Evidence for a Spectral Interpretation of the Zeros of

L-functions from One-Parameter Families of Elliptic Curves

Steven Miller

Princeton University

Salt Lake City: October 27^{th}, 2002

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

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Fundamental Problem: Spacing Between Events

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

Often need to normalize by average spacing.

Example 1: Spacings Between Primes / Prime Pairs.

Example 2: Spacings Between Energy Levels of Nuclei.

Example 3: Spacings Between Eigenvalues of Matrices.

Example 4: Spacings Between Zeros of L-Functions.

Elliptic Curves

Consider E : y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}, a_{i} ∈ ℚ and its L-function

L(s, E) = ∏_{p|Δ} (1 − a_{p}p^{−s})^{−1} ∏_{p †Δ} (1 − a_{p}p^{−s} + p^{1−2s})^{−1}

By GRH: All non-trivial zeros on the critical line, can talk about spacings between zeros.

Rational solutions form a group: E(ℚ) = ℤ^{r} ⨁ T, T is the torsion points, r is the geometric rank.

Birch and Swinnerton-Dyer Conjecture: Geometric rank equals the analytic rank, the order of vanishing of L(s, E) at s = \frac{1}{2}.

One-parameter families: a_{i} = a_{i}(t) ∈ ℤ[t].

Random Matrix Theory

Consider the group of N × N matrices from one of the classical compact groups: unitary, symplectic, orthogonal.

One assigns probability measures to matrices from various groups. By explicitly calculating properties associated to an individual matrix and integrating over the group, one can often use the group average to make good predictions about the expected behaviour of statistics from a generic, randomly chosen element.

More generally, can consider other spaces: GUE / GOE: Hermitian / Symmetric matrices with Gaussian probabilities for entries.

Measures of Spacings: n-Level Correlations

{α_{j}} be 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}

Instead of using a box, can use a smooth test function.

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

Let f(x) = ∏_{i} f_{i}(x_{i}), f_{i} even Schwartz functions whose Fourier Transforms are compactly supported.

D_{n,E}(f) = ∑_{{j_{1},...,jn \atop distinct}} f_{1}(L_{E}γ_{E}^{(j_{1})}) ⋯ f_{n}(L_{E}γ_{E}^{(j_{n})})

1. individual zeros contribute in limit

2. most of contribution is from low zeros

3. average over similar curves (family)

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

Normalization of Zeros

How should we normalize the zeros of the curves in our family?

1. Local Data (hard): using some natural measure from the curve

2. Global Data (easy): using an average from the family Hope: for f a good even test function with

compact support, as |F| → ∞,

\frac{1}{|F|} ∑_{E∈F} D_{n,E}(f) = \frac{1}{|F|} ∑_{E∈F} ∑_{{j_{1},...,jn \atop j_{i}≠±j_{k}}} ∏_{i} f_{i}(\frac{log N_{E}}{2π}γ_{E}^{(j_{i})})

→ ∫ ⋯ ∫ f(x)W_{n,G(F)}(x)dx
= ∫ ⋯ ∫ f̂(u)W_{n,G(F)}^{}(u)du.

Much of the work is handling the dependence on the conductors.

1-Level Densities

Katz and Sarnak calculate the n-level densities for the classical compact groups. Unlike the correlations, the densities are different for different groups.

The Fourier Transforms for the 1-level densities are

W_{1,O^{+}}^{}(u) = δ_{0}(u) + \frac{1}{2}η(u)
W_{1,O}^{}(u) = δ_{0}(u) + \frac{1}{2}
W_{1,O^{−}}^{}(u) = δ_{0}(u) − \frac{1}{2}η(u) + 1

W_{1,Sp}^{}(u) = δ_{0}(u) − \frac{1}{2}η(u)

W_{1,U}^{}(u) = δ_{0}(u)

where δ_{0}(u) is the Dirac Delta functional and η(u) is 1, \frac{1}{2}, and 0 for |u| less than 1, 1, and greater than 1.

2-Level Densities

We give the effect of the Fourier Transform of the densities on test functions supported in σ_{1} + σ_{2} < 1, where σ_{i} is the support of f_{i}.

Let c(G) = 0, \frac{1}{2} or 1 for G = SO(even), O, and SO(odd). For G one of these three groups we have

∫ ∫ f̂_{1}(u_{1})f̂_{2}(u_{2})Ŵ_{2,G}(u)du_{1}du_{2} = [f̂_{1}(0) + \frac{1}{2}f_{1}(0)][f̂_{2}(0) + \frac{1}{2}f_{2}(0)]
+ 2 ∫ |u|f̂_{1}(u)f̂_{2}(u)du − 2f̂_{1}f_{2}(0)
−f_{1}(0)f_{2}(0)
+ c(G)f_{1}(0)f_{2}(0).

For G = U we have

∫ ∫ f̂_{1}(u_{1})f̂_{2}(u_{2})W_{2,U}^{}(u)du_{1}du_{2} = f̂_{1}(0)f̂_{2}(0) + ∫ |u|f̂_{1}(u)f̂_{2}(u)du − f̂_{1}f_{2}(0),

and for G = Sp, we have

∫ ∫ f̂_{1}(u_{1})f̂_{2}(u_{2})Ŵ_{2,G}(u)du_{1}du_{2} = [f̂_{1}(0) + \frac{1}{2}f_{1}(0)][f̂_{2}(0) + \frac{1}{2}f_{2}(0)]
+ 2 ∫ |u|f̂_{1}(u)f̂_{2}(u)du − 2f̂_{1}f_{2}(0)
− f_{1}(0)f_{2}(0)
−f_{1}(0)f̂_{2}(0) − f̂_{1}(0)f_{2}(0) + 2f_{1}(0)f_{2}(0).

These densities are all distinguishable for functions with arbitrarily small support.

For the orthogonal groups, the densities (in this range) depend only on the distribution of the signs of the fuctionnal eqs.

Explicit Formula

Relates sums of test functions over zeros to sums over primes of a_{E}(p) and a_{E}^{2}(p).

∑_{γ_{E}^{(j)}} G(\frac{log N_{E}}{2π} γ_{E}^{(j)}) = Ĝ(0) + G(0)

− 2 ∑_{p} \frac{log p}{log N_{E}} \frac{1}{p}Ĝ(\frac{log p}{log N_{E}} )a_{E}(p)

− 2 ∑_{p} \frac{log p}{log N_{E}} \frac{1}{p^{2}} Ĝ(\frac{2 log p}{log N_{E}} )a_{E}^{2}(p)

+ O(\frac{log log N_{E}}{log N_{E}} ).

Modified Explicit Formula:

∑_{γ_{E}^{(j)}} G(\frac{log X}{2π} γ_{E}^{(j)}) = \frac{log N_{E}}{log X} Ĝ(0) + G(0)

− 2 ∑_{p} \frac{log p}{log X} \frac{1}{p}Ĝ(\frac{log p}{log X} )a_{E}(p)

− 2 ∑_{p} \frac{log p}{log X} \frac{1}{p^{2}} Ĝ(\frac{2 log p}{log X})a_{E}^{2}(p)

+ O(\frac{log log X}{log X} ).

Some Previous Results

1. Orthogonal: Iwaniec-Luo-Sarnak: 1-level density for holomorphic even weight k cuspidal newforms of square-free level N (SO(even) and SO(odd) if split by sign).

2. Symplectic: Rubinstein: n-level densities for twists L(s, χ_{d}) of the zeta-function.

Main Tools:

1. Averaging Formulas (Petersson formula in ILS, Orthogonality of characters in Rubinstein).

2. Constancy of conductors.

Elliptic Curve Conductors:

C(t) = ∏_{p|Δ(t)} p^{f_{p}(t)}

1-Level Expansion

D_{1,F}(f) = \frac{1}{|F|} ∑_{E∈F} ∑_{j} f(\frac{log N_{E}}{2π} γ_{E}^{(j)})
= \frac{1}{|F|} ∑_{E∈F} f̂(0) + f_{i}(0)

− \frac{2}{|F|} ∑_{E∈F} ∑_{p} \frac{log p}{log N_{E}} \frac{1}{p}f̂(\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}} f̂(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(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(p_{1}···p_{n})} a_{t_{1}}^{r_{1}}(p_{1}) ⋯ a_{t_{n}}^{r_{n}}(p_{n}) = A_{r_{1},F}(p_{1}) ⋯ A_{r_{n},F}(p_{n}).

Needed 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 (Silverman and Rosen):

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

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

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

New Results

Rational Surfaces Density Theorem: Consider a 1parameter family of elliptic curves of rank r over ℚ(t) that is a rational surface. Assume GRH, j(t) non-constant, and the ABC (or Sq-Free Sieve) conjecture if Δ(t) has an irreducible polynomial factor of degree ≥ 4. Let m = deg C(t) and f_{i} be an even Schwartz function of small support σ_{i} (σ_{1} < min(\frac{1}{2}, \frac{2}{3m}) for the 1-level density, σ_{1} + σ_{2} < \frac{1}{3m} for the 2-level density). Possibly after passing to a subsequence, we observe two pieces. The first equals the expected contribution from r zeros at the critical point (agreeing with what B-SD suggests). The second is

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

where N(F, −1) is the percent of curves with odd sign.

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

Examples

Constant-Sign Families:

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

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

3. y^{2} = x^{3} + tx^{2} − (t + 3)x + 1, t^{2} + 3t + 9 Sq-Free: all odd. First two rank 0 over ℚ(t); third is rank 1. Only as-

sume GRH for first two; add B-SD to interpret third.

Family of Rank 6 over ℚ(t) (modulo reasonable conjs):

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 & = & 8916100448256000000 \\ B & = & −811365140824616222208 \\ C & = & 26497490347321493520384 \\ D & = & −343107594345448813363200 \\ a & = & 16660111104 \\ b & = & −1603174809600 \\ c & = & 2149908480000 \end{aligned}

Sieving

∑_{t=N ; {D(t) \atop sqfree}}^{2N} S(t) = ∑_{d=1}^{N^{k/2}} μ(d) ∑_{{D(t)≡0(d^{2}) \atop t∈[N,2N]}} S(t)

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

Handle first piece by progressions (need progressions to evaluate sums of a_{t}(p)).

Handle second piece by Cauchy-Schwartz: The number of t in the second sum (by ABC or SqFree Sieve Conj) is o(N). Can show ∑_{t=N}^{2N} S^{2}(t) = O(N). Then

∑_{t∈T} S(t) ≪ ( ∑_{t∈T} S^{2}(t))^{\frac{1}{2}} · ( ∑_{t∈T} 1)^{\frac{1}{2}}

≪ ( ∑_{t∈[N,2N]} S^{2}(t))^{\frac{1}{2}} · o(\sqrt{N}).

Partial Summation

Notation: 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))

O(p^{R}) is the error from using Hasse to bound the partial sums: p^{R} = p^{1+\frac{r}{2}}.

Difficult Piece

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

Taylor Expansion not enough. Use Bounded Variation: conductors must be

monotone.

∑_{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})})|

Handling the Conductors

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

Let P be the product of the bad primes.

By Tate’s Algorithm, can determine f_{p}(t), which depends on the coefficients a_{i}(t) mod powers of p.

Apply Tate’s Algorithm to E_{t_{1}} to determine f_{p}(t_{1}) for the bad primes. m large, f_{p}(τ) = f_{p}(P^{m}t + t_{1}) = f_{p}(t_{1}) for p|P.

m enormous, for bad primes, the order of p dividing D(P^{m}t + t_{1}) is independent of t. So can find integers st C(τ) = c_{bad} \frac{D_{1}^{2}(τ)}{c_{1}} \frac{D_{2}(τ)}{c_{2}}, D(τ) square-free.

Application: Bounding Excess Rank

D_{1,F}(f_{1}) = f̂_{1}(0) + \frac{1}{2}f_{1}(0) + rf_{1}(0).

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

Rf_{1}(0)P_{R} ≤ f̂_{1}(0) + \frac{1}{2}f_{1}(0), R > 1.

Note the family rank r has been cancelled from both sides.

By using the 2-level density, however, we get squares of the rank on the left hand side. The advantage is we get a cross term rR. The disadvantage is our support is smaller. Once R is large, the 2-level density yields better results.

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 behaviour agrees with the predicted behaviour. 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