Brown University Algebra Seminar

Random Matrix Theory Models for zeros of L-functions near the central point (and applications to elliptic curves)

Steven J. Miller Brown University September 13th, 2004, Providence, RI

http://www.math.brown.edu/∼sjmiller/math/talks/talks.html

Origins of Random Matrix Theory

Classical Mechanics: 3 Body Problem Intractable. Heavy nuclei like Uranium (200+ protons / neutrons) even worse! Info by shooting high-energy neutrons into nucleus. Fundamental Equation: Quantum Mechanics

Hψ_{n} = E_{n}ψ_{n}

Similar to stat mech, leads to considering eigenvalues of ensembles of matrices. Real Symmetric (GOE), Complex Hermitian (GUE), Classical Compact Groups.

Random Matrix Ensembles

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & ⋯ & a_{1N} \\ a_{12} & a_{22} & a_{23} & ⋯ & a_{2N} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ a_{1N} & a_{2N} & a_{3N} & ⋯ & a_{N} \end{matrix} ) = A^{T}

Let p(x) be a probability density.

p(x) ≥ 0
∫_{ℝ} p(x)dx = 1.

Often assume p(x) has finite moments:

k^{th}-moment = ∫_{ℝ} x^{k}p(x)dx.

Define

Prob(A) = ∏_{1≤i<j≤N} p(a_{ij}).

Eigenvalue Distribution

Key to Averaging:

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

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} · 1
∑_{i=1}^{N} λ_{i}^{2}(A) ∼ N^{2}

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

Eigenvalue Distribution (cont)

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

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

Obtain:

k^{th}-moment = ∫ x^{k}μ_{A,N}(x)dx
= \frac{1}{N} ∑_{i=1}^{N} \frac{λ_{i}^{k}(A)}{(2\sqrt{N})^{k}}
= \frac{Trace(A^{k})}{2^{k}N^{\frac{k}{2}+1}}

Semi-Circle Law

N × N real symmetric matrices, entries i.i.d.r.v. from fixed p(x).

Semi-Circle Law: Assume p has mean 0, variance 1, other moments finite. Then

μ_{A,N}(x) → \frac{2}{π}\sqrt{1 − x^{2}} with probability 1

Trace formula converts sums over eigenvalues to sums over entries of A.

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

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

Proof: 2^{nd}-Moment

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

Substituting into expansion gives

\frac{1}{2^{2}N^{2}} ∫_{ℝ} ⋯ ∫_{ℝ} ∑_{i,j=1}^{N} a_{ji}^{2} · p(a_{11})da_{11} ⋯ p(a_{NN})da_{NN}

Integration factors as

∫_{a_{ij}∈ℝ} a_{ij}^{2}p(a_{ij})da_{ij} · ∏_{{(k,l)≠(ij) \atop k<l}} ∫_{a_{kl}∈ℝ} p(a_{kl})da_{kl} = 1.

Have N^{2} summands, answer is \frac{1}{4}. Key: Averaging Formula, Trace Lemma.

Measures of Spacings: n-Level Correlations

{α_{j}} increasing sequence, 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}

Observations and Results:

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

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

3. n-level correlations for all automorphic cupsidal L-functions (Rudnick-Sarnak)

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

5. insensitive to any finite set of zeros

Measures of Spacings: n-Level Density and Families

ϕ(x) = ∏_{i} ϕ_{i}(x_{i}), ϕ_{i} even Schwartz functions, ϕ̂_{i} compactly supported.

D_{n,f}(ϕ) = ∑_{j1,...,jn ; distinct} ϕ_{1}(L_{f}γ_{f}^{(j_{1})}) ⋯ ϕ_{n}(L_{f}γ_{f}^{(j_{n})})

L_{f} = Conductor, the scale factor for low zeros.

1. individual zeros contribute in limit

2. most of contribution is from low zeros

3. average over similar curves (family)

D_{n,F}(ϕ) = \frac{1}{|F|} ∑_{f∈F} D_{n,f}(ϕ).

Limiting Behavior

As N → ∞,

\frac{1}{|F_{N}|} ∑_{f∈F_{N}} D_{n,f}(ϕ) = \frac{1}{|F_{N}|} ∑_{f∈F_{N}} ∑_{{j_{1},...,jn \atop j_{i}≠±j_{k}}} ∏_{i} ϕ_{i}(\frac{γ_{f}^{(j_{i})} log L_{f}}{2π} )
→ ∫ ⋯ ∫ ϕ(x)W_{n,G(F)}(x)dx
→ ∫ ⋯ ∫ ϕ̂(y)Ŵ_{n,G(F)}(y)dy.

Conj: Distribution of Low Zeros agrees with a classical compact group.

Correspondences

Similarities b/w Nuclear Physics and L-Functions:

Zeros ⟷ Energy Levels
Support ⟷ Neutron Energy.

Conjecture: Zeros near central point in a family of L-functions behave like eigenvalues near 1 of a classical compact group (Unitary, Symplectic, Orthogonal).

Some Number Theory Results

• Orthogonal: Iwaniec-Luo-Sarnak: 1-level density for H_{k}^{±}(N), N square-free; Hughes-Miller: n-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: 1, 2-level for one-parameter families of elliptic curves.

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

• Unitary:

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

Main Tools

• Explicit Formula: Relates sums over zeros to sums over primes.

• Averaging Formulas: Orthogonality of characters, Petersson formula.

• Control of conductors: Monotone.

1-Level Densities

Fourier Transforms for 1-level densities:

\begin{aligned}Ŵ_{1,SO(even)}(u) & = & δ(u) + \frac{1}{2}η(u) \\ Ŵ_{1,O}(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) \end{aligned}

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}

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 (−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) + χ̄(p)]p^{−\frac{1}{2}}

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

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

Expansion

{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) + χ̄(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) + χ̄^{2}(p)]p^{−1}
+ O(\frac{1}{log m}).

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) + χ̄(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) + χ̄^{2}(p)}{p}.

∑_{χ≠χ_{0}}[χ^{2}(p) + χ̄^{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}.

Elliptic Curves

Conductors grow rapidly. Results for small support, where Orthogonal densities indistinguishable. Study 1 and 2-Level Densities.

D_{n,f}(ϕ) = ∑_{j1,...,jn ; distinct} ϕ_{1}(L_{f}γ_{f}^{(j_{1})}) ⋯ ϕ_{n}(L_{f}γ_{f}^{(j_{n})})
D_{n,F}(ϕ) = \frac{1}{|F|} ∑_{f∈F} D_{n,f}(ϕ).

2-Level Densities

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

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

∫ ∫ ϕ̂_{1}(u_{1})ϕ̂_{2}(u_{2})Ŵ_{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).

Elliptic Curves E : y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}, a_{i} ∈ ℚ

Often can write E : y^{2} = x^{3} + Ax + B. Let N_{p} be the number of solns mod p:

N_{p} = ∑_{x(p)} [1 + \binom{\underline{x^{3} + Ax + B}}{p}] = p + ∑_{x(p)} \binom{\underline{x^{3} + Ax + B}}{p}

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

a_{E}(p) = − ∑_{x mod p} \binom{\underline{x^{3} + Ax + B}}{p}

Elliptic Curves: Arithmetic Progression

One-parameter families:

E_{t} : 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.

Elliptic Curves (cont)

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

By GRH: All zeros on the critical line. Rational solutions: E(ℚ) = ℤ^{r} ⨁ T. Birch and Swinnerton-Dyer Conjecture: Geometric rank r equals analytic rank (order of vanishing at central point).

Comments on Previous Results

• explicit formula relating zeros and Fourier coeffs;

• averaging formulas for the family;

• conductors easy to control (constant or monotone)

Elliptic curve E_{t}: discriminant Δ(t), conductor N_{E_{t}} = C(t) is

C(t) = ∏_{p|Δ(t)} p^{f_{p}(t)}

Normalization of Zeros

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

\frac{1}{|F_{N}|} ∑_{E∈F_{N}} D_{n,E}(ϕ) = \frac{1}{|F_{N}|} ∑_{E∈F_{N}} ∑_{{j_{1},...,jn \atop j_{i}≠±j_{k}}} ∏_{i} ϕ_{i}(\frac{log N_{E}}{2π}γ_{E}^{(j_{i})})
→ ∫ ⋯ ∫ ϕ(x)W_{n,G(F)}(x)dx
→ ∫ ⋯ ∫ ϕ̂(y)Ŵ_{n,G(F)}(y)dy.

Conj: Distribution of Low Zeros agrees with Orthogonal Densities.

1-Level Expansion

D_{1,F}(ϕ) = \frac{1}{|F|} ∑_{E∈F} ∑_{j} ϕ(\frac{log N_{E}}{2π}γ_{E}^{(j)})

= \frac{1}{|F|} ∑_{E∈F} ϕ̂(0) + ϕ_{i}(0)

− \frac{2}{|F|} ∑_{E∈F} ∑_{p} \frac{log p}{log N_{E}} \frac{1}{p}ϕ̂(\frac{log p}{log N_{E}})a_{E}(p)

− \frac{2}{|F|} ∑_{E∈F} ∑_{p} \frac{log p}{log N_{E}} \frac{1}{p^{2}}ϕ̂(2\frac{log p}{log N_{E}})a_{E}^{2}(p) + O(\frac{log log N_{E}}{log N_{E}})

Want to move \frac{1}{|F|} ∑_{E∈F}, 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|} ∑_{E∈F} ∏_{i=1}^{2} \frac{1}{p_{i}^{r_{i}}}g_{i}(\frac{log p_{i}}{log N_{E}})a_{E}^{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

(1) : A_{1,F}(p) = −rp + O(1)
(2) : 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}(ϕ) = \frac{1}{|F|} ∑_{E∈F} ∑_{{j1,...,jn \atop ji≠±jk}} ∏_{i} ϕ_{i}(\frac{log N_{E}}{2π}γ_{E}^{(j_{i})})

D_{n,F}^{(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 irr poly factor of deg ≥ 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}O & if half odd \\ SO(even) & if all even \\ SO(odd) & if all odd \end{matrix}

1 and 2-level densities confirm Katz-Sarnak, B-SD predictions for small support.

Examples

Constant-Sign Families:

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

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

3. 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))

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

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), depend 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, 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 st C(τ) = c_{bad}\frac{D_{1}^{2}(τ)}{c_{1}} \frac{D_{2}(τ)}{c_{2}}, D(τ) sq-free.

Excess Rank

One-parameter family, rank r over ℚ(t), RMT ⟹ 50% rank r, r+1.

For many families, observe

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

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

Data on Excess Rank

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a6

Family: a_{1} : 0 to 10, rest −10 to 10.

Percent with rank 0 = 28.60% Percent with rank 1 = 47.56% Percent with rank 2 = 20.97% Percent with rank 3 = 2.79% Percent with rank 4 = .08%

14 Hours, 2,139,291 curves (2,971 singular, 248,478 distinct).

Data on Excess Rank y^{2} + y = x^{3} + tx

Each data set 2000 curves from start.

\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.

Excess Rank Calculations Families with y^{2} = f_{t}(x); D(t) SqFree

\underline{Family}  \underline{t Range} \underline{Num t} \underline{r}  \underline{r} \underline{r + 1} \underline{r + 2} \underline{r + 3}
+4(4t + 2) [2, 2002] 1622 0  95.44  4.56
−4(4t + 2) [2, 2002] 1622 0 70.53  29.35
9t + 1  [2, 247]  169 0 71.01  28.99
t^{2} + 9t + 1 [2, 272]  169 1 71.60  27.81
t(t − 1) [2, 2002] 643 0 40.44 48.68 10.26 0.62
(6t + 1)x^{2} [2, 101]  93 1 34.41 47.31 17.20 1.08
(6t + 1)x [2, 77]  66 2 30.30 50.00 16.67 3.03

1. x^{3} + 4(4t + 2)x, 4t + 2 Sq-Free, odd.

2. x^{3} − 4(4t + 2)x, 4t + 2 Sq-Free, even.

3. x^{3} + 2^{4}(−3)^{3}(9t + 1)^{2}, 9t + 1 Sq-Free, even.

4. x^{3} + tx^{2} − (t + 3)x + 1, t^{2} + 3t + 9 Sq-Free, odd.

5. x^{3} + (t + 1)x^{2} + tx, t(t − 1) Sq-Free, rank 0.

6. x^{3} + (6t + 1)x^{2} + 1, 4(6t + 1)^{3} + 27 Sq-Free, rank 1.

7. x^{3} − (6t + 1)^{2}x + (6t + 1)^{2}, (6t + 1)[4(6t + 1)^{2} − 27] Sq-Free, rank 2.

Excess Rank Calculations Families with y^{2} = f_{t}(x); All D(t)

\underline{Family}  \underline{t Range} \underline{Num t} \underline{r}  \underline{r} \underline{r + 1} \underline{r + 2} \underline{r + 3}
+4(4t + 2) [2, 2002] 2001 0 6.45 85.76 3.95 3.85
−4(4t + 2) [2, 2002] 2001 0 63.52 9.90 25.99 .50
9t + 1  [2, 247]  247 0 55.28 23.98 20.73
t^{2} + 9t + 1 [2, 272]  271 1 73.80  25.83
t(t − 1) [2, 2002] 2001 0 42.03 48.43 9.25 0.30
(6t + 1)x^{2} [2, 101]  100 1 32.00 50.00 17.00 1.00
(6t + 1)x [2, 77]  76 2 32.89 50.00 14.47 2.63

1. x^{3} + 4(4t + 2)x, 4t + 2 Sq-Free, odd.

2. x^{3} − 4(4t + 2)x, 4t + 2 Sq-Free, even.

3. x^{3} + 2^{4}(−3)^{3}(9t + 1)^{2}, 9t + 1 Sq-Free, even.

4. x^{3} + tx^{2} − (t + 3)x + 1, t^{2} + 3t + 9 Sq-Free, odd.

5. x^{3} + (t + 1)x^{2} + tx, t(t − 1) Sq-Free, rank 0.

6. x^{3} + (6t + 1)x^{2} + 1, 4(6t + 1)^{3} + 27 Sq-Free, rank 1.

7. x^{3} − (6t + 1)^{2}x + (6t + 1)^{2}, (6t + 1)[4(6t + 1)^{2} − 27] Sq-Free, rank 2.

Orthogonal Random Matrix Model

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

dϵ_{0}(θ) ∝ ∏_{j<k}(cos θ_{k} − cos θ_{j})^{2} ∏_{j} dθ_{j}

Model: forced zeros independent (suggested by Function Field analogue)

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

Orthogonal Random Matrix Models

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

dϵ_{0}(θ) ∝ ∏_{j<k}(cos θ_{k} − cos θ_{j})^{2} ∏_{j} dθ_{j}

Interaction Model: NOT SUGGESTED BY FUNCTION FIELD Sub-ensemble of SO(2N) with the last 2n of the 2N eigenvalues equal +1:

dε_{2n}(θ) ∝ ∏_{j<k}(cos θ_{k} − cos θ_{j})^{2} ∏_{j}(1 − cos θ_{j})^{2n} ∏_{j} dθ_{j},

with 1 ≤ j, k ≤ N − n. Independent Model: SUGGESTED BY FUNCTION FIELD

A_{2N,2n} = \{(\begin{matrix}g \\ & I_{2n} \end{matrix}) : g ∈ SO(2N − 2n)\}

Random Matrix Models and One-Level Densities

Fourier transform of 1-level density:

ρ̂_{0}(u) = δ(u) + \frac{1}{2}η(u).

Fourier transform of 1-level density (Rank 2, Independent):

ρ̂_{2,Ind}(u) = [δ(u) + \frac{1}{2}η(u) + 2] .

Fourier transform of 1-level density (Rank 2, Interaction):

ρ̂_{2,Int}(u) = [δ(u) + \frac{1}{2}η(u) + 2] + 2(|u| − 1)η(u).

Testing RMT Model

For small support, 1-level densities for Elliptic Curves agree with ρ_{r,Indep}. Curve E, conductor N_{E}, expect first zero \frac{1}{2} + iγ_{E}^{(1)} with γ_{E}^{(1)} ≈ \frac{1}{log N_{E}}. If r zeros at central point, if repulsion of zeros is of size \frac{c_{r}}{log N_{E}}, might detect in 1-level density:

\frac{1}{|F_{N}|} ∑_{E∈F_{N}} ∑_{j} ϕ(\frac{γ_{E}^{(j)} log N_{E}}{2π}).

Corrections of size ϕ (x_{0} + c_{r}) − ϕ(x_{0}) ≈ ϕ′ (x(x_{0}, c_{r})) · 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: [0, 0, 0, −t^{2}, 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

Summary

• Similar behavior in different systems.

• Find correct scale.

• Average over similar elements.

• Need an Explicit Formula.

• Different statistics tell different stories.

• Evidence for B-SD, RMT interpretation of zeros • Need more data.

Appendices

The first two appendices list various standard conjectures. The second provides (at least conjecturally) when a family should have equidistribution of signs of functional equations. Experimental evidence is provided in the third appendix, which is on the distribution of signs of elliptic curves in a one-parameter family. Testing whether or not a generic family is equidistributed in sign. We looked at 1000 consecutive elliptic curves, and calculated the excess of positive over negative. We did this many times, and created a histogram plot. The fluctuations look Gaussian! The third appendix gives the formula to numerically approximate the analytic rank of an elliptic curve. For a curve of conductor N_{E}, one needs about \sqrt{N_{E}} log N_{E} Fourier coefficients. The fourth appendix gives some estimates on bounding the number of curves in a family with given rank.

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} + tx

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} + tx

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|} ∑_{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} .

Appendix V: 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) + χ̄(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) + χ̄^{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 (cont):

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}}.

Summary:

Agrees with Unitary for σ < 2.

We proved:

Lemma:

• m square-free odd integer with r = r(m) factors;

• m = ∏_{i=1}^{r} m_{i};

• M_{2} = ∏_{i=1}^{r}(m_{i} − 2). Consider the family F_{m} of primitive characters mod m. Then

First Sum ≪ \frac{1}{M_{2}}2^{r}m^{\frac{1}{2}σ}
Second Sum ≪ \frac{1}{M_{2}}3^{r}m^{\frac{1}{2}}.

Dirichlet Characters: m ∈ [N, 2N] Square-free

F_{N} all primitive characters with conductor odd square-free integer in [N, 2N]. At least N/ log^{2} N primes in the interval. At least N · \frac{N}{log^{2} N} = N^{2} log^{−2} N primitive characters:

M ≥ N^{2} log^{−2} N ⇒ \frac{1}{M} ≤ \frac{log^{2} N}{N^{2}} .

Bounds

S_{1,m} ≪ \frac{1}{M} 2^{r(m)}m^{\frac{1}{2}σ}
S_{2,m} ≪ \frac{1}{M} 3^{r(m)}m^{\frac{1}{2}}.

2^{r(m)} = τ(m), the number of divisors of m, and 3^{r(m)} ≤ τ^{2}(m). While it is possible to prove

∑_{n≤x} τ^{l}(n) ≪ x(log x)^{2^{l}−1}

the crude bound

τ(n) ≤ c(ϵ)n^{ϵ}

yields the same region of convergence.

First Sum Bound

S_{1} = ∑_{{m=N \atop m squarefree}}^{2N} S_{1,m}
≪ ∑_{m=N}^{2N} \frac{1}{M} 2^{r(m)}m^{\frac{1}{2}σ}
≪ \frac{1}{M} N^{\frac{1}{2}σ} ∑_{m=N}^{2N} τ(m)
≪ \frac{1}{M} N^{\frac{1}{2}σ}c(ϵ)N^{1+ϵ}
≪ \frac{log^{2} N}{N^{2}} N^{\frac{1}{2}σ}c(ϵ)N^{1+ϵ}
≪ c(ϵ)N^{\frac{1}{2}σ+ϵ−1} log^{2} N.

No contribution if σ < 2. Second sum handled similarly.

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