Ranks of One-Parameter Families of Elliptic Curves Over ℚ(t) and Thoughts on the Excess Rank

Question.

Steven Miller

Princeton University

Boston College: March 10^{th}, 2003

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

2

Outline of the Talk

• Review of 1 and 2-Level Densities.

• Constructing 1-Parameter Families with rank over ℚ(t).

• n-Level Density and Bounding Average Rank.

• Numerically Calculating (Approximating) Ranks.

• Potential Lower Order Density Corrections.

• Calculations of Excess Rank in Families.

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].

n-Level Density: I

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.

n-Level Density: II

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})})

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.

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} ).

New Results

Rational Surfaces Density Theorem: 1-parameter family, rank r over ℚ(t) and a rational surface. Assume

• GRH;

• j(t) non-constant;

• ABC (or Sq-Free Sieve) if Δ(t) has an irred. factor of degree ≥ 4. Let m = deg C(t), f_{i} even Schwartz with support σ_{i},

σ_{1} + σ_{2} < \frac{1}{3m}. 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}

Mestre’s Construction

Mestre ([Mes1], [Mes2]): families of rank 11 and 12 over ℚ(t); within such a family, Nagao [Na1] finds a curve with rank ≥ 21.

Consider 6-tuple of integers a_{i}. Let q(x) = ∏_{i=1}^{6}(x − a_{i}), p(t, x) = q(x − t)q(x + t). ∃g(t, x) of degree 6 in x and r(t, x) of degree at most

5 in x such that p(t, x) = g^{2}(t, x) − r(t, x).

Consider y^{2} = r(t, x). If r(t, x) is of degree 3 or 4 in x, we obtain an elliptic curve, with points P_{±i}(t) = ( ± t + a_{i}, g(±t + a_{i})).

If r(t, x) has degree 4, change variables to make the coefficient of x^{4} a perfect square (see [Mor], [Na1]). Two 6-tuples that work (see [Na1]) are (−17, −16, 10, 11, 14, 17) and (399, 380, 352, 47, 4, 0).

Get families with rank up to 14 over ℚ(t).

Shioda [Sh] gives explicit constructions for rank 2, 4, 6, 7 and 8 over ℚ(t) and generators of the Mordell-Weil groups.

Rosen-Silverman Theorem

A_{E}(p) = \frac{1}{p} ∑_{t=0}^{p−1} a_{t}(p) = \frac{1}{p}A_{1,F}(p).

The elliptic surface y^{2} = x^{3} + A(t)x + B(t), t ∈ ℤ, is rational if and only if one of the following holds

1. 0 < max{3deg A(t), 2deg B(t)} < 12.

2. 3deg A(t) = 2deg B(t) = 12, ord_{t=0}t^{12}Δ(t^{−1}) = 0

Thm [R-S]: Let E : y^{2} = x^{3} + A(t)x + B(t), and assume Tate’s conjecture (known for rational surfaces) for the surface. Then

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

We construct E with A_{E}(p) = −r + O(\frac{1}{p}). The PNT implies r is the rank over ℚ(t).

Quadratic Legendre Sums

LEMMA: Quadratic Legendre Sums: Assume a and b are not both zero mod p and p > 2. Then

∑_{t=0}^{p−1} \binom{\underline{at^{2} + bt + c}}{p} = \{(p − 1)\binom{\underline{a}}{p} if p | b^{2} − 4ac −\binom{\underline{a}}{p} otherwise

Moral: We can handle linear and quadratic Legendre sums; for cubic best is Hasse, giving O(\sqrt{p}).

Rank 6 Rational Surfaces over ℚ(t)

y^{2} = f_{t}(x) = x^{3}t^{2} + 2g(x)t − h(x)
g(x) = x^{3} + ax^{2} + bx + c, c ≠ 0
h(x) = (A − 1)x^{3} + Bx^{2} + Cx + D
D_{t}(x) = g(x)^{2} + x^{3}h(x).

D_{t}(x) is one-fourth the discriminant of the quadratic (in t) polynomial x^{3}t^{2} + 2g(x)t − h(x).

The number of distinct, non-zero roots of D_{t}(x) will control the rank.

a_{t}(p) = − ∑_{x(p)} \binom{\underline{f_{t}(x)}}{p} = − ∑_{x(p)} \binom{\underline{x^{3}t^{2} + 2g(x)t − h(x)}}{p}.

Study −A_{1,F}(p) = ∑_{x=0}^{p−1} ∑_{t=0}^{p−1} \binom{\underline{f_{t}(x)}}{p}.

x = 0, the t-sum vanishes if c ≠ 0, as it is just ∑_{t=0}^{p−1} \binom{\underline{2ct−D}}{p}.

Rank 6: II

Studying ∑_{t(p)} ∑_{x≢0(p)} \binom{\underline{x^{3}t^{2} + 2g(x)t − h(x)}}{p}.

For x ≢ 0, have

∑_{t=0}^{p−1} \binom{\underline{x^{3}t^{2} + 2g(x)t − h(x)}}{p} = \{(p − 1)\binom{\underline{x^{3}}}{p} if p | D_{t}(x) −1\binom{\underline{x^{3}}}{p} otherwise

Find coefficients a, b, c, A, B, C, D with D_{t}(x) six distinct, non-zero perfect square roots.

Gives 6(p − 1) − 1(−6) + 0 = 6p.

Rank 6: III

For 1 ≤ i ≤ 6, let r_{i} = ρ_{i}^{2}.

D_{t}(x) = g(x)^{2} + x^{3}h(x)
= (x^{3} + ax^{2} + bx + c)^{2} + x^{3} ((A − 1)x^{3} + Bx^{2} + Cx + D)
= Ax^{6} + (B + 2a)x^{5} + (C + a^{2} + 2b)x^{4} + (D + 2ab + 2c)x^{3}

+ (2ac + b^{2})x^{2} + (2bc)x + c^{2}

= A(x^{6} + \frac{B + 2a}{A}x^{5} + \frac{C + a^{2} + 2b}{A}x^{4} + \frac{D + 2ab + 2c}{A}x^{3}

+ \frac{2ac + b^{2}}{A}x^{2} + \frac{2bc}{A}x + \frac{c^{2}}{A})

= A(x^{6} + R_{5}x^{5} + R_{4}x^{4} + R_{3}x^{3} + R_{2}x^{2} + R_{1}x + R_{0})
= A(x − r_{1})(x − r_{2})(x − r_{3})(x − r_{4})(x − r_{5})(x − r_{6})
= AΦ(x).

Determining Constants a, …, D

Can matching the x^{5}, x^{4}, x^{3} terms (B, C, D). We must simultaneously solve

2ac + b^{2} = R_{2}A
2bc = R_{1}A
c^{2} = R_{0}A.

(0.0.1)

Send A → Aw^{2}, rescaling b and c by w. Have 2ac + b^{2}w = R_{2}Aw. Taking w = 2c simplifies

to a + b^{2} = R_{2}A. Take A = 4R_{0}. Then c^{2} = 4R_{0}^{2}, or c = 2R_{0}. 2bc = R_{1}A yields b = R_{1}. We now send A → Aw^{2} = (4R_{0}) · (2c)^{2} = 64R_{0}^{3}.

Determining Constants a, …, D: II

\begin{aligned}c^{2} & = & 64R_{0}^{4} & → & c & = & 8R_{0}^{2} \\ 2bc & = & 64R_{0}^{3}R_{1} & → & b & = & 4R_{0}R_{1} \\ 2ac + b^{2} & = & 64R_{0}^{3}R_{2} & → & a & = & 4R_{0}R_{2} − R_{1}^{2}. \end{aligned}

Take r_{i} = ρ_{i}^{2} = i^{2}. Then

AΦ(x) = A(x − 1)(x − 4)(x − 9)(x − 16)(x − 25)(x − 36)
= A(x^{6} − 91x^{5} + 3003x^{4} − 44473x^{3}

+296296x^{2} − 773136x + 518400)

= A(x^{6} + \frac{B + 2a}{A}x^{5} + \frac{C + a^{2} + 2b}{A}x^{4} + \frac{D + 2ab + 2c}{A}x^{3}

+ \frac{2ac + b^{2}}{A}x^{2} + \frac{2bc}{A}x + \frac{c^{2}}{A})

= A(x^{6} + R_{5}x^{5} + R_{4}x^{4} + R_{3}x^{3} + R_{2}x^{2} + R_{1}x + R_{0}).

We have

R_{0} = 518400, R_{1} = −773136, R_{2} = 296296. Hence

\begin{aligned}A & = & 64R_{0}^{3} & = & 8916100448256000000 \\ c & = & 8R_{0}^{2} & = & 2149908480000 \\ b & = & 4R_{0}R_{1} & = & −1603174809600 \\ a & = & 4R_{0}R_{2} − R_{1}^{2} & = & 16660111104 \end{aligned}

Determining Constants a, …, D: III

\begin{aligned}B & = & R_{5}A − 2a & = & −811365140824616222208 \\ C & = & R_{4}A − a^{2} − 2b & = & 26497490347321493520384 \\ D & = & R_{3}A − 2ab − 2c & = & −343107594345448813363200 \end{aligned}

We convert y^{2} = f_{t}(x) to y^{2} = F_{t}(x) in Weierstrass normal form.

y → \frac{y}{t^{2}+2t−A+1}, x → \frac{x}{t^{2}+2t−A+1}, multiply through by (t^{2} + 2t − A + 1)^{2}.

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

Except for finitely many primes, F_{t} will give same sums (modulo lower order terms).

Is a rational surface; Rosen-Silverman Theorem is applicable.

Future Work

By using cubics, quartics, and higher degrees in t, can we force sums to be large?

Will not be possible (in general) to evaluate most ∑_{t(p)} for fixed x.

Might hope to get a large contribution from x where the discriminant (in t) vanishes, and hope everything else "cancels".

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 eventu-

ally 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_{j_{1}}})f_{2}(Lγ_{E_{j_{2}}}). 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} .

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.

Potential Lower Order Density Corrections

D_{1,F}(f) = f̂(0) + f(0) − 2\frac{1}{|F|} ∑_{E∈F} ∑_{p} \frac{log p}{log N_{E}} \frac{1}{p}f̂(\frac{log p}{log N_{E}} )a_{t}(p)

−2\frac{1}{|F|} ∑_{E∈F} ∑_{p} \frac{log p}{log N_{E}} \frac{1}{p^{2}} f̂(2\frac{log p}{log N_{E}} )a_{t}(p)^{2}.

j(t) non-const ⇒ A_{2,F}(p) = ∑_{t(p)} a_{t}(p)^{2} = p^{2} + O(p^{\frac{3}{2}}).

The main term of A_{2,F}(p) contributes −\frac{1}{2}f(0) in the limit. For a great many families, we can do better than Michel’s O(p^{\frac{3}{2}}).

Have errors of size \frac{1}{log N} and \frac{log log N}{log N} from the Explicit Formula.

The conductor dependence in the Gamma factors of the Explicit Formula is easily managed. The real difficulty is handling the primes which divide the discriminant and the sums of a_{E}^{m}(p), m ≥ 3.

For convenience, rescale each curve’s zeros by the average log-conductor.

Observed A_{2,F}(p) Corrections

\underline{Family:y^{2} =}  \underline{A_{1,F}(p)}  \underline{A_{2,F}(p)}
All Curves  0  p^{3} − p^{2}
x^{3} + 2^{4}(−3)^{3}(9t + 1)^{2}  0  \{{2p^{2}−2p p≡2(3) \atop 0  p≡1(3)}
x^{3} ± 4(4t + 2)x  0  \{{2p^{2}−2p p≡1(4) \atop 0  p≡3(3)}
x^{3} + (t + 1)x^{2} + tx  0  p^{2} − 2p − 1
x^{3} + x^{2} + 2t + 1  0  p^{2} − 2p − \binom{\underline{−3}}{p}
x^{3} + tx^{2} + 1  −p  p^{2} − c_{1}(p)p
x^{3} − t^{2}x + t^{2}  −2p  p^{2} − p − c_{2}(p)p − c_{3}(p)
x^{3} − t^{2}x + t^{4}  −2p  p^{2} − p − c_{2}(p)p − c_{3}(p)
c_{1}(p) = \{ 3p + 1  p ≡ 2(3) ; 3ph_{3,p}(2) + 1 − p ∑_{x(p)} \binom{\underline{4x^{3}+1}}{p} p ≡ 1(3)
c_{2}(p) = [ ∑_{x(p)} \binom{\underline{x^{3} − x}}{p}]^{2}
c_{3}(p) = \binom{\underline{−3}}{p} + \binom{\underline{3}}{p},

and h_{3,p}(2) is one if 2 is a cube mod p and zero otherwise.

Contributions from A_{2,F}(p) Corrections Correction from −m_{F}p, m_{F} > 0. Have \frac{N}{p} full sums

and one partial sum. Substituting yields

C_{2} = \frac{2}{N} ∑_{p} \frac{log p}{log M} f̂(2\frac{log p}{log M} )\frac{1}{p^{2}} \frac{N}{p} m_{F}p
= \frac{2m_{F}}{log M} ∑_{p} f̂(2\frac{log p}{log M} )\frac{log p}{p^{2}}.

Correction of size \frac{1}{log N}. Consider

f̂(u) = \frac{1}{2}(\frac{1}{2}σ − \frac{1}{2}|u|), |u| ≤ σ
f(x) = \frac{sin^{2}(2π\frac{1}{2}σx)}{(2πx)^{2}}.

Gives

Ave Rank ≤ \frac{1}{σ} + r + \frac{1}{2} + (\frac{.986}{σ} − \frac{2.966}{σ^{2} log M} )\frac{m_{F}}{log M} .

Observed A_{2,F}(p) Corrections

Family of all elliptic curves, σ = \frac{4}{7}. Ignoring the correction term, AveRank ≤ 2.25. Correction is .05.

Extra Contributions: m_{F} = 1

\begin{matrix}M & σ = \frac{4}{7} & σ = 1 & σ = 2 & σ = ∞ \\ 10^{6} & .077 & .056 & .032 & 0 \\ 10^{12} & .051 & .032 & .017 & 0 \\ 10^{18} & .036 & .022 & .011 & 0 \\ 10^{24} & .025 & .015 & .007 & 0 \end{matrix}

Fermigier observes average rank ≈ (r + \frac{1}{2}) + .40.

Cond ≈ 10^{12}: σ = 1: AveRank ≤ (r + \frac{1}{2}) + 1 + .03.

Cond ≈ 10^{12}: σ = 2: AveRank ≤ (r + \frac{1}{2}) + \frac{1}{2} + .02.

Hopeless to get support ≥ 2. ILS obtain such large support for some of their families, but only because of great averaging formulas. No good analogue, and our conductors grow very quickly.

All Curves:

Complete sums of a_{a,b}^{m}(p) vanish for m odd, and

∑_{a=0}^{p−1} ∑_{b=0}^{p−1} a_{a,b}^{4}(p) = 3p − 3p^{2} − 2p^{3} + 2p^{4}

∑_{a=0}^{p−1} ∑_{b=0}^{p−1} a_{a,b}^{6}(p) = 5p + 4p^{2} − 9p^{3} − 5p^{4} + 5p^{5}

∑_{a=0}^{p−1} ∑_{b=0}^{p−1} a_{a,b}^{8}(p) = 7p + 13p^{2} + 8p^{3} − 28p^{4} − 14p^{5} + 14p^{6}.

Explicit Formula gives sums of α_{E}^{m}(p) + β_{E}^{m}(p), m ≥ 3.

For p| ∖ N_{E}, α_{E}(p) + β_{E}(p) = a_{E}(p) and α_{E}(p)β_{E}(p) = p. For p| ∖ N_{E},

α_{E}^{3}(p) + β_{E}^{3}(p) = a_{E}^{3}(p) − 3pa_{E}(p)
α_{E}^{4}(p) + β_{E}^{4}(p) = a_{E}^{4}(p) − 4pa_{E}^{2}(p) + 2p^{2}
α_{E}^{5}(p) + β_{E}^{5}(p) = a_{E}^{5}(p) − 5p(α_{E}^{3}(p) − β_{E}^{3}(p)) + 10p^{2}a_{E}(p)
α_{E}^{6}(p) + β_{E}^{6}(p) = a_{E}^{6}(p) − 6pa_{E}^{4}(p) + 9p^{2}a_{E}^{2}(p) − 2p^{3}
α_{E}^{7}(p) + β_{E}^{7}(p) = a_{E}^{7}(p) − 7p(α_{E}^{5}(p) + β_{E}^{5}(p)) − 21p^{2}(α_{E}^{3}(p) + β_{E}^{3}(p)) − 35p^{3}a_{E}(p)
α_{E}^{8}(p) + β_{E}^{8}(p) = a_{E}^{8}(p) − 8pa_{E}^{6}(p) + 20p^{2}a_{E}^{4}(p) − 16p^{3}a_{E}^{2}(p) + 2p^{4}.

Have \frac{1}{N^{5}} \frac{N^{2}}{p} \frac{N^{3}}{p} complete sums, multiply by \frac{1}{p^{2m}} (and other factors) and sum over the primes.

The main contribution to the potential density correction is from m = 2. Complete sum of a_{E}^{2}(p) gives p^{3} − p^{2}, with the p^{2} term leading to a sum of \frac{1}{p^{2}}.

All Curves (cont)

∑_{a=0}^{p−1} ∑_{b=0}^{p−1} a_{a,b}^{4}(p) = 3p − 3p^{2} − 2p^{3} + 2p^{4}
α_{E}^{4}(p) + β_{E}^{4}(p) = a_{E}^{4}(p) − 4pa_{E}^{2}(p) + 2p^{2}

∑_{a=0}^{p−1} ∑_{b=0}^{p−1} a_{a,b}^{6}(p) = 5p + 4p^{2} − 9p^{3} − 5p^{4} + 5p^{5}
α_{E}^{6}(p) + β_{E}^{6}(p) = a_{E}^{6}(p) − 6pa_{E}^{4}(p) + 9p^{2}a_{E}^{2}(p) − 2p^{3}

∑_{a=0}^{p−1} ∑_{b=0}^{p−1} a_{a,b}^{8}(p) = 7p + 13p^{2} + 8p^{3} − 28p^{4} − 14p^{5} + 14p^{6}.
α_{E}^{8}(p) + β_{E}^{8}(p) = a_{E}^{8}(p) − 8pa_{E}^{6}(p) + 20p^{2}a_{E}^{4}(p) − 16p^{3}a_{E}^{2}(p) + 2p^{4}.

When m = 4, potential term of size \frac{1}{p^{2}}, as \frac{1}{p^{2·2}} exactly balances p^{2+2}. Note, the test function will be evaluated at \frac{3 log p}{log M} and not \frac{2 log p}{log M}.

The p^{4} terms (which yield the main term) exactly cancel. Thus, the m = 4 term contributes a sum of size \frac{1}{p^{3}} and not of size \frac{1}{p^{2}}.

For m ≥ 6, the contributions will be of size \frac{1}{p^{3}} or less.

Hence, subject to proving the expansion formulas, for the family of all elliptic curves (no sieving, rescaling by the average logconductor), the m ≥ 1 terms contribute a potential lower order correction, where we sum over the primes terms of size \frac{1}{p^{2}}.

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.

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 critical 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 one-parameter families we investigate are rational surfaces, where Tate’s conjecture is known. See, for example, [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 oneparameter 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.

Appendix III: Quadratic Legendre Sums

Lemma: For p > 2,

S(n) = ∑_{x=0}^{p−1} \binom{\underline{n_{1} + x}}{p}\binom{\underline{n_{2} + x}}{p} = \{ p − 1 if p | n_{1} − n_{2} −1 otherwise

Proof: Shifting x by −n_{2}, we need only prove the lemma when n_{2} = 0. Assume (n, p) = 1 as otherwise the result is trivial. For (a, p) = 1 we have:

S(n) = ∑_{x=0}^{p−1} \binom{\underline{n + x}}{p}\binom{\underline{x}}{p}  (0.0.2)

= ∑_{x=0}^{p−1} \binom{\underline{n + a^{−1}x}}{p}\binom{\underline{a^{−1}x}}{p}

= ∑_{x=0}^{p−1} \binom{\underline{an + x}}{p}\binom{\underline{x}}{p} = S(an)

Hence

S(n) = \frac{1}{p − 1} ∑_{a=1}^{p−1} ∑_{x=0}^{p−1} \binom{\underline{an + x}}{p}\binom{\underline{x}}{p}  (0.0.3)

= \frac{1}{p − 1} ∑_{a=0}^{p−1} ∑_{x=0}^{p−1} \binom{\underline{an + x}}{p}\binom{\underline{x}}{p} − \frac{1}{p − 1} ∑_{x=0}^{p−1} \binom{\underline{x}}{p}^{2}

= \frac{1}{p − 1} ∑_{x=0}^{p−1} \binom{\underline{x}}{p} ∑_{a=0}^{p−1} \binom{\underline{an + x}}{p} − 1
= 0 − 1 = −1

Where do we use p > 2? We used ∑_{a=0}^{p−1} \binom{\underline{an+x}}{p} = 0 for (n, p) = 1. This is true for all odd primes (as there are \frac{p−1}{2} quadratic residues, \frac{p−1}{2} non-residues, and 0); for p = 2, there is one quadratic residue, no non-residues, and 0.

Lemma: Quadratic Legendre Sums: Assume a and b are not both zero mod p and p > 2. Then

∑_{t=0}^{p−1} \binom{\underline{at^{2} + bt + c}}{p} = \{{(p − 1)\binom{\underline{a}}{p} if p | b^{2} − 4ac \atop −\binom{\underline{a}}{p} otherwise}

Proof: Assume a ≢ 0(p) as otherwise the proof is trivial. Let δ = 4^{−1}(b^{2} − 4ac). Then

∑_{t=0}^{p−1} \binom{\underline{at^{2} + bt + c}}{p} = ∑_{t=0}^{p−1} \binom{\underline{a^{−1}}}{p}\binom{\underline{a^{2}t^{2} + bat + ac}}{p}  (0.0.4)

= ∑_{t=0}^{p−1} \binom{\underline{a}}{p}\binom{\underline{t^{2} + bt + ac}}{p}

= ∑_{t=0}^{p−1} \binom{\underline{a}}{p}\binom{\underline{t^{2} + bt + 4^{−1}b^{2} + ac − 4^{−1}b^{2}}}{p}

= ∑_{t=0}^{p−1} \binom{\underline{a}}{p}\binom{\underline{(t + 2^{−1}b)^{2} − 4^{−1}(b^{2} − 4ac)}}{p}

= ∑_{t=0}^{p−1} \binom{\underline{a}}{p}\binom{\underline{t^{2} − δ}}{p}

= \binom{\underline{a}}{p} ∑_{t=0}^{p−1} \binom{\underline{t^{2} − δ}}{p}

If δ ≡ 0(p) we get p − 1. If δ ≡ η^{2}, η ≠ 0, then by the previous Lemma

∑_{t=0}^{p−1} \binom{\underline{t^{2} − δ}}{p} = ∑_{t=0}^{p−1} \binom{\underline{t − η}}{p}\binom{\underline{t + η}}{p} = −1.

We note that ∑_{t=0}^{p−1} \binom{\underline{t^{2}−δ}}{p} is the same for all non-square δ’s (let g be a generator of the multiplicative group, δ = g^{2k+1}, change variables by t → g^{k}t). Denote this sum by S, the set of non-zero squares mod p by R, and the non-squares mod p by N. Since ∑_{δ=0}^{p−1} \binom{\underline{t^{2}−δ}}{p} = 0 we have

∑_{δ=0}^{p−1} ∑_{t=0}^{p−1} \binom{\underline{t^{2} − δ}}{p} = ∑_{t=0}^{p−1} \binom{\underline{t^{2}}}{p} + ∑_{δ∈R} ∑_{t=0}^{p−1} \binom{\underline{t^{2} − δ}}{p} + ∑_{δ∈N} ∑_{t=0}^{p−1} \binom{\underline{t^{2} − δ}}{p}
= (p − 1) + \frac{p − 1}{2}(−1) + \frac{p − 1}{2}S = 0.

Hence S = −1, proving the lemma.

Appendix IV: Contributions from Sub-Families

A strong possible explanation for the observed excess rank is the presence of sub-families of higher rank. These families’ contributions will be dwarfed in the limit, but noticeable for small N.

Consider an elliptic curve E : y^{2} = x^{3} + ax + b over ℚ and its family of twists E_{d} : dy^{2} = x^{3} + ax + b by square-free d. Assuming the Restricted Sign conjecture for twists, Gouvéa and Mazur [GM] prove that there exist positive constants C_{0}(ϵ, E) and C_{1}(ϵ, E) such that, if N ≥ C_{0}(ϵ, E), the number of squarefree d ≤ N with the rank of E_{d} even and at least 2 is at least C_{1}(ϵ, E)N^{\frac{1}{2} −ϵ}.

Similarly, Mai [Mai] considered E_{d} : x^{3} + y^{3} = d for cube-free d. Assuming the Restricted Sign conjecture for cubic twists, he proved there exist positive constants C_{2}(ϵ, E) and C_{3}(ϵ, E) such that, if N ≥ C_{2}(ϵ, E), the number of cube-free |d| ≤ N with the rank of E_{d} even and at least 2 is at least C_{3}(ϵ, E)N^{\frac{2}{3} −ϵ}.

Stewart and Top [ST] generalize these results and remove the dependence on the Restricted Sign conjecture, though at the cost of weaker bounds. They prove:

Theorem 0.0.1 (Cubic Twists). For the family x^{3} + y^{3} = d, there exists a universal constant C_{4} such that, for cube-free |d| ≤ N, if N ≥ 657, at least C_{4}N^{\frac{1}{6}} curves have rank at least 3.

Theorem 0.0.2 (Quadratic Twists). Let E be an elliptic curve over ℚ with j(E) ≠ 0, 1728, and let E_{d} be a quadratic twist. There exist constants C_{5}(E) and C_{6}(E) such that, for square-free |d| ≤ N, if N ≥ C_{5}(E), then at least C_{6}(E)N^{\frac{1}{7}} · log^{−2} N have rank even and at least 2.

In all of the above results, there is very slow decay with respect to N. The cardinality in these families is a multiple of N. Thus, we have contributions ranging from N^{\frac{1}{7}} /N to N^{\frac{1}{2}} /N to N^{\frac{2}{3}} /N. Taking N = 100, 1000 and 10000 yields

\begin{matrix}\underline{N} & \underline{N^{− \frac{6}{7}}} & \underline{N^{− \frac{1}{2}}} & \underline{N^{− \frac{2}{3}}} \\ 100 & .0193 & .1000 & .2154 \\ 1000 & .0027 & .0316 & .1000 \\ 10000 & .0004 & .0100 & .0464 \end{matrix}

In Fermigier’s [Fe2] investigations, N ranges from 250 to 1000. Thus, it is very likely, depending on the size of the constants (and the true values of the exponents) that there may be higher rank sub-families of cardinality N^{c} lurking within our families, 0 < c < 1. While not contributing in the limit, they will be very noticeable for small values of N.

To determine the analytic rank (the order of vanishing of L(s, E) at the critical point s = 1) requires studying sums of the coefficients a_{E}(n) for n ≤ \sqrt{N_{E}} log N_{E}. See, for example, [Cr]. As the conductors grow polynomially in our families, it already requires several hours to investigate families for t up to 1000. It thus seems unlikely that we will be able to attain large enough ranges of t to get past the contributions of these possibly lower cardinality sub-families.

Appendix V: Heath-Brown and Brumer Bounds

Some very quick notes about the details of the proof for the family of elliptic curves (the exponential decay).

We have the following expansion of \binom{\underline{x}}{p}:

\binom{\underline{x}}{p} = G_{p}^{−1} ∑_{c=1}^{p} \binom{\underline{c}}{p}e\binom{\underline{cx}}{p},  (0.0.5)

where G_{p} = ∑_{a(p)} \binom{\underline{a}}{p}e\binom{\underline{a}}{p}, which equals \sqrt{p} for p ≡ 1(4) and i\sqrt{p} for p ≡ 3(4). See, for example, [BEW].

For the curve y^{2} = f_{E}(x), a_{E}(p) = − ∑_{x(p)} \binom{\underline{f_{E}(x)}}{p}. We expand the x-sum by using Gauss sums, namely

a_{E}(p) = G_{p}^{−1} ∑_{x(p)} ∑_{c=1}^{p} \binom{\underline{c}}{p}e\binom{\underline{cf_{E}(x)}}{p}.  (0.0.6)

For the family of all elliptic curves, y^{2} = x^{3} + ax + b, when we need to calculate high moments of the U sum we will be led to

∑_{a,b} ∑_{t_{1}(p_{1})} ⋯ ∑_{t_{2k}(p_{2k})} ∑_{x_{1}(p_{1})} ⋯ ∑_{x_{2k}(p_{2k})} ∏_{i=1}^{2k} (\frac{t_{i}}{p_{i}} ) e_{p_{i}}(t_{i}x_{i}^{3} + t_{i}x_{i}a + t_{i}b). (0.0.7)

We have complete sums as a and b run through p_{1} ⋯ p_{2k}. |a| ≤ T^{\frac{1}{3}} , each tuple of x_{i}s will give a ∏_{i} p_{i}, and we will be left with having to bound

T^{\frac{1}{3}} ∏_{i} p_{i} ∑_{t_{1}(p_{1})} ⋯ ∑_{t_{2k}(p_{2k})} |∑_{b} e ([\frac{t_{1}}{p_{1}} + ⋯ + \frac{t_{2k}}{p_{2k}} ] b)| .  (0.0.8)

The main contribution will be when there is at least one prime that occurs only once. In this case, the exponent is not an integer, since its denominator will be divisible by that prime. We then have a geometric series, which can be bounded well (where the bound will depend on the fractional part of \frac{t_{1}}{p_{1}} + ⋯ + \frac{t_{2k}}{p_{2k}}.

It is crucial here that we have something of the form y^{2} = x^{3} + ax + b, and we can average over b. This allows us to use the Gauss sum expansion and obtain good cancellation. We are being very crude in estimating the x_{i} sums. For a general family of elliptic curves, we will not have such a separation of parameters. Thus, in general we would have a 1-parameter family, say y^{2} = x^{3}+A(t)x+B(t), and we would not be able to execute the summation over t without having to worry about the values of the x_{i}s.

Bibliography

[BEW] B. Berndt, R. Evans and K. Williams, Gauss and Jacobi Sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 21, Wiley-Interscience Publications, John Wiley & Sons, Inc., New York, 1998.

[Bi] B. Birch, How the number of points of an elliptic curve over a fixed prime field varies, J. London Math. Soc. 43, 1968, 57 − 60.

[BS] B. Birch and N. Stephens, The parity of the rank of the Mordell-Weil group, Topology 5, 1966, 295 − 299.

[BSD1] B. Birch and H. Swinnerton-Dyer, Notes on elliptic curves. I, J. reine angew. Math. 212, 1963, 7 − 25.

[BSD2] B. Birch and H. Swinnerton-Dyer, Notes on elliptic curves. II, J. reine angew. Math. 218, 1965, 79 − 108.

[BCDT] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc. 14, no. 4, 2001, 843 − 939.

[Br] A. Brumer, The average rank of elliptic curves I, Invent. Math. 109, 1992, 445 − 472.

[BHB3] A. Brumer and R. Heath-Brown, The average rank of elliptic curves III, preprint.

[BHB5] A. Brumer and R. Heath-Brown, The average rank of elliptic curves V, preprint.

[BM] A. Brumer and O. McGuinness, The behaviour of the Mordell-Weil group of elliptic curves, Bull. AMS 23, 1991, 375 − 382.

[CW] J. Coates and A. Wiles, On the conjecture of Birch and Swinnterton-Dyer, Invent. Math. 39, 1977, 43 − 67.

[Cr] Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, 1992.

[Di] F. Diamond, On deformation rings and Hecke rings, Ann. Math. 144, 1996, 137 − 166.

[Fe1] S. Fermigier, Zéros des fonctions L de courbes elliptiques, Exper. Math. 1, 1992, 167 − 173.

[Fe2] S. Fermigier, Étude expérimentale du rang de familles de courbes elliptiques sur ℚ, Exper. Math. 5, 1996, 119 − 130.

[FP] E. Fouvrey and J. Pomykala, Rang des courbes elliptiques et sommes d’exponentelles, Monat. Math. 116, 1993, 111 − 125.

[GM] F. Gouvéa and B. Mazur, The square-free sieve and the rank of elliptic curves, J. Amer. Math. Soc. 4, 1991, 45 − 65.

[Go] D. Goldfeld, Conjectures on elliptic curves over quadratic fields, Number Theory (Proc. Conf. in Carbondale, 1979), Lecture Notes in Math. 751, Springer-Verlag, 1979, 108 − 118.

[Gr] Granville, ABC Allows Us to Count Squarefrees, International Mathematics Research Notices 19, 1998, 991 − 1009.

[He] H. Helfgott, On the distribution of root numbers in families of elliptic curves, preprint.

[Ho] C. Hooley, Applications of Sieve Methods to the Theory of Numbers, Cambridge University Press, Cambridge, 1976.

[ILS] H. Iwaniec, W. Luo and P. Sarnak, Low lying zeros of families of Lfunctions, Inst. Hautes Études Sci. Publ. Math. 91, 2000, 55 − 131.

[Kn] A. Knapp, Elliptic Curves, Princeton University Press, Princeton, 1992.

[KS1] N. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, AMS Colloquium Publications 45, AMS, Providence, 1999.

[KS2] N. Katz and P. Sarnak, Zeros of zeta functions and symmetries, Bull. AMS 36, 1999, 1 − 26.

[Ko] V. Kolyvagin, On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curves, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 429−436.

[Mai] L. Mai, The analytic rank of a family of elliptic curves, Canadian Journal of Mathematics 45, 1993, 847 − 862.

[Mes1] J. Mestre, Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Mathematica 58, 1986, 209 − 232.

[Mes2] J. Mestre, Courbes elliptiques de rang ≥ 11 sur Q(t), C. R. Acad. Sci. Paris, ser. 1, 313, 1991, 139 − 142.

[Mes3] J. Mestre, Courbes elliptiques de rang ≥ 12 sur Q(t), C. R. Acad. Sci. Paris, ser. 1, 313, 1991, 171 − 174.

[Mi] P. Michel, Rang moyen de familles de courbes elliptiques et lois de Sato-Tate, Monat. Math. 120, 1995, 127 − 136.

[Mil] S. J. Miller, 1- and 2-Level Densities for Families of Elliptic Curves: Evidence for the Underlying Group Symmetries, P.H.D. Thesis, Princeton University, 2002, http://www.math.princeton.edu/∼sjmiller/thesis/thesis.pdf.

[Mor] Mordell, Diophantine Equations, Academic Press, New York, 1969.

[Na1] K. Nagao, On the rank of elliptic curve y^{2} = x^{3} − kx, Kobe J. Math. 11, 1994, 205 − 210.

[Na2] K. Nagao, Construction of high-rank elliptic curves, Kobe J. Math. 11, 1994, 211 − 219.

[Na3] K. Nagao, ℚ(t)-rank of elliptic curves and certain limit coming from the local points, Manuscr. Math. 92, 1997, 13 − 32.

[Ri] Rizzo, Average root numbers for a non-constant family of elliptic curves, preprint.

[Ro] D. Rohrlich, Variation of the root number in families of elliptic curves, Compos. Math. 87, 1993, 119 − 151.

[RSi] M. Rosen and J. Silverman, On the rank of an elliptic surface, Invent. Math. 133, 1998, 43 − 67.

[RS] Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Journal of Math. 81, 1996, 269 − 322.

[Sh] T. Shioda, Construction of elliptic curves with high-rank via the invariants of the Weyl groups, J. Math. Soc. Japan 43, 1991, 673 − 719.

[Si1] J. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 106, Springer-Verlag, Berlin - New York, 1986.

[Si2] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics 151, Springer-Verlag, Berlin - New York, 1994.

[Si3] J. Silverman, The average rank of an algebraic family of elliptic curves, J. reine angew. Math. 504, 1998, 227 − 236.

[St1] N. Stephens, A corollary to a conjecture of Birch and Swinnerton-Dyer, J. London Math. Soc. 43, 1968, 146 − 148.

[St2] N. Stephens, The diophantine equation X^{3}+Y^{3} = DZ^{3} and the conjectures of Birch and Swinnerton-Dyer, J. reine angew. Math. 231, 1968, 16 − 162.

[ST] C. Stewart and J. Top, On ranks of twists of elliptic curves and power-free values of binary forms, Journal of the American Mathematical Society 40, number 4, 1995.

[Ta] J. Tate, Algebraic cycles and the pole of zeta functions, Arithmetical Algebraic Geometry, Harper and Row, New York, 1965, 93 − 110.

[TW] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141, 1995, 553 − 572.

[Wa] L. Washington, Class numbers of the simplest cubic fields, Math. Comp. 48, number 177, 1987, 371 − 384.

[Wi] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math 141, 1995, 443 − 551.