NOTES ON STATISTICS, PROBABILITY and MATHEMATICS

Birch Swinnerton Dyer Conjecture:

1. The L-function and its Euler Product:

\[L_p(s, E) = 1 - \frac{a_p}{p^{s}} + p^{1-2s}\]

where

\[a_p = p + 1 - |E(\mathbb{F}_p)|\]

We can express \(|E(\mathbb{F}_p)|\) as:

\[|E(\mathbb{F}_p)| = p + 1 - a_p\]

Now, consider the fraction \(\frac{|E(\mathbb{F}_p)|}{p}\):

\[\frac{|E(\mathbb{F}_p)|}{p} = \frac{p + 1 - a_p}{p} = 1 - \frac{a_p}{p}+ \frac{1}{p} \]

2. The Connection to Cardinalities:

3. The Heuristic Approach:

\[\prod_{p \le x} \frac{|E(\mathbb{F}_p)|}{p} \approx C \, (\log x)^r\]

\[\log\left(\prod_{p \leq X}\; \frac {N_p}p \right)=\log C + r\; \log\left (\log X\right)\]

4. Approximating \(L_p(1, E)\):

Why are we are interested in the behavior of \(L(s, E)\) at \(s=1\). Let’s evaluate \(L_p(s, E)\) at \(s=1\):

\[L_p(1, E) = 1 - \frac{a_p}{p^{s}} + p^{1-2s} = 1 - \frac{a_p}{p} + \frac{1}{p}\]

Notice that:

\[L_p(1, E) = \frac{|E(\mathbb{F}_p)|}{p}\]

Therefore:

\[L(1, E) = \prod_{p} \frac{1}{\frac{|E(\mathbb{F}_p)|}{p}} = \prod_{p} \frac{p}{|E(\mathbb{F}_p)|}\]

Then, the inverse of that would be:

\[1/L(1,E) = \prod_{p} \frac{|E(\mathbb{F}_p)|}{p}\]

5. The Partial Product:

When we consider the product \(\prod_{p \le x} \frac{|E(\mathbb{F}_p)|}{p}\), we are essentially taking a partial product of the Euler product representation of \(1/L(1,E)\).

Summary:

6. Why \(L_p(s, E) = 1 - a_p p^{-s} + p^{1-2s}\):

The exponential in the zeta function of an elliptic curve over a finite field arises from the way we want to encode the number of points on the curve over all finite extensions of the base field. It’s a clever way to package this information into a generating function with desirable properties.

Here’s a breakdown of why the exponential is used:

1. Encoding Point Counts over Extensions:

We want to encode the number of points on the elliptic curve \(E\) over all finite extensions \(\mathbb{F}_{q^n}\) of the base field \(\mathbb{F}_q\). These point counts \(|E(\mathbb{F}_{q^n})|\) for \(n=1, 2, 3, \dots\) form a sequence of numbers.

2. Generating Function Approach:

A natural way to package this sequence is to create a generating function, which is a power series where the coefficients are the terms of the sequence. However, we don’t directly use \(|E(\mathbb{F}_{q^n})|\) as the coefficients. Instead, we use \(|E(\mathbb{F}_{q^n})|/n\).

3. The Logarithmic Derivative:

Consider the logarithmic derivative of the zeta function: \[\frac{d}{dT} \log Z(T, E/\mathbb{F}_q) = \frac{Z'(T, E/\mathbb{F}_q)}{Z(T, E/\mathbb{F}_q)}\]

We want this logarithmic derivative to be a simple power series that directly encodes the point counts.

4. The Exponential Relation:

It turns out that if we define the zeta function as:

$$Z(T, E/\mathbb{F}_q) = \exp\left(\sum_{n=1}^\infty \frac{|E(\mathbb{F}_{q^n})|}{n}T^n\right)$$

Then its logarithmic derivative is: \[\frac{Z'(T, E/\mathbb{F}_q)}{Z(T, E/\mathbb{F}_q)} = \sum_{n=1}^\infty |E(\mathbb{F}_{q^n})|T^{n-1}\] In essence, the exponential is not an arbitrary choice. It’s used to create a generating function whose logarithmic derivative directly encodes the number of points on the elliptic curve over all finite extensions of the base field, simplifying the analysis of the zeta function and revealing its key properties.

This zeta function encodes information about the number of points on \(E\) over all finite extensions of \(\mathbb{F}_p\).

It can be shown that the zeta function is a rational function of the form:

\[Z(T, E/\mathbb{F}_p) = \frac{1 - a_p T + pT^2}{(1-T)(1-pT)}\]

This is a key result from the theory of elliptic curves over finite fields.

The local factor \(L_p(s, E)\) is related to the numerator of the zeta function:

\[L_p(s, E) = 1 - a_p p^{-s} + p^{1-2s}\]

To see this, make the substitution \(T = p^{-s}\) in the numerator of the zeta function:

\[1 - a_p T + pT^2 = 1 - a_p p^{-s} + p (p^{-s})^2 = 1 - a_p p^{-s} + p^{1-2s}\]

6. The BSD conjecture:

The Birch and Swinnerton-Dyer (BSD) conjecture, when focused on the first term of the L-function’s Taylor series expansion around \(s=1\) (i.e., \(L(1)\)), can be stated as follows:

Simplified BSD Conjecture (Focusing on L(1)):

“Let \(E\) be an elliptic curve defined over the rational numbers \(\mathbb{Q}\), and let \(L(s, E)\) be its L-function. Then:

More Precise Statement:

“Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\), and let \(r\) be the rank of the Mordell-Weil group \(E(\mathbb{Q})\). Let \(L(s, E)\) be the L-function of \(E\). Then, the order of vanishing of \(L(s, E)\) at \(s=1\) is equal to \(r\).

In particular:

Key Points:

In simpler terms:

In sagemath the curves plotted above are 800.e1:

# Define the elliptic curve over the rational numbers
E = EllipticCurve([-5, 0])

# Print the elliptic curve
print("Elliptic curve defined by y^2 = x^3 - 5x over Q:")
print(E)

# Find the rank of the elliptic curve
rank = E.rank()
print("The rank of the elliptic curve y^2 = x^3 - 5x over Q is:", rank)

# Find the points of finite order (torsion points)
torsion_points = E.torsion_points()
print("Torsion points on the elliptic curve:")
for point in torsion_points:
    print(point)

# Find the generators of the Mordell-Weil group
generators = E.gens()
print("Generators of the Mordell-Weil group:")
for gen in generators:
    print(gen)

E = EllipticCurve([-5, 0])
L = E.lseries()

L_at_1 = L(1)
print(L_at_1)

Lfunc = L.taylor_series(1,4)
print(Lfunc)

---

Elliptic curve defined by y^2 = x^3 - 5x over Q:
Elliptic Curve defined by y^2 = x^3 - 5*x over Rational Field
The rank of the elliptic curve y^2 = x^3 - 5x over Q is: 1
Torsion points on the elliptic curve:
(0 : 1 : 0)
(0 : 0 : 1)
Generators of the Mordell-Weil group:
(-1 : 2 : 1)
0.000000000000000
0.00 + 2.2*z - 2.0*z^2 + 0.56*z^3 + 1.0*z^4 - 1.6*z^5 + O(z^6)

and 5077.a1:

# Define the elliptic curve over the rational numbers
E = EllipticCurve([0, 0, 1, -7, 6])

print(E)

# Find the rank of the elliptic curve
rank = E.rank()
print("The rank of the elliptic curve y^2 = x^3 - 5x over Q is:", rank)

# Find the points of finite order (torsion points)
torsion_points = E.torsion_points()
print("Torsion points on the elliptic curve:")
for point in torsion_points:
    print(point)

# Find the generators of the Mordell-Weil group
generators = E.gens()
print("Generators of the Mordell-Weil group:")
for gen in generators:
    print(gen)

E = EllipticCurve([-5, 0])
L = E.lseries()

L_at_1 = L(1)
print(L_at_1)

Lfunc = L.taylor_series(1,4)
print(Lfunc)

---

Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
The rank of the elliptic curve y^2 = x^3 - 5x over Q is: 3
Torsion points on the elliptic curve:
(0 : 1 : 0)
Generators of the Mordell-Weil group:
(-2 : 3 : 1)
(-7/4 : 25/8 : 1)
(1 : -1 : 1)
0.000000000000000
0.00 + 2.2*z - 2.0*z^2 + 0.56*z^3 + 1.0*z^4 - 1.6*z^5 + O(z^6)

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NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.