(Change slides with left and right arrows. Type "m" to see all slides.)

The "Explicit Formulas" in analytic number theory deal with
*arithmetically interesting quantities*, often given as
partial sums-the summands corresponding to primes $p$-up
to some cutoff value $X$. We'll call them *"Sums of local data"*.

A *"Sum of local data"* is a sum of contributions for each prime $p\leq X$:
*local* considerations at the prime $p$.

\delta(X) := \sum_{p\le X}g(p)

where the rules of the game require the values $g(p)$
to be determined by only
We will be concentrating on *sums of local data*
attached to elliptic curves without CM over $\mathbf{Q}$,

\delta_E(X):=\sum_{p\le X}g_E(p)

where the weighting function
p \mapsto g_E(p)

is a specific function of $p$ and $a_E(p)$,
the $p$-th Fourier coefficient of the eigenform of weight two parametrizing the elliptic curve.
We will be interested in *issues of bias.*

Examine computations of these biases, following
the classical "Explicit Formula," and the work of:

Sarnak, Granville, Rubenstein, Watkins, Martin, Fiorilli, Conrey-Snaith, ...

ROUGHLY — half the Fourier coefficients $a_E(p)$ are positive and half negative.

That is: there are roughly as many $p$'s for which the number of rational points of $E$ over $\mathbf{F}_p$ is

That is: there are roughly as many $p$'s for which the number of rational points of $E$ over $\mathbf{F}_p$ is

greater than $p+1$

as there are primes for which it is
less than $p+1$.

$\text{Curve}$ | $\text{Positive } a_E(p)\text{ for }p<10^7$ | $\text{Negative }a_E(p)\text{ for }p<10^7$ |
---|---|---|

11a (rank 0) | 332169 | 332119 |

32a (rank 0; CM) | 166054 | 166126 |

37a (rank 1) | 332127 | 332240 |

389a (rank 2) | 332317 | 332022 |

5077a (rank 3) | 331706 | 332632 |

So let's study finer statistical issues related to this symmetric
distribution. For example, we can ask the *raw question:*
which of these classes of primes are winning the race, and how often?

I.e., what can one say about:

I.e., what can one say about:

\Delta_E(X) = \frac{\log(X)}{\sqrt{X}} \text{ times }

\#\{p \text{ such that } |E(\mathbf{F}_p)| > p+1\}

minus
\#\{p \text{ such that } |E(\mathbf{F}_p)| < p+1\}?

\gamma_E(p) = \begin{cases}
0 & \text{if $p$ is a bad or supersingular prime for $E$},\\
-1 & \text{if $E$ has more than $p+1$ points rational over $\mathbf{F}_p$},\\
+1 & \text{if less}
\end{cases}

\Delta_E(X) = \frac{\log(X)}{\sqrt{X}} \sum_{p\leq X} \gamma_E(p)

Rank 0 curve 11a ($p<1000$):

Rank 0 curve 11a ($p < 10^6$):

Rank 1 curve 37a ($p < 10^6$):

Rank 2 curve 389a ($p < 10^6$):

Rank 3 curve 5077a ($p < 10^6$):

$\Delta_E(X): =\sum_{p\le X}\gamma_E(p)$

Recall that to say that
*possesses a limiting distribution $\mu_\delta$ with respect to the multiplicative measure $dX/X$*
means that for continuous bounded functions $f$ on $\mathbf{R}$ we have:
*mean* of $\delta(X)$ is by definition:

\delta(X) = \sum_{p\le X}g(X)

\lim_{X \to {\infty}}\ {\frac{1}{\log(X)}}\int_0^Xf(\delta(x))\frac{dx}{x} = \int_{\mathbf{R}}f(x)d\mu_\delta(x).

The
{\mathcal E} : = \lim_{X \to {\infty}}\ {\frac{1}{\log(X)}}\int_0^X\delta(x)\frac{dx}{x} = \int_{\mathbf{R}}d\mu_\delta(x).

In the work of Sarnak and Fiorilli, another measure for understanding "bias behavior" is given by what one might call *the percentage of positive support* (relative to the multiplicative measure $dX/X$). Namely:

\begin{align*}
{\mathcal P} & := \lim {\rm inf}_{X\to \infty}{\frac{1}{\log(X)}}\int_{2\le x \le X; \delta(x)\le 0}dx/x\\
\quad &= \lim {\rm sup}_{X\to \infty}{\frac{1}{\log(X)}}\int_{2\le x \le X; \delta(x)\le 0}dx/x
\end{align*}

It is indeed a conjecture, in specific instances interesting to us, that these limits ${\mathcal E} $ and ${\mathcal P}$ exist.
(Discuss a beautiful result of Fiorilli about ${\mathcal P}$)

\text{mean of $\delta(x)$} : = \lim_{X \to {\infty}}\ {\frac{1}{\log(X)}}\int_0^X\delta(x)\frac{dx}{x} = \int_{\mathbf{R}}d\mu_\delta(x).

Consider weighting functions $p\mapsto g_E(p)$ that have the property that:

Any such $p \mapsto g_E(p)$ represents a version of a "bias race".

To illustrate specific features of the "Explicit Formula" we focus on three examples of such races for an elliptic curve $E$.

- for all primes $p$, the number $g_E(p)$ is an
*odd*function of the value $a_E(p)$ - the
*sum of local data*\delta_E(X) := \sum_{p\leq X} g_E(p)has—or can be convincingly conjectured to have—a finite*mean*.

Any such $p \mapsto g_E(p)$ represents a version of a "bias race".

To illustrate specific features of the "Explicit Formula" we focus on three examples of such races for an elliptic curve $E$.

\text{mean of } \delta(x) : = \lim_{X \to {\infty}}\ {\frac{1}{\log(X)}}\int_0^X\delta(x) \frac{dx}{x}; \qquad
\gamma_E(p) = \begin{cases}
0 & \text{if $p$ is a bad or supersingular prime for $E$},\\
-1 & \text{if $E$ has more than $p+1$ points rational over $\mathbf{F}_p$},\\
+1 & \text{if less}
\end{cases}

## RAW |
$\Delta_E(X): = \frac{\log(X)}{\sqrt{X}} \sum_{p\le X}\gamma_E(p)$ |

## MEDIUM-RARE |
${\mathcal D}_E(X):= {\frac{\log(X)}{\sqrt X}}\sum_{p \le X}{\frac{a_E(p)}{\sqrt p}}$ |

## WELL-DONE |
${D}_E(X):= {\frac{1}{\log(X)}} \sum_{p \le X}{\frac{a_E(p)\log p}{ p}}$ |

The fun here is that there are clean conjectures for the values of the
*means* (relative to $dX/X$)
*variances*:

—i.e., the *biases*—

of the three
"sums of local data" and clean expectations of
their
(Use mouse to hover over definition above to see a conjecture.)

-r,

where $r=r_E$ is the
1-2r

{\frac{2}{\pi}}- {\frac{16}{3\pi}}r + {\frac{4}{\pi}} \sum_{k=1}^{\infty} (-1)^{k+1}\left[{\frac{1}{2k+1}} + {\frac{1}{2k+3}}\right]r({2k+1}),

where
$r(n) := r_{f_E}(n) =$ the order of vanishing of $L(\text{symm}^n f_E, s)$
at $s=1/2$, with $f_E$ the newform corresponding to $E$ and $s=1/2$
is the central point.
$\Delta_E(X): =\frac{\log(X)}{\sqrt{X}} \sum_{p\le X}\gamma_E(p), \quad
{\mathcal D}_E(X):= {\frac{\log(X)}{\sqrt X}}\sum_{p \le X}{\frac{a_E(p)}{\sqrt p}} \to 1-2r,\quad
{D}_E(X):= {\frac{1}{\log(X)}} \sum_{p \le X}{\frac{a_E(p)\log p}{ p}} \to -r$

- The (conjectured) distinction in the variances of the three formats:
- The raw data has
*infinite variance* - The medium-rare and well-done data have
*finite variance*

- The raw data has
- The numbers n\mapsto r_E(n) = \text{ the order of vanishing of }L(\text{symm}^n f_E, s) \text{ at }s=1/2(for $n$ odd) conjecturally determine
*all biases*! - We have the beginnings of some data for those numbers, $n\mapsto r_E(n)$, but
*nothing systematic*. - And no firm conjectures yet.

Numerically, instead of simply looking at examples of curves of various ranks, we instead look
for curves with interesting $r_E(n)$ and focus on the *mean*...

\text{mean of $\delta(x)$} : = \lim_{X \to {\infty}}\ {\frac{1}{\log(X)}}\int_0^X\delta(x)\frac{dx}{x} = \int_{\mathbf{R}}d\mu_\delta(x);\qquad
r_E(n) = \text{ the order of vanishing of }L(\text{symm}^n f_E, s)
\text{ at }s=1/2

If $g(t)$ is a continuous function on $[-1,+1]$ with—appropriately defined—Fourier coefficients $\{c_n\}_n$, then the *mean* of the sum of local data

\delta(X) := \sum_{p\leq X} g(a(p)/(2\sqrt{p}))

is conjecturally
\sum_{n=1}^{\infty} c_n(2 r_E(n) + (-1)^n).

Thus
\left\{\text{ Means of }\delta(X)'s\right\} \longleftrightarrow \left\{ r_E(n)'s \right\}

$E$ | rank | RAW mean | MEDIUM mean | $\to 1-2r?$ | WELL mean | $\to -r?$ |
---|---|---|---|---|---|---|

11a | 0 | 0.647 | 0.598 | 1 | 0.155 | 0 |

14a | 0 | 0.752 | 0.554 | 1 | 0.114 | 0 |

37a | 1 | -1.412 | -1.967 | -1 | -0.816 | -1 |

43a | 1 | -0.366 | -1.906 | -1 | -0.792 | -1 |

389a | 2 | -2.658 | -4.293 | -3 | -1.663 | -2 |

433a | 2 | -4.055 | -4.167 | -3 | -1.617 | -2 |

5077a | 3 | -5.228 | -6.598 | -5 | -2.507 | -3 |

11197a | 3 | -4.428 | -6.289 | -5 | -2.360 | -3 |

raw mean

medium-rare mean

well-done mean

Sum of local data = the "bias" + Oscillatory term + Error term

D_E(X) := \frac{1}{\log(X)}\sum_{p\leq X} \frac{a_{E}(p)\log p}{p}

the Explicit Formula gives $D_E(X)$ as a sum of three contributions:
-r_E + S_E(X) + O(1/\log(X))

where the "Oscillatory term" $S_E(X)$ is the wild card (even assuming GRH)
and we take it to be the limit ($Y\to\infty$) of these generalized
trigonometric sums:
S_E(X,Y) = \frac{1}{\log(X)} \sum_{|\gamma|\leq Y} \frac{X^{i\gamma}}{i\gamma}

(sum over imaginary parts of complex zeros of $L(f_E,s)$ above $s=1/2$)

S_E(X,Y) = \frac{1}{\log(X)} \sum_{|\gamma|\leq Y} \frac{X^{i\gamma}}{i\gamma}
\quad\text{(sum over zeros)}

It has been tentatively conjectured that
*explicit*.

\lim_{X,Y\to\infty} S_E(X,Y) = 0,

but for computations it would be good to know
something more
r_E(n) = \text{ the order of vanishing of }L(\text{symm}^n f_E, s)
\text{ at }s=1/2,
\qquad S_E(X,Y) = \frac{1}{\log(X)} \sum_{|\gamma|\leq Y} \frac{X^{i\gamma}}{i\gamma}

- the distribution of the $r_{E}(n)$'s
- the convergence of $\lim_{X,Y\to\infty} S_E(X,Y) = 0$
- the conditional biases—and multivariate distributions—related to the zeroes of $L$-functions of tensor products of symmetric powers of two (or more) automorphic forms