density by

∑_{n} ϕ (γ_{n} log C)

• Individual zeros γ_{n} contribute in limit

• Most of contribution is from low zeros

• Average over similar L-functions (family)

Let ϕ be an even Schwartz function whose Fourier Transform is compactly supported. Let L(s) be an L-function with zeros \frac{1}{2} + iγ (γ ∈ ℝ) and conductor C. Define the 1-level density by

∑_{n} ϕ (γ_{n} log C)

• Individual zeros γ_{n} contribute in limit

• Most of contribution is from low zeros

• Average over similar L-functions (family)

To any geometric family, Katz-Sarnak predict the 1-level density depends only on a symmetry group (a classical compact group) attached to the family.

Random Matrix Ensembles and Number Theory

• Zeros far away from s = \frac{1}{2} well-modelled by GUE.

Random Matrix Ensembles and Number Theory

• Zeros far away from s = \frac{1}{2} well-modelled by GUE. • Choose one L-Function, look at high zeros.

• One L-function has enough freedom to average.

Random Matrix Ensembles and Number Theory

• Zeros far away from s = \frac{1}{2} well-modelled by GUE. • Choose one L-Function, look at high zeros.

• One L-function has enough freedom to average.

• Story different for zeros near s = \frac{1}{2}.

Random Matrix Ensembles and Number Theory

• Zeros far away from s = \frac{1}{2} well-modelled by GUE. • Choose one L-Function, look at high zeros.

• One L-function has enough freedom to average.

• Story different for zeros near s = \frac{1}{2}.

• One L-function no longer suffices for averaging.

• Look at many similar L-functions.

• Hope L-functions’ zeros near s = \frac{1}{2} behave similarly.

Random Matrix Ensembles and Number Theory

• Zeros far away from s = \frac{1}{2} well-modelled by GUE. • Choose one L-Function, look at high zeros.

• One L-function has enough freedom to average.

• Story different for zeros near s = \frac{1}{2}.

• One L-function no longer suffices for averaging.

• Look at many similar L-functions.

• Hope L-functions’ zeros near s = \frac{1}{2} behave similarly.

• Analogy with Random Matrix Theory:

Random Matrix Ensembles and Number Theory

• Zeros far away from s = \frac{1}{2} well-modelled by GUE. • Choose one L-Function, look at high zeros.

• One L-function has enough freedom to average.

• Story different for zeros near s = \frac{1}{2}.

• One L-function no longer suffices for averaging.

• Look at many similar L-functions.

• Hope L-functions’ zeros near s = \frac{1}{2} behave similarly.

• Analogy with Random Matrix Theory:

• RMT: pick many N × N matrices at random, average, study limit as N → ∞.

• NT: pick many L-functions in a family with conductors about the same size, average, study what happens as conductors go to infinity.

Random Matrix Ensembles

Real Symmetric, Complex Hermitian Matrices:

1. λ ∈ ℝ.

2. Randomness: upper triangular entries independently chosen from p; freedom to choose p.

Random Matrix Ensembles

Real Symmetric, Complex Hermitian Matrices:

1. λ ∈ ℝ.

2. Randomness: upper triangular entries independently chosen from p; freedom to choose p.

Classical Compact Groups:

1. λ = e^{iθ}, θ ∈ (−π, π] ⊂ ℝ.

2. Randomness: Haar measure; canonical choice.

3. Subgroups: Orthogonal Matrices (Q^{T} Q = I):

SO(even) : e^{iθ} : ⋯ ≤ −θ_{2} ≤ −θ_{1} ≤ 0 ≤ θ_{1} ≤ θ_{2} ≤ ⋯
SO(odd) : e^{iθ} : ⋯ ≤ −θ_{2} ≤ −θ_{1} ≤ θ_{0} = 0 ≤ θ_{1} ≤ θ_{2} ≤ ⋯

Theorems for Families of Elliptic Curves Family E : y^{2} = x^{3} + A(T)x + B(T), specialized curves E_{t}

If family E has rank r(E): As conductors go to infinity:

• Results suggest E_{t} has at least r(E) zeros at s = \frac{1}{2};

• Behavior of remaining zeros near s = \frac{1}{2} agree with eigenvalues near 1 of orthogonal groups.

Theorems for Families of Elliptic Curves Family E : y^{2} = x^{3} + A(T)x + B(T), specialized curves E_{t}

If family E has rank r(E): As conductors go to infinity:

• Results suggest E_{t} has at least r(E) zeros at s = \frac{1}{2};

• Behavior of remaining zeros near s = \frac{1}{2} agree with eigenvalues near 1 of orthogonal groups.

Method of Proof

• Explicit Formula: (Analogue of Eigenvalue Trace Lemma)

• Relates Sums over Zeros to Sums over Primes.

• Gives correct scale.

• Summation Formulas: (Analogue of Integrating Matrix Entries) • Sums of Legendre Symbols.

• Like Central Limit Theorem, just first two moments matter:

∑_{t mod p} a_{p,t} and ∑_{t mod p} a_{p,t}^{2}.

Predictions from Random Matrix Theory Family E of Elliptic Curves with rank r(E)

Families of Elliptic Curves well-modelled by Orthogonal Groups: zeros near s = \frac{1}{2} look like eigenvalues near 1.

Families of Elliptic Curves well-modelled by Orthogonal Groups: zeros near s = \frac{1}{2} look like eigenvalues near 1.

As conductor C_{t} goes to infinity, expect half the elliptic curves E_{t} to have rank r(E), half to have rank r(E) + 1.

Families of Elliptic Curves well-modelled by Orthogonal Groups: zeros near s = \frac{1}{2} look like eigenvalues near 1.

As conductor C_{t} goes to infinity, expect half the elliptic curves E_{t} to have rank r(E), half to have rank r(E) + 1.

As conductor C_{t} goes to infinity, for each E_{t} expect the r(E) family zeros to be independent of the other zeros of E_{t} near s = \frac{1}{2}.

Families of Elliptic Curves well-modelled by Orthogonal Groups: zeros near s = \frac{1}{2} look like eigenvalues near 1.

As conductor C_{t} goes to infinity, expect half the elliptic curves E_{t} to have rank r(E), half to have rank r(E) + 1.

As conductor C_{t} goes to infinity, for each E_{t} expect the r(E) family zeros to be independent of the other zeros of E_{t} near s = \frac{1}{2}.

In particular, the distribution of the first zero above s = \frac{1}{2} should be independent of r(E).

Excess Rank

One-parameter family, rank r(E) over ℚ(T). For each t ∈ ℤ, consider curves E_{t}. RMT ⟹ 50% rank r(E), 50% rank r(E) + 1.

Excess Rank

One-parameter family, rank r(E) over ℚ(T). For each t ∈ ℤ, consider curves E_{t}. RMT ⟹ 50% rank r(E), 50% rank r(E) + 1.

For many families, observe

Percent with rank r(E)  = 32% Percent with rank r(E) + 1 = 48%
Percent with rank r(E) + 2 = 18% Percent with rank r(E) + 3 = 2%

Excess Rank

One-parameter family, rank r(E) over ℚ(T). For each t ∈ ℤ, consider curves E_{t}. RMT ⟹ 50% rank r(E), 50% rank r(E) + 1.

For many families, observe

Percent with rank r(E)  = 32% Percent with rank r(E) + 1 = 48%
Percent with rank r(E) + 2 = 18% Percent with rank r(E) + 3 = 2%

Problem: small data sets, sub-families, convergence rate log(conductor)?

Interval  Primes  Twin Primes Pairs
[1, 10]  2, 3, 5, 7 (40%)  (3, 5), (5, 7) (20%)
[11, 20] 11, 13, 17, 19 (40%) (11, 13), (17, 19) (20%)

Small data can be misleading! Remember ∑_{n≤N} \frac{1}{n} ∼ log N.

Data on Excess Rank y^{2} = x^{3} + 16T x + 32

Each data set runs over 2000 consecutive t-values.

\underline{t-Start} \underline{Rk 0} \underline{Rk 1} \underline{Rk 2} \underline{Rk 3} \underline{Time (hrs)}
-1000 39.4 47.8 12.3 0.6  <1
1000 38.4 47.3 13.6 0.6  <1
4000 37.4 47.8 13.7 1.1  1
8000 37.3 48.8 12.9 1.0  2.5
24000 35.1 50.1 13.9 0.8  6.8
50000 36.7 48.3 13.8 1.2  51.8

Last set has conductors of size 10^{11}, but on logarithmic scale still small.

SO(even) Random Matrix Models

RMT: 2N eigenvalues, in pairs e^{±iθ_{j}}, probability measure on [0, π]^{N}:

∏_{j<k}(cos θ_{k} − cos θ_{j})^{2} ∏_{j} dθ_{j}

SO(even) Random Matrix Models

RMT: 2N eigenvalues, in pairs e^{±iθ_{j}}, probability measure on [0, π]^{N}:

∏_{j<k}(cos θ_{k} − cos θ_{j})^{2} ∏_{j} dθ_{j}

Independent Model: 2r Eigenvalues at 1

\{(\begin{matrix}g \\ & I_{2r} \end{matrix}) : g ∈ SO(2N − 2r)\}

SO(even) Random Matrix Models

RMT: 2N eigenvalues, in pairs e^{±iθ_{j}}, probability measure on [0, π]^{N}:

∏_{j<k}(cos θ_{k} − cos θ_{j})^{2} ∏_{j} dθ_{j}

Independent Model: 2r Eigenvalues at 1

\{(\begin{matrix}g \\ & I_{2r} \end{matrix}) : g ∈ SO(2N − 2r)\}

Interaction Model: 2r Eigenvalues at 1 Sub-ensemble of SO(2N) with 2r eigenvalues forced to be +1:

∏_{j<k}(cos θ_{k} − cos θ_{j})^{2} ∏_{j}(1 − cos θ_{j})^{2r} ∏_{j} dθ_{j},

with 1 ≤ j, k ≤ N − r.

Comparing the two Random Matrix Models

For small support, 1-level densities for Elliptic Curves agree with Independent Model.

Comparing the two Random Matrix Models

For small support, 1-level densities for Elliptic Curves agree with Independent Model. Elliptic Curve E, conductor C, expect first zero above s = \frac{1}{2} to be \frac{1}{2} + iγ with γ ∼ \frac{1}{log C}.

Comparing the two Random Matrix Models

For small support, 1-level densities for Elliptic Curves agree with Independent Model. Elliptic Curve E, conductor C, expect first zero above s = \frac{1}{2} to be \frac{1}{2} + iγ with γ ∼ \frac{1}{log C}. If r zeros at central point, if repulsion of zeros is of size \frac{c_{r}}{log C}, would detect in zeros near central point:

∑_{n} ϕ (γ_{n} log C) .

Corrections of size

ϕ (x + c_{r}) − ϕ(x) ≈ ϕ′ (x) · c_{r}.

Theoretical Distribution of First Normalized Zero First normalized eigenvalue: 230,400 from SO(6) with Haar Measure First normalized eigenvalue: 322,560 from SO(7) with Haar Measure Rank 0 Curves: 1st Normalized Zero

(Far left and right bins just for formatting)

750 curves, log(cond) ∈ [3.2, 12.6]; mean = 1.04

750 curves, log(cond) ∈ [12.6, 14.9]; mean = .88 Rank 2 Curves: 1st Normalized Zero

665 curves, log(cond) ∈ [10, 10.3125]; mean = 2.30 665 curves, log(cond) ∈ [16, 16.5]; mean = 1.82

Rank 2 Curves: y^{2} = x^{3} − T^{2}x + T^{2}: 1st Normalized Zero

35 curves, log(cond) ∈ [7.8, 16.1]; mean = 2.24 34 curves, log(cond) ∈ [16.2, 23.3]; mean = 2.00

PART VI CONCLUSIONS

Correspondences

Similarities between Nuclei and Primes:

Energy Levels ⟷ Zeros of L-Functions
Neutron Energy ⟷ Support of Test Functions
Different Elements: U, Pu, ... ⟷ Different L-Functions

Summary

• Find correct scale to compare different systems.

• Similar behavior in different systems.

• Need a Trace Lemma.

• Average over similar elements.

• Need more data.

Open Problems in Number Theory

Identifying Classical Compact Group: Given a reasonable family of L-functions, determine the corresponding symmetry group.

Montgomery-Odlyzko Law: Show that zeros of L-functions at height T → ∞ behave like eigenvalues of N × N matrices with N ∼ log \frac{T}{2π}.

Finite Height / Finite Family Size: Know correct model for high zeros (N = log \frac{T}{2π}); what is the correct model for zeros near the central point as we move through the family (ordered by conductor)?

APPENDIX I: Dirichlet L-Functions Dirichlet Characters: m Prime

(ℤ/mℤ)^{∗} is cyclic of order m − 1 with generator g. Let ζ_{m−1} = e^{2πi/(m−1)}. The principal character χ_{0} is given by

χ_{0}(k) = \{{1 (k, m) = 1 \atop 0 (k, m) > 1.}

The m − 2 primitive characters are determined (by multiplicativity) by action on g. As each χ : (ℤ/mℤ)^{∗} → ℂ^{∗}, for each χ there exists an l such that χ(g) = ζ_{m−1}^{l}. Hence

for each l, 1 ≤ l ≤ m − 2 we have

χ_{l}(k) = \begin{cases}ζ_{m−1}^{la} & k ≡ g^{a}(m) \\ 0 & (k, m) > 0 \end{cases}

Dirichlet L-Functions

Let χ be a primitive character mod m. Let

c(m, χ) = ∑_{k=0}^{m−1} χ(k)e^{2πik/m}.

c(m, χ) is a Gauss sum of modulus \sqrt{m}.

L(s, χ) = ∏_{p}(1 − χ(p)p^{−s})^{−1}
Λ(s, χ) = π^{−\frac{1}{2}(s+ϵ)}Γ (\frac{s + ϵ}{2}) m^{\frac{1}{2}(s+ϵ)}L(s, χ),

where

ϵ = \begin{cases}0 & if χ(−1) = 1 \\ 1 & if χ(−1) = −1 \end{cases}
Λ(s, χ) = (−i)^{ϵ}\frac{c(m, χ)}{\sqrt{m}}Λ(1 − s, χ̄).

Explicit Formula

Let ϕ be an even Schwartz function with compact support in (−σ, σ). Let χ be a non-trivial primitive Dirichlet character of conductor m.

∑ ϕ (γ\frac{log(\frac{m}{π})}{2π}) = ∫_{−∞}^{∞} ϕ(y)dy

− ∑_{p} \frac{log p}{log(m/π)}ϕ̂(\frac{log p}{log(m/π)}) [χ(p) + \overline{χ}(p)]p^{−\frac{1}{2}}

− ∑_{p} \frac{log p}{log(m/π)}ϕ̂(2\frac{log p}{log(m/π)}) [χ^{2}(p) + \overline{χ}^{2}(p)]p^{−1}

+O (\frac{1}{log m}) .

Expansion

{χ_{0}} ∪ {χ_{l}}_{1≤l≤m−2} are all the characters mod m. Consider the family of primitive characters mod a prime m (m − 2 characters):

∫_{−∞}^{∞} ϕ(y)dy

− \frac{1}{m − 2} ∑_{χ≠χ_{0}} ∑_{p} \frac{log p}{log(m/π)}ϕ̂(\frac{log p}{log(m/π)}) [χ(p) + \overline{χ}(p)]p^{−\frac{1}{2}}
− \frac{1}{m − 2} ∑_{χ≠χ_{0}} ∑_{p} \frac{log p}{log(m/π)}ϕ̂(2\frac{log p}{log(m/π)}) [χ^{2}(p) + \overline{χ}^{2}(p)]p^{−1}
+ O (\frac{1}{log m}) .

Note can pass Character Sum through Test Function.

Character Sums

∑_{χ} χ(k) = {m − 1 k ≡ 1(m) 0 otherwise

For any prime p ≠ m

∑_{χ≠χ_{0}} χ(p) = \begin{cases}m − 1 − 1 & p ≡ 1(m) \\ −1 & otherwise \end{cases}

Substitute into

\frac{1}{m − 2} ∑_{χ≠χ_{0}} ∑_{p} \frac{log p}{log(m/π)}ϕ̂(\frac{log p}{log(m/π)}) [χ(p) + \overline{χ}(p)]p^{−\frac{1}{2}}

First Sum

\frac{−2}{m − 2} ∑_{p}^{m^{σ}} \frac{log p}{log(m/π)}ϕ̂(\frac{log p}{log(m/π)}) p^{−\frac{1}{2}}

+ 2\frac{m − 1}{m − 2} ∑_{p≡1(m)}^{m^{σ}} \frac{log p}{log(m/π)}ϕ̂(\frac{log p}{log(m/π)}) p^{−\frac{1}{2}}
≪ \frac{1}{m} ∑_{p}^{m^{σ}} p^{−\frac{1}{2}} + ∑_{p≡1(m)}^{m^{σ}} p^{−\frac{1}{2}}
≪ \frac{1}{m} ∑_{k}^{m^{σ}} k^{−\frac{1}{2}} + ∑_{{k≡1(m) \atop k≥m+1}}^{m^{σ}} k^{−\frac{1}{2}}

≪ \frac{1}{m} ∑_{k}^{m^{σ}} k^{−\frac{1}{2}} + \frac{1}{m} ∑_{k}^{m^{σ}} k^{−\frac{1}{2}}
≪ \frac{1}{m}m^{σ/2}.

No contribution if σ < 2.

Second Sum

\frac{1}{m − 2} ∑_{χ≠χ_{0}} ∑_{p} \frac{log p}{log(m/π)}ϕ̂(2\frac{log p}{log(m/π)}) \frac{χ^{2}(p) + \overline{χ}^{2}(p)}{p}.

∑_{χ≠χ_{0}}[χ^{2}(p) + \overline{χ}^{2}(p)] = \begin{cases}2(m − 2) & p ≡ ±1(m) \\ −2 & p ≢ ±1(m) \end{cases}

Up to O (\frac{1}{log m}) we find that

≪ \frac{1}{m − 2} ∑_{p}^{m^{σ/2}} p^{−1} + \frac{2m − 2}{m − 2} ∑_{p≡±1(m)}^{m^{σ/2}} p^{−1}

≪ \frac{1}{m − 2} ∑_{k}^{m^{σ/2}} k^{−1} + ∑_{{k≡1(m) \atop k≥m+1}}^{m^{σ/2}} k^{−1} + ∑_{{k≡−1(m) \atop k≥m−1}}^{m^{σ/2}} k^{−1}

≪ \frac{log(m^{σ/2})}{m − 2} + \frac{1}{m} ∑_{k}^{m^{σ/2}} k^{−1} + \frac{1}{m} ∑_{k}^{m^{σ/2}} k^{−1} + O (\frac{1}{m})
≪ \frac{log m}{m} + \frac{log m}{m} + \frac{log m}{m}.

Dirichlet Characters: m Square-free

Fix an r and let m_{1}, …, m_{r} be distinct odd primes.

m = m_{1}m_{2} ⋯ m_{r}
M_{1} = (m_{1} − 1)(m_{2} − 1) ⋯ (m_{r} − 1) = ϕ(m)
M_{2} = (m_{1} − 2)(m_{2} − 2) ⋯ (m_{r} − 2).

M_{2} is the number of primitive characters mod m, each of conductor m. A general primitive character mod m is given by χ(u) = χ_{l_{1}}(u)χ_{l_{2}}(u) ⋯ χ_{l_{r}}(u). Let F = {χ : χ = χ_{l_{1}}χ_{l_{2}} ⋯ χ_{l_{r}}}.

\frac{1}{M_{2}} ∑_{p} \frac{log p}{log(m/π)}ϕ̂(\frac{log p}{log(m/π)}) p^{−\frac{1}{2}} ∑_{χ∈F}[χ(p) + \overline{χ}(p)]

\frac{1}{M_{2}} ∑_{p} \frac{log p}{log(m/π)}ϕ̂(2\frac{log p}{log(m/π)}) p^{−1} ∑_{χ∈F}[χ^{2}(p) + \overline{χ}^{2}(p)]

Characters Sums:

∑_{l_{i}=1}^{m_{i}−2} χ_{l_{i}}(p) = \begin{cases}m_{i} − 1 − 1 & p ≡ 1(m_{i}) \\ −1 & otherwise \end{cases}

Define

δ_{m_{i}}(p, 1) = \begin{cases}1 & p ≡ 1(m_{i}) \\ 0 & otherwise \end{cases}

Then

∑_{χ∈F} χ(p) = ∑_{l_{1}=1}^{m_{1}−2} ⋯ ∑_{l_{r}=1}^{m_{r}−2} χ_{l_{1}}(p) ⋯ χ_{l_{r}}(p)

= ∏_{i=1}^{r} ∑_{l_{i}=1}^{m_{i}−2} χ_{l_{i}}(p)

= ∏_{i=1}^{r} (−1 + (m_{i} − 1)δ_{m_{i}}(p, 1)) .

Expansion Preliminaries:

k(s) is an s-tuple (k_{1}, k_{2}, …, k_{s}) with k_{1} < k_{2} < ⋯ < k_{s}. This is just a subset of (1, 2, …, r), 2^{r} possible choices for k(s).

δ_{k(s)}(p, 1) = ∏_{i=1}^{s} δ_{m_{k_{i}}}(p, 1).

If s = 0 we define δ_{k(0)}(p, 1) = 1 ∀p. Then

∏_{i=1}^{r} (−1 + (m_{i} − 1)δ_{m_{i}}(p, 1))
= ∑_{s=0}^{r} ∑_{k(s)}(−1)^{r−s}δ_{k(s)}(p, 1) ∏_{i=1}^{s}(m_{k_{i}} − 1)

First Sum:

≪ ∑_{p}^{m^{σ}} p^{−\frac{1}{2}} \frac{1}{M_{2}} (1 + ∑_{s=1}^{r} ∑_{k(s)} δ_{k(s)}(p, 1) ∏_{i=1}^{s}(m_{k_{i}} − 1)) .

As m/M_{2} ≤ 3^{r}, s = 0 sum contributes

S_{1,0} = \frac{1}{M_{2}} ∑_{p}^{m^{σ}} p^{−\frac{1}{2}} ≪ 3^{r}m^{\frac{1}{2}σ−1},

hence negligible for σ < 2. Now we study

S_{1,k(s)} = \frac{1}{M_{2}} ∏_{i=1}^{s}(m_{k_{i}} − 1) ∑_{p}^{m^{σ}} p^{−\frac{1}{2}}δ_{k(s)}(p, 1)

≪ \frac{1}{M_{2}} ∏_{i=1}^{s}(m_{k_{i}} − 1) ∑_{n≡1(m_{k(s)})}^{m^{σ}} n^{−\frac{1}{2}}

≪ \frac{1}{M_{2}} ∏_{i=1}^{s}(m_{k_{i}} − 1)\frac{1}{∏_{i=1}^{s}(m_{k_{i}})} ∑_{n}^{m^{σ}} n^{−\frac{1}{2}}
≪ 3^{r}m^{\frac{1}{2}σ−1}.

First Sum (continued):

There are 2^{r} choices, yielding

S_{1} ≪ 6^{r}m^{\frac{1}{2}σ−1},

which is negligible as m goes to infinity for fixed r if σ < 2. Cannot let r go to infinity. If m is the product of the first r primes,

log m = ∑_{k=1}^{r} log p_{k}
= ∑_{p≤r} log p ≈ r

Therefore

6^{r} ≈ m^{log 6} ≈ m^{1.79}.

Second Sum Expansions:

∑_{l_{i}=1}^{m_{i}−2} χ_{l_{i}}^{2}(p) = \begin{cases}m_{i} − 1 − 1 & p ≡ ±1(m_{i}) \\ −1 & otherwise \end{cases}

∑_{χ∈F} χ^{2}(p)

= ∑_{l_{1}=1}^{m_{1}−2} ⋯ ∑_{l_{r}=1}^{m_{r}−2} χ_{l_{1}}^{2}(p) ⋯ χ_{l_{r}}^{2}(p)

= ∏_{i=1}^{r} ∑_{l_{i}=1}^{m_{i}−2} χ_{l_{i}}^{2}(p)

= ∏_{i=1}^{r} (−1 + (m_{i} − 1)δ_{m_{i}}(p, 1) + (m_{i} − 1)δ_{m_{i}}(p, −1))

Second Sum Bounds:

Handle similarly as before. Say

p ≡ 1 mod m_{k_{1}}, …, m_{k_{a}}
p ≡ −1 mod m_{k_{a}+1}, …, m_{k_{b}}

How small can p be? +1 congruences imply p ≥ m_{k_{1}} ⋯ m_{k_{a}} + 1. −1 congruences imply p ≥ m_{k_{a+1}} ⋯ m_{k_{b}} − 1. Since the product of these two lower bounds is greater than ∏_{i=1}^{b}(m_{k_{i}} − 1), at least one

must be greater than (∏_{i=1}^{b}(m_{k_{i}} − 1))^{\frac{1}{2}}.

There are 3^{r} pairs, yielding

Second Sum = ∑_{s=0}^{r} ∑_{k(s)} ∑_{j(s)} S_{2,k(s),j(s)} ≪ 9^{r}m^{−\frac{1}{2}}.

APPENDIX II: Random Graphs

Random Graphs (with Peter Sarnak, Yakov Sinai)

• Dustin Steinhauer

• Randy Qian

• Felice Kuan

• Leo Goldmakher

Fat Thin Families

Need a family F A T enough to do averaging. Need a family THIN enough so that everything isn’t averaged out. Real Symmetric Matrices have \frac{N(N+1)}{2} independent entries.

Random Graphs

Degree of a vertex = number of edges leaving the vertex. Adjacency matrix: a_{ij} = number edges between Vertex i and Vertex j.

A = ( \begin{matrix}0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 2 \\ 1 & 0 & 2 & 0 \end{matrix} )

These are Real Symmetric Matrices.

McKay’s Law

Density of Eigenvalues for d-regular graphs

f(x) = \begin{cases}\sqrt[\frac{d}{2π(d^{2}−x^{2})}]{4(d − 1) − x^{2}} & |x| ≤ 2\sqrt{d − 1} \\ 0 & otherwise. \end{cases}

d = 3.

McKay’s Law

d = 6.

Fat Thin: fat enough to average, thin enough to get something different than Semi-circle.

3-Regular, 2000 Vertices and GOE

159

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