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Zeros Near the Central Point of Elliptic Curve L-Functions

Steven J. Miller Brown University

Brandeis, April 1^{st}, 2005 http://www.math.brown.edu/∼sjmiller

Acknowledgments

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

• 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.

• Understand zeros of Elliptic Curve L-functions near the central point.

PART I

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! 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)

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

Origins of Random Matrix Theory (continued)

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

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}.

• 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.

PART II 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.

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 (A^{∗} = A).

Explicit Formula: Analogue of the 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.

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}.

Normalized zeros: study γ_{n+1} log γ_{n+1} − γ_{n} log γ_{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} L_{p}(p^{−s}), Re(s) ≫ 0, L_{p}(x) = polynomial.

• Functional Equation:

Λ(s) := (Γ − Factors) · L(s)
Λ(s) = 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}

Measures of Spacings: n-Level Correlations

{α_{j}} an increasing sequence of numbers, B ⊂ ℝ^{n−1} a compact box. Define the n-level correlation by

lim_{N→∞} \frac{#\{ (α_{j_{1}} − α_{j_{2}}, …, α_{j_{n−1}} − α_{j_{n}}) ∈ B, j_{i} ≠ j_{k} ≤ N\}}{N}

lim_{N→∞} \frac{#\{ (α_{j_{1}} − α_{j_{2}}, …, α_{j_{n−1}} − α_{j_{n}}) ∈ B, j_{i} ≠ j_{k} ≤ N\}}{N}

Results on Zeros (Assuming GRH):

• Normalized spacings of ζ(s) starting at 10^{20} (Odlyzko)

• Pair and triple correlations of ζ(s) (Montgomery, Hejhal)

• n-level correlations for all automorphic cupsidal L-fns (Rudnick-Sarnak)

• n-level correlations for the classical compact groups (Katz-Sarnak)

• insensitive to any finite set of zeros

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

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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. 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. 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})

Local data: a_{p} = p − N_{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}}.

From Breuil, Conrad, Diamond, Taylor and Wiles:

Λ(E, s) = (2π)^{−s}C_{E}^{s/2}Γ(s)L(E, s)
Λ(E, s) = ϵ_{E}Λ(E, 2 − s), ϵ_{E} = ±1.

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].

Have a FAMILY of L-Functions:

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

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

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

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?

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.

• Insensitive to finitely many zeros.

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.

• Insensitive to finitely many zeros.

• 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.

• Insensitive to finitely many zeros.

• 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.

• Insensitive to finitely many zeros.

• 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.

• Insensitive to finitely many zeros.

• 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, N → ∞.

• NT: pick many L-functions in a family, C_{t} → ∞.

Random Matrix Ensembles

Real Symmetric, Complex Hermitian Matrices:

• λ ∈ ℝ.

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

Random Matrix Ensembles

Real Symmetric, Complex Hermitian Matrices:

• λ ∈ ℝ.

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

Classical Compact Groups:

• λ = e^{iθ}, θ ∈ (−π, π] ⊂ ℝ.

• Randomness: Haar measure; canonical choice.

• 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} ≤ ⋯

Measures of Spacings: 1-Level Density and Families

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

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

D_{n,f}(ϕ) = ∑_{{j1,...,jn \atop ji≠±jk}} ϕ_{1} (γ_{j_{1}}\frac{log C_{f}}{2π} ) ⋯ ϕ_{n} (γ_{j_{n}}\frac{log C_{f}}{2π} )

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

D_{n,f}(ϕ) = ∑_{{j1,...,jn \atop ji≠±jk}} ϕ_{1} (γ_{j_{1}}\frac{log C_{f}}{2π} ) ⋯ ϕ_{n} (γ_{j_{n}}\frac{log C_{f}}{2π} )

• Individual zeros contribute in limit

• Most of contribution is from low zeros

• Average over similar L-functions (family)

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

D_{n,f}(ϕ) = ∑_{{j1,...,jn \atop ji≠±jk}} ϕ_{1} (γ_{j_{1}}\frac{log C_{f}}{2π} ) ⋯ ϕ_{n} (γ_{j_{n}}\frac{log C_{f}}{2π} )

• Individual zeros contribute in limit

• Most of contribution is from low zeros

• Average over similar L-functions (family)

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

Normalization of Zeros

Local (hard) vs Global (easy). As N → ∞:

\frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} D_{n,E_{t}}(ϕ) = \frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∑_{{j1,...,jn \atop ji≠±jk}} ∏_{i} ϕ_{i} (γ_{t,j_{i}}\frac{log C_{t}}{2π})
→ ∫ ⋯ ∫ ϕ(x)W_{n,G(F)}(x)dx
→ ∫ ⋯ ∫ ϕ̂(u)Ŵ_{n,G(F)}(u)du.

Conj: Distribution of Low Zeros agrees with Orthogonal Densities.

Some Number Theory Results

• Orthogonal: Iwaniec-Luo-Sarnak: 1-level density for H_{k}^{±}(N), N square-free; Dueñez-Miller: 1, 2-level {ϕ × sym^{2}f : f ∈ H_{k}(1)}, ϕ even Maass; Miller, Young: families of elliptic curves. Güloğlu: 1-level for {Sym^{r}f : f ∈ H_{k}(1)}, r odd.

• Symplectic: Rubinstein: n-level densities for L(s, χ_{d}); Dueñez-Miller: 1-level for {ϕ × f : f ∈ H_{k}(1)}, ϕ even Maass. Güloğlu: 1-level for {Sym^{r}f : f ∈ H_{k}(1)}, r even.

• Unitary:

Hughes-Rudnick, Miller: Families of Primitive Dirichlet Characters.

1-Level Densities

Fourier Transforms for 1-level densities:

Ŵ_{1,SO(even)}(u) = δ(u) + \frac{1}{2}η(u)
Ŵ_{1,SO}(u)  = δ(u) + \frac{1}{2}
Ŵ_{1,SO(odd)}(u) = δ(u) − \frac{1}{2}η(u) + 1
Ŵ_{1,Sp}(u)  = δ(u) − \frac{1}{2}η(u)
Ŵ_{1,U}(u)  = δ(u)

where δ(u) is the Dirac Delta functional and

η(u) = \{ \begin{matrix}1 if |u| < 1 \\ \frac{1}{2} & if |u| = 1 \\ 0 if |u| > 1 \end{matrix}

2-Level Densities

c(G) = \{ \begin{matrix}0 if G = SO(even) \\ \frac{1}{2} & if G = SO \\ 1 if G = SO(odd) \end{matrix}

For G = SO(even), SO or SO(odd):

∫ ∫ ϕ̂_{1}(u_{1})ϕ̂_{2}(u_{2})W_{2,G}^{}(u)du_{1}du_{2} = [ϕ̂_{1}(0) + \frac{1}{2}ϕ_{1}(0)][f̂_{2}(0) + \frac{1}{2}ϕ_{2}(0)]

+ 2 ∫ |u|ϕ̂_{1}(u)ϕ̂_{2}(u)du −2ϕ̂_{1}ϕ_{2}(0) − ϕ_{1}(0)ϕ_{2}(0) + c(G)ϕ_{1}(0)ϕ_{2}(0).

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

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

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:

∑_{γ} ϕ (γ\frac{log C}{2π}) .

Comparing the two Random Matrix Models

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:

∑_{γ} ϕ (γ\frac{log C}{2π}) .

Corrections of size

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

Motivation: Dirichlet Characters: m Prime

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

\frac{1}{m − 2} ∑_{χ≠χ_{0}} ∑_{γ_{χ}} ϕ (γ_{χ}\frac{log(\frac{m}{π})}{2π}) = ∫_{−∞}^{∞} 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}) .

Can pass Character Sum through Test Function.

Elliptic Curves: Arithmetic Progression

One-parameter families:

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

We have

a_{t}(p) = − ∑_{x mod p} \binom{\underline{x^{3} + A(t)x + B(t)}}{p} = a_{t+mp}(p)

Can handle sums of a_{t}(p) for t in arithmetic progression.

Comments on Other Number Theory Results

• explicit formula relating zeros and Fourier coefficients;

• averaging formulas for the family;

• conductors easy to control (constant or monotone)

• explicit formula relating zeros and Fourier coefficients;

• averaging formulas for the family;

• conductors easy to control (constant or monotone)

Elliptic curve E_{t}: discriminant Δ(t), conductor C_{t} is

C_{t} = ∏_{p|Δ(t)} p^{f_{p}(t)}

1-Level Expansion

D_{1,F_{N}}(ϕ) = \frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∑_{j} ϕ (γ_{t,j} \frac{log C_{t}}{2π})

= \frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} [ϕ̂(0) + ϕ_{i}(0)]

− \frac{2}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∑_{p} \frac{1}{p}\frac{log p}{log C_{t}}ϕ̂(\frac{log p}{log C_{t}}) a_{t}(p)

− \frac{2}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∑_{p} \frac{1}{p^{2}} \frac{log p}{log C_{t}}ϕ̂(2\frac{log p}{log C_{t}}) a_{t}^{2}(p) + O (\frac{log log N}{log N} )

D_{1,F_{N}}(ϕ) = \frac{1}{|F_{N}|} ∑_{E∈F_{N}} ∑_{j} ϕ (γ_{E}^{(j)}\frac{log C_{t}}{2π})

= \frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} [ϕ̂(0) + ϕ_{i}(0)]

− \frac{2}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∑_{p} \frac{1}{p}\frac{log p}{log C_{t}}ϕ̂(\frac{log p}{log C_{t}}) a_{t}(p)

− \frac{2}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∑_{p} \frac{1}{p^{2}} \frac{log p}{log C_{t}}ϕ̂(2\frac{log p}{log C_{t}}) a_{t}^{2}(p) + O (\frac{log log N}{log N} )

Want to move \frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}}, leads us to study

A_{r,F}(p) = ∑_{t mod p} a_{t}^{r}(p), r = 1 or 2.

2-Level Expansion

Need to evaluate terms like

\frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∏_{i=1}^{2} \frac{1}{p_{i}^{r_{i}}} g_{i} (\frac{log p_{i}}{log C_{t}}) a_{t}^{r_{i}}(p_{i}).

\frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∏_{i=1}^{2} \frac{1}{p_{i}^{r_{i}}} g_{i} (\frac{log p_{i}}{log C_{t}}) a_{t}^{r_{i}}(p_{i}).

Analogue of Petersson / Orthogonality: If p_{1}, …, p_{n} are distinct primes

∑_{t mod p_{1}···p_{n}} a_{t}^{r_{1}}(p_{1}) ⋯ a_{t}^{r_{n}}(p_{n}) = A_{r_{1},F}(p_{1}) ⋯ A_{r_{n},F}(p_{n}).

Input

For many families

• A_{1,F}(p) = −rp + O(1)
• A_{2,F}(p) = p^{2} + O(p^{3/2})

Input

For many families

• A_{1,F}(p) = −rp + O(1)
• A_{2,F}(p) = p^{2} + O(p^{3/2})

Rational Elliptic Surfaces (Rosen and Silverman): If rank r over ℚ(T):

lim_{X→∞} \frac{1}{X} ∑_{p≤X} −\frac{A_{1,F}(p) log p}{p} = r

Input

For many families

• A_{1,F}(p) = −rp + O(1)
• A_{2,F}(p) = p^{2} + O(p^{3/2})

Rational Elliptic Surfaces (Rosen and Silverman): If rank r over ℚ(T):

lim_{X→∞} \frac{1}{X} ∑_{p≤X} −\frac{A_{1,F}(p) log p}{p} = r

Surfaces with j(T) non-constant (Michel):

A_{2,F}(p) = p^{2} + O (p^{3/2}) .

DEFINITIONS

D_{n,F_{N}}(ϕ) = \frac{1}{|F_{N}|} ∑_{E_{t}∈F_{N}} ∑_{{j1,...,jn \atop ji≠±jk}} ∏_{i} ϕ_{i} (γ_{t,j_{i}}\frac{log C_{t}}{2π})

D_{n,F_{N}}^{(r)}(ϕ): n-level density with contribution of r zeros at central point removed.

F_{N}: Rational one-parameter family, t ∈ [N, 2N], conductors monotone.

ASSUMPTIONS

1-parameter family of Ell Curves, rank r over ℚ(T), rational surface. Assume

• GRH;

• j(T) non-constant;

• Sq-Free Sieve if Δ(T) has irred. poly. factor of degree ≥ 4.

ASSUMPTIONS

1-parameter family of Ell Curves, rank r over ℚ(T), rational surface. Assume

• GRH;

• j(T) non-constant;

• Sq-Free Sieve if Δ(T) has irred. poly. factor of degree ≥ 4.

Pass to positive percent sub-seq where conductors polynomial of degree m. ϕ_{i} even Schwartz, support σ_{i}:

• σ_{1} < min (\frac{1}{2}, \frac{2}{3m}) for 1-level.

• σ_{1} + σ_{2} < \frac{1}{3m} for 2-level.

MAIN RESULT

Theorem (M–): Under previous conditions, as N → ∞, n = 1, 2:

D_{n,F_{N}}^{(r)}(ϕ) ⟶ ∫ ϕ(x)W_{G}(x)dx,

where

G = \{ \begin{matrix}SO & if half odd \\ SO(even) & if all even \\ SO(odd) & if all odd \end{matrix}

1 and 2-level densities confirm Katz-Sarnak, Birch and Swinnderton-Dyer predictions for small support.

• Agree with Independent Model, note universality;

• Dependence on F through lower order correction terms.

Examples

Constant-Sign Families:

• y^{2} = x^{3} + 2^{4}(−3)^{3}(9t + 1)^{2}, 9t + 1 Square-Free: all even.

• y^{2} = x^{3} ± 4(4t + 2)x, 4t + 2 Square-Free: + all odd, − all even.

• y^{2} = x^{3} + tx^{2} − (t + 3)x + 1, t^{2} + 3t + 9 Square-Free: all odd.

First two rank 0 over ℚ(T), third is rank 1.

Without 2-Level Density, couldn’t say which orthogonal group.

Examples (cont)

Rational Surface of Rank 6 over ℚ(T):

y^{2} = x^{3} + (2at − B)x^{2} + (2bt − C)(t^{2} + 2t − A + 1)x
+(2ct − D)(t^{2} + 2t − A + 1)^{2}

\begin{aligned}A & = & 8, 916, 100, 448, 256, 000, 000 \\ B & = & −811, 365, 140, 824, 616, 222, 208 \\ C & = & 26, 497, 490, 347, 321, 493, 520, 384 \\ D & = & −343, 107, 594, 345, 448, 813, 363, 200 \\ a & = & 16, 660, 111, 104 \\ b & = & −1, 603, 174, 809, 600 \\ c & = & 2, 149, 908, 480, 000 \end{aligned}

Need GRH, Sq-Free Sieve to handle sieving.

Sketch of Proof

1. Sieving (Arithmetic Progressions)

2. Partial Summation (Complete Sums)

3. Controlling Conductors (Monotone).

Sieving

∑_{{t=N \atop D(t)sqfree}}^{2N} S(t) = ∑_{d=1}^{N^{k/2}} μ(d) ∑_{{D(t)≡0(d2) \atop t∈[N,2N]}} S(t)
= ∑_{d=1}^{log^{l} N} μ(d) ∑_{{D(t)≡0(d2) \atop t∈[N,2N]}} S(t) + ∑_{d≥log^{l} N}^{N^{k/2}} μ(d) ∑_{{D(t)≡0(d2) \atop t∈[N,2N]}} S(t).

Handle first by progressions. Handle second by Cauchy-Schwartz: The number of t in the second sum (by Sq-Free Sieve Conj) is o(N):

Sieving (cont)

∑_{d=1}^{log^{l} N} μ(d) ∑_{{D(t)≡0(d2) \atop t∈[N,2N]}} S(t)

t_{i}(d) roots of D(t) ≡ 0 mod d^{2}.

t_{i}(d), t_{i}(d) + d^{2}, …, t_{i}(d) + [\frac{N}{d^{2}}] d^{2}.

If (d, p_{1}p_{2}) = 1, go through complete set of residue classes \frac{N/d^{2}}{p_{1}p_{2}} times.

Partial Summation

ã_{d,i,p}(t′) = a_{t(d,i,t′)}(p), G_{d,i,P} (u) is related to the test functions, d and i from progressions. Applying Partial Summation

S(d, i, r, p) = ∑_{t′=0}^{[N/d^{2}]} ã_{d,i,p}^{r}(t′)G_{d,i,p}(t′)
= (\frac{[N/d^{2}]}{p}A_{r,F}(p) + O (p^{R})) G_{d,i,p}([N/d^{2}])

− ∑_{u=0}^{[N/d^{2}]−1} (\frac{u}{p}A_{r,F}(p) + O (p^{R})) (G_{d,i,p}(u) − G_{d,i,p}(u + 1))

Difficult Piece: Fourth Sum I

∑_{u=0}^{[N/d^{2}]−1} O(P^{R}) (G_{d,i,P} (u) − G_{d,i,P} (u + 1))

∑_{u=0}^{[N/d^{2}]−1} O(P^{R}) (G_{d,i,P} (u) − G_{d,i,P} (u + 1))

Taylor Series of G_{d,i,P} (u) − G_{d,i,P} (u + 1) gives \frac{N}{d^{2}} \frac{P^{R}}{P^{r} log N}. \frac{1}{|F_{N}|} ∑_{i,d} gives O(\frac{P^{R}}{P^{r} log N}).

Problem is in summing over the primes, as we no longer have \frac{1}{|F_{N}|}.

Fourth Sum: II

If exactly one of the r_{j}’s is non-zero, then

∑_{u=0}^{[N/d^{2}]−1} |G_{d,i,P} (u) − G_{d,i,P} (u + 1)|

= ∑_{u=0}^{[N/d^{2}]−1} |g (\frac{log p}{log C(t_{i}(d) + ud^{2})}) − g (\frac{log p}{log C(t_{i}(d) + (u + 1)d^{2})}) |

Fourth Sum: II

If exactly one of the r_{j}’s is non-zero, then

∑_{u=0}^{[N/d^{2}]−1} |G_{d,i,P} (u) − G_{d,i,P} (u + 1)|

= ∑_{u=0}^{[N/d^{2}]−1} |g (\frac{log p}{log C(t_{i}(d) + ud^{2})}) − g (\frac{log p}{log C(t_{i}(d) + (u + 1)d^{2})}) |

If conductors monotone, for fixed i, d and p, small independent of N (bounded variation).

Fourth Sum: II

If exactly one of the r_{j}’s is non-zero, then

∑_{u=0}^{[N/d^{2}]−1} |G_{d,i,P} (u) − G_{d,i,P} (u + 1)|

= ∑_{u=0}^{[N/d^{2}]−1} |g (\frac{log p}{log C(t_{i}(d) + ud^{2})}) − g (\frac{log p}{log C(t_{i}(d) + (u + 1)d^{2})}) |

If conductors monotone, for fixed i, d and p, small independent of N (bounded variation). If two of the r_{j}’s are non-zero:

|a_{1}a_{2} − b_{1}b_{2}| = |a_{1}a_{2} − b_{1}a_{2} + b_{1}a_{2} − b_{1}b_{2}|
≤ |a_{1}a_{2} − b_{1}a_{2}| + |b_{1}a_{2} − b_{1}b_{2}|
= |a_{2}| · |a_{1} − b_{1}| + |b_{1}| · |a_{2} − b_{2}|

Handling the Conductors: I

y^{2} + a_{1}(T)xy + a_{3}(T)y = x^{3} + a_{2}(T)x^{2} + a_{4}(T)x + a_{6}(T)

C(t) = ∏_{p|Δ(t)} p^{f_{p}(t)}
D_{1}(t) = primitive irred poly factors Δ(t) and c_{4}(t) share
D_{2}(t) = remaining primitive irred poly factors of Δ(t)
D(t) = D_{1}(t)D_{2}(t)

D(t) sq-free, C(t) like D_{1}^{2}(t)D_{2}(t) except for a finite set of bad primes.

Handling the Conductors: II y^{2} + a_{1}(T)xy + a_{3}(T)y = x^{3} + a_{2}(T)x^{2} + a_{4}(T)x + a_{6}(T)

Let P be the product of the bad primes. Tate’s Algorithm gives f_{p}(t), depends only on a_{i}(t) mod powers of p. Apply Tate’s Algorithm to E_{t_{1}}. Get f_{p}(t_{1}) for p|P. For m large and p|P:

f_{p}(τ) = f_{p}(P^{m}t + t_{1}) = f_{p}(t_{1}),

and order of p dividing D(P^{m}t + t_{1}) is independent of t. Get integers such that if D(τ) is sq-free then C(τ) = c_{bad}\frac{D_{1}^{2}(τ)}{c_{1}} \frac{D_{2}(τ)}{c_{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 from Independent Model.

• Application: Bounding average rank in a family (use positive test function).

PART III

NUMERICAL DATA: THEORY vs. EXPERIMENT 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.

As C_{t} → ∞ expect half the E_{t} to have rank r(E), half to have rank r(E)+1.

As C_{t} → ∞ 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%

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

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%)

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 ∑_{p≤x} \frac{1}{p} ∼ log log x.

Data on Excess Rank

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

Family: a_{1} : 0 to 10, rest −10 to 10. 14 Hours, 2,139,291 curves (2,971 singular, 248,478 distinct).

Percent with rank r  = 28.60% Percent with rank r + 1 = 47.56%
Percent with rank r + 2 = 20.97% Percent with rank r + 3 = 2.79%
Percent with rank r + 4 = .08%

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.

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 Heavy 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

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)?

APPENDICES

116

Appendix I: Standard Conjectures

Generalized Riemann Hypothesis (for Elliptic Curves) Let L(s, E) be the (normalized) L-function of the elliptic curve E. Then the non-trivial zeros of L(s, E) satisfy Re(s) = \frac{1}{2}.

Birch and Swinnerton-Dyer Conjecture [BSD1], [BSD2] Let E be an elliptic curve of geometric rank r over ℚ (the Mordell-Weil group is ℤ^{r} ⊕ T, T is the subset of torsion points). Then the analytic rank (the order of vanishing of the L-function at the central point) is also r.

Tate’s Conjecture for Elliptic Surfaces [Ta] Let E/ℚ be an elliptic surface and L_{2}(E, s) be the L-series attached to Hét^{2}(E/\overline{ℚ}, ℚ_{l}). Then L_{2}(E, s) has a meromorphic continuation to C and satisfies −ord_{s=2}L_{2}(E, s) = rank NS(E/ℚ), where NS(E/ℚ) is the ℚ-rational part of the Néron-Severi group of E. Further, L_{2}(E, s) does not vanish on the line Re(s) = 2.

Most of the 1-param families we investigate are rational surfaces, where Tate’s conjecture is known. See [RSi].

Appendix II: Equidistribution of Signs

ABC Conjecture Fix ϵ > 0. For co-prime positive integers a, b and c with c = a + b and N(a, b, c) = ∏_{p|abc} p, c ≪_{ϵ} N(a, b, c)^{1+ϵ}. The full strength of ABC is never needed; rather, we need a consequence of ABC, the Square-Free Sieve (see [Gr]): Square-Free Sieve Conjecture Fix an irreducible polynomial f(t) of degree at least 4. As N → ∞, the number of t ∈ [N, 2N] with f(t) divisible by

p^{2} for some p > log N is o(N).

For irreducible polynomials of degree at most 3, the above is known, complete with a better error than o(N) ([Ho], chapter 4). Restricted Sign Conjecture (for the Family F) Consider a one-parameter family F of elliptic curves. As N → ∞, the signs of the curves E_{t} are

equidistributed for t ∈ [N, 2N].

The Restricted Sign conjecture often fails. First, there are families with constant j(E_{t}) where all curves have the same sign. Helfgott [He] has recently related the Restricted Sign conjecture to the Square-Free Sieve conjecture and standard conjectures on sums of Moebius:

Polynomial Moebius Let f(t) be a non-constant polynomial such that no fixed square divides f(t) for all t. Then ∑_{t=N}^{2N} μ(f(t)) = o(N). The Polynomial Moebius conjecture is known for linear f(t). Helfgott shows the Square-Free Sieve and Polynomial Moebius imply the Restricted Sign conjecture for many families. More precisely, let M(t) be

the product of the irreducible polynomials dividing Δ(t) and not c_{4}(t).

Theorem: Equidistribution of Sign in a Family [He]: Let F be a one-parameter family with a_{i}(t) ∈ ℤ[t]. If j(E_{t}) and M(t) are non-constant, then the signs of E_{t}, t ∈ [N, 2N], are equidistributed as N → ∞. Further, if we restrict to good t, t ∈ [N, 2N] such that D(t) is good (usually square-free), the signs are still equidistributed in the limit.

Distribution of Signs: y^{2} = x^{3} + (T + 1)x^{2} + T x

Histogram plot:D(t) sq-free, first 2 · 10^{6} such t. Histogram plot: All t ∈ [2, 2 · 10^{6}].

Distribution of signs: y^{2} = x^{3} + (T + 1)x^{2} + T x

Histogram plot: All t ∈ [2, 5 · 10^{7}]

The observed behavior agrees with the predicted behavior. Note as the number of curves increase (comparing the plot of 5 · 10^{7} points to 2 · 10^{6} points), the fit to the Gaussian improves. Graphs by Atul Pokharel

Appendix III: Numerically Approximating Ranks: Preliminaries

Cusp form f, level N, weight 2:

f(−1/Nz) = −ϵNz^{2}f(z)
f(i/y\sqrt{N}) = ϵy^{2}f(iy/\sqrt{N}).

Define

L(f, s) = (2π)^{s}Γ(s)^{−1} ∫_{0}^{i∞}(−iz)^{s}f(z)\frac{dz}{z}
Λ(f, s) = (2π)^{−s}N^{s/2}Γ(s)L(f, s) = ∫_{0}^{∞}f(iy/√ ; \overline{N})y^{s−1}dy.

Get

Λ(f, s) = ϵΛ(f, 2 − s), ϵ = ±1.

To each E corresponds an f, write ∫_{0}^{∞} = ∫_{0}^{1} + ∫_{1}^{∞} and use transformations.

Algorithm for L^{r}(s, E): I

Λ(E, s) = ∫_{0}^{∞}f(iy/√ ; \overline{N})y^{s−1}dy
= ∫_{0}^{1}f(iy/√ ; \overline{N})y^{s−1}dy + ∫_{1}^{∞}f(iy/√ ; \overline{N})y^{s−1}dy
= ∫_{1}^{∞}f(iy/√ ; \overline{N})(y^{s−1} + ϵy^{1−s})dy.

Differentiate k times with respect to s:

Λ^{(k)}(E, s) = ∫_{1}^{∞}f(iy/√ ; \overline{N})(log y)^{k}(y^{s−1} + ϵ(−1)^{k}y^{1−s})dy.

At s = 1,

Λ^{(k)}(E, 1) = (1 + ϵ(−1)^{k}) ∫_{1}^{∞}f(iy/√ ; \overline{N})(log y)^{k}dy.

Trivially zero for half of k; let r be analytic rank.

Algorithm for L^{r}(s, E): II

Λ^{(r)}(E, 1) = 2 ∫_{1}^{∞}f(iy/√ ; \overline{N})(log y)^{r}dy
= 2 ∑_{n=1}^{∞} a_{n} ∫_{1}^{∞} e^{−2πny/√ ; \overline{N}}(log y)^{r}dy.

Integrating by parts

Λ^{(r)}(E, 1) = \frac{√ ; \overline{N}}{π} ∑_{n=1}^{∞} \frac{a_{n}}{n} ∫_{1}^{∞} e^{−2πny/√ ; \overline{N}}(log y)^{r−1} \frac{dy}{y}.

We obtain

L^{(r)}(E, 1) = 2r! ∑_{n=1}^{∞} \frac{a_{n}}{n}G_{r} (\frac{2πn}{√ ; \overline{N}}) ,

where

G_{r}(x) = \frac{1}{(r − 1)!} ∫_{1}^{∞} e^{−xy}(log y)^{r−1} \frac{dy}{y}.

Expansion of G_{r}(x)

G_{r}(x) = P_{r} (log \frac{1}{x}) + ∑_{n=1}^{∞} \frac{(−1)^{n−r}}{n^{r} · n!}x^{n}

P_{r}(t) is a polynomial of degree r, P_{r}(t) = Q_{r}(t − γ).

Q_{1}(t) = t;
Q_{2}(t) = \frac{1}{2}t^{2} + \frac{π^{2}}{12};
Q_{3}(t) = \frac{1}{6}t^{3} + \frac{π^{2}}{12}t − \frac{ζ(3)}{3};
Q_{4}(t) = \frac{1}{24}t^{4} + \frac{π^{2}}{24}t^{2} − \frac{ζ(3)}{3}t + \frac{π^{4}}{160};
Q_{5}(t) = \frac{1}{120}t^{5} + \frac{π^{2}}{72}t^{3} − \frac{ζ(3)}{6}t^{2} + \frac{π^{4}}{160}t − \frac{ζ(5)}{5} − \frac{ζ(3)π^{2}}{36}.

For r = 0,

Λ(E, 1) = \frac{√ ; \overline{N}}{π} ∑_{n=1}^{∞} \frac{a_{n}}{n} e^{−2πny/√ ; \overline{N}}.

Need about \sqrt{N} or \sqrt{N} log N terms.

Appendix IV: Bounding Excess Rank

D_{1,F}(ϕ_{1}) = ϕ̂_{1}(0) + \frac{1}{2}ϕ_{1}(0) + rϕ_{1}(0).

To estimate the percent with rank at least r + R, P_{R}, we get

Rϕ_{1}(0)P_{R} ≤ ϕ̂_{1}(0) + \frac{1}{2}ϕ_{1}(0), R > 1.

Note the family rank r has been cancelled from both sides. The 2-level density gives squares of the rank on the left, get a cross term rR. The disadvantage is our support is smaller. Once R is large, the 2-level density yields better results. We now give more details.

n-Level Density and Excess Rank Bounds

For n = 1 and 2, consider the test functions

f̂_{i}(u) = \frac{1}{2}(\frac{1}{2}σ_{n} − \frac{1}{2}|u|), |u| ≤ σ
f_{i}(x) = \frac{sin^{2}(2π\frac{1}{2}σ_{n}x)}{(2πx)^{2}}.

Expect σ_{2} = \frac{σ_{1}}{2}; only able to prove for σ_{2} = \frac{σ_{1}}{4}. Note f_{i}(0) = \frac{σ_{n}^{2}}{4}, f̂_{i}(0) = f_{i}(0)\frac{1}{σ_{n}}.

Assume B-SD, Equidistribution of Sign

Notation

Family with rank r, D_{1,F}(f) = f̂(0) + \frac{1}{2}f(0) + rf(0). By even (odd) we mean a curve whose rank r_{E} has r_{E} − r even (odd). P_{0}: probability even curve has rank ≥ r + 2a_{0}. P_{1}: probability odd curve has rank ≥ r + 1 + 2b_{0}.

D_{1,F}(f) = \frac{1}{|F_{N}|} ∑_{E∈F} ∑_{γ_{E}} f (\frac{log N_{E}}{2π}γ_{E}) ,

γ_{E} is the imaginary part of the zeros.

Average Rank: 1-Level Bounds

\frac{1}{|F|} ∑_{E∈F} r_{E}f(0) ≤ f̂_{1}(0) + \frac{1}{2}f_{1}(0) + rf_{1}(0)
\frac{1}{|F|} ∑_{E∈F} r_{E} ≤ \frac{1}{σ_{1}} + \frac{1}{2} + r.

• All Curves: r = 0, σ = \frac{4}{7}, giving 2.25 (Brumer, Heath-Brown: [Br], [BHB3], [BHB5])

• 1-Parameter Families: (deg(N(t)) + r + \frac{1}{2})·(1 + o(1)) (Silverman [Si3]).

Hope 1-Level Density true for σ → ∞.

Would yield average rank is r + \frac{1}{2}.

Excess Rank: 1-Level Bounds

Assume half even, half odd. Even curves: 1 − P_{0} have rank ≤ r + 2a_{0} − 2; replace ranks with r. P_{0} have rank ≥ r + 2a_{0}; replace with

r + 2a_{0}. Odd curves: 1 − P_{1} contributing r + 1. P_{1} contributing r + 1 + 2b_{0}.

\frac{1}{σ_{1}} + \frac{1}{2} + r ≥ \frac{1}{2}[(1 − P_{0})r + P_{0}(r + 2a_{0})]

+\frac{1}{2}[(1 − P_{1})(r + 1) + P_{1}(r + 1 + 2b_{0})]

\frac{1}{σ_{1}} ≥ a_{0}P_{0} + b_{0}P_{1}.

1-Level Density Bounds for Excess Rank

P_{0} ≤ \frac{1}{a_{0}σ_{1}}
P_{1} ≤ \frac{1}{b_{0}σ_{1}}
Prob\{rank ≥ r + 2a_{0}\} ≤ \frac{1}{a_{0}σ_{1}}.

2-Level Bounds:

D_{2,F}(f) = D_{2,F}^{∗}(f) − 2D_{1,F}(f_{1}f_{2}) + f_{1}(0)f_{2}(0)N(F, −1)
D_{2,F}^{∗}(f) = ∏_{i=1}^{2} [f̂_{i}(0) + \frac{1}{2}f_{i}(0)] + 2 ∫ |u|f̂_{1}(u)f̂_{2}(u)du

+rf̂_{1}(0)f_{2}(0) + rf_{1}(0)f̂_{2}(0) + (r^{2} + r)f_{1}(0)f_{2}(0)

D_{1,F}(f) = f̂(0) + \frac{1}{2}f(0) + rf(0).

D_{2,F}^{∗}(f) is over all zeros. Gives

\frac{1}{|F|} ∑_{E∈F} r_{E}^{2} ≤ \frac{1}{σ_{2}^{2}} + \frac{1}{σ_{2}} + \frac{1}{4} + \frac{1}{3} + \frac{2r}{σ_{2}} + r^{2} + r
= \frac{1}{σ_{2}^{2}} + \frac{2r + 1}{σ_{2}} + \frac{1}{12} + r^{2} + r + \frac{1}{2}.

Excess Rank: 2-Level Bounds: I

Similar proof yields Theorem: First 2-Level Density Bounds

P_{0} ≤ \frac{_{\frac{1}{2σ_{2}^{2}}} + \frac{1}{24} + \frac{r+\frac{1}{2}}{σ_{2}}}{a_{0}(a_{0} + r)}
P_{1} ≤ \frac{_{\frac{1}{2σ_{2}^{2}}} + \frac{1}{24} + \frac{r+\frac{1}{2}}{σ_{2}}}{b_{0}(b_{0} + r + 1)} .

For σ_{2} = \frac{σ_{1}}{4}, r = 0, a_{1} = 1: worse than 1-level density. For fixed σ_{2} = \frac{σ_{1}}{4} and r, as we increase a_{0} we eventually do get a better bound. Proportional to \frac{1}{(a_{0}σ_{1})^{2}} instead of \frac{1}{a_{0}σ_{1}}.

Excess Rank: 2-Level Bounds: II

Use D_{2,F}(f) instead of D_{2,F}^{∗}(f). r_{E} = number of zeros of curve E. Sum over j_{1} ≠ j_{2}. r_{E} even, get r_{E}(r_{E} − 2) (each zero matched with r_{E} − 2 others). r_{E} odd: (r_{E} − 1)(r_{E} − 2) + (r_{E} − 1) = r_{E}(r_{E} − 2) + 1. Theorem: Second 2-Level Density Bounds

P_{0} ≤ \frac{\frac{1}{2σ_{2}^{2}} + \frac{1}{24} + \frac{r}{σ_{2}} − \frac{1}{6σ_{2}}}{a_{0}(a_{0} + r − 1)}
P_{1} ≤ \frac{\frac{1}{2σ_{2}^{2}} + \frac{1}{24} + \frac{r}{σ_{2}} − \frac{1}{6σ_{2}}}{b_{0}(b_{0} + r)} ,

where a_{0} ≠ 1 if r = 0. σ_{2} = \frac{σ_{1}}{4} and r = 0, better for a_{0} > \frac{σ_{1}^{2}+8σ_{1}+192}{24σ_{1}}. r = 1, better for a_{0} > \frac{σ_{1}^{2}+80σ_{1}+192}{24σ_{1}}. Decay is proportional to \frac{1}{(a_{0}σ_{1})^{2}}. Note the numerator is never negative; at least \frac{1}{18}.

Excess Rank: 2-Level Bounds: IIIa

r_{E} = r + z_{E}.

∑_{j_{1}} ∑_{j_{2}} f_{1}(Lγ_{E_{j1}})f_{2}(Lγ_{E_{j2}}). Let j_{1} be one of the r family zeros, varying j_{2} gives f_{1}(0)D_{1,E}(f_{2}). Interchanging j_{1} and j_{2} we get a contribution of D_{1,E}(f_{1})f_{2}(0) for each of the r family.

Only double counting when j_{1} and j_{2} are both a family zero. Subtract off r^{2}f_{1}(0)f_{2}(0). For the other z_{E} zeros: already taken into account contribution from j_{1} one of the z_{E} zeros and j_{2} one of the r family zeros (and vice-versa).

Thus, for a given curve, a lower bound of the contribution from all pairs (j_{1}, j_{2}) is

rf_{1}(0)D_{1,E}(f_{2}) + rD_{1,E}(f_{1})f_{2}(0) − r^{2}f_{1}(0)f_{2}(0) + z_{E}^{2}.

Excess Rank: 2-Level Bounds: IIIb

Summing over all E ∈ F and simplifying gives

\frac{1}{|F|} ∑_{E∈F} z_{E}^{2} ≤ \frac{1}{σ_{2}^{2}} + \frac{1}{σ_{2}} + \frac{1}{12} + \frac{1}{2}.

Similar calculation gives Theorem: Third 2-Level Density Bounds

P_{0} ≤ \frac{\frac{1}{2σ_{2}^{2}} + \frac{1}{2σ_{2}} + \frac{1}{24}}{a_{0}^{2}}
P_{1} ≤ \frac{\frac{1}{2σ_{2}^{2}} + \frac{1}{2σ_{2}} + \frac{1}{24}}{b_{0} + b_{0}^{2}}

σ_{2} = \frac{σ_{1}}{4}: beats 1-level for a_{0} > \frac{σ_{1}^{2}+48σ_{1}+192}{24σ_{1}}. r ≠ 0: beats first 2-level once a_{0} > \frac{σ_{1}^{2}+48σ_{1}+192}{96σ_{1}}. r ≥ 1: beats second 2-level once a_{0} > \frac{3(r−1)}{3r−2} \frac{σ_{1}^{2}+48σ_{1}+192}{96σ_{1}}.

Heath-Brown & Brumer

Family of all elliptic curves E_{a,b}:

F_{T} = \{y^{2} = x^{3} + ax + b; |a| ≤ T^{\frac{1}{3}}, |b| ≤ T^{\frac{1}{2}}.

From 1-Level Expansion, get

r(E_{a,b}) ≤ 2 + \frac{log T}{log X} − 2 ∑_{p≤X} a_{P}(E_{a,b})h (\frac{log p}{log X} ) + O (\frac{1}{log X} ) .

If r(E_{a,b}) ≥ r ≥ 3 + 2\frac{log T}{log X}, then |U(E_{a,b}, X)| ≥ \frac{log T}{2} . Led to

#\{E_{a,b} ∈ F_{T} : r(E_{a,b}) ≥ r\} · (\frac{log T}{2} )^{2k} ≤ ∑_{E_{a,b}∈F} |U(E_{a,b}, X)|^{2k}.

Find X = T^{\frac{1}{10k}}, k = [\frac{r−3}{20}]. Yields

Prob(rank(E_{a,b}) ≥ r) ≪ (11r)^{−\frac{r}{20}}
rank(E_{a,b}) ≤ 17\frac{log T}{log log T} .

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