The Lumenary · Research pathways

Arithmetic geometry and elliptic curves

Name: ______________________

Date: ______________________

You will learn to

Define elliptic curves, construct the chord-and-tangent group law, and compute torsion points with the Nagell–Lutz theorem
State the Mordell–Weil theorem, prove its descent step, and prove the Hasse bound from the theory of the Frobenius endomorphism
Count points of elliptic curves over small finite fields and use the Weil zeta function to count over extensions
Explain the statements of the Weil conjectures, the modularity theorem, the route from modularity to Fermat's Last Theorem, and the Birch and Swinnerton-Dyer conjecture, separating theorems from conjectures

1.
On $E: y^2 = x^3 + 1$ the point $P = (2, 3)$ has order $6$ , and $2P = (0, 1)$ . What is the order of the point $(0, 1)$ ?
2.
Count the points of $E: y^2 = x^3 + 2$ over $\mathbb{F}_7$ , including the point at infinity. (The non-zero squares mod $7$ are $1, 2, 4$ .)
3.
For $E: y^2 = x^3 + x + 1$ over $\mathbb{F}_5$ we found $\#E(\mathbb{F}_5) = 9$ , so $a = 5 + 1 - 9 = -3$ . Use $\#E(\mathbb{F}_{q^n}) = q^n + 1 - \alpha^n - \beta^n$ , where $\alpha, \beta$ are the roots of $T^2 - aT + q$ , to find $\#E(\mathbb{F}_{25})$ .
4.
Let $E: y^2 = x^3 + 1$ , so $a = 0$ , $b = 1$ and $4a^3 + 27b^2 = 27$ . By the Nagell–Lutz theorem, a torsion point $(x, y) \neq O$ has integer coordinates with $y = 0$ or $y^2 \mid 27$ . List every integer $y$ that is allowed.
5.
In the proof of Hasse's theorem we showed $m^2 - a\,mn + q\,n^2 = \deg(m - n\phi) \ge 0$ for all integers $m, n$ , and concluded $a^2 \le 4q$ . Which justification of the last step is correct?
- A.A quadratic form that is non-negative on $\Z^2$ is non-negative on $\R^2$ , so its discriminant $a^2 - 4q$ is at most $0$ .
- B.Because $\deg$ is multiplicative: $\deg(\psi\chi) = \deg\psi \cdot \deg\chi$ .
- C.Because $1 - \phi$ is separable.
- D.Because $a$ is an integer, so $a^2 - 4q$ is an integer.
6.
Let $E: y^2 = x^3 - 2$ . For which fields $K$ is $E(K)$ not finitely generated? (Choose all that apply.)
- A. $K = \Q$
- B. $K = \Q(i)$
- C. $K = \R$
- D. $K = \mathbb{F}_{7}$
- E. $K = \overline{\Q}$ , the algebraic numbers
7.
The lesson's Lean theorem chord_third_point has hypotheses h₁ : y₁ ^ 2 = x₁ ^ 3 + a * x₁ + b, h₂ : y₂ ^ 2 = x₂ ^ 3 + a * x₂ + b, hx : x₁ ≠ x₂ and hℓ : y₂ - y₁ = ℓ * (x₂ - x₁). What would go wrong if we deleted hx?
- A.The statement would become false: with $x_1 = x_2$ and $y_1 = y_2$ , hℓ holds for every $\ell$ , and for most $\ell$ the point $(\ell^2 - 2x_1, \dots)$ is not on the curve.
- B.Nothing: the hypothesis is never used in the proof.
- C.It would still be true, but Lean could not prove it without hx.
- D.Lean would reject the statement as ill-typed.
8.
Write count_points(p, a, b) returning $\#E(\mathbb{F}_p)$ for $y^2 = x^3 + ax + b$ over $\mathbb{F}_p$ ( $p$ an odd prime), including the point at infinity. Then use it to check the Hasse bound for every curve over $\mathbb{F}_p$ , $p \lt 50$ .
Computational — Practice
9.
Which of these statements about elliptic curves $E$ over $\Q$ are proved theorems (as of this lesson)? Choose all that apply.
- A.If $L(E, s)$ has a simple zero at $s = 1$ , then $E(\Q)$ has rank $1$ .
- B.The rank of $E(\Q)$ equals the order of vanishing of $L(E, s)$ at $s = 1$ , for every $E$ .
- C.Every $E/\Q$ is modular.
- D.The Tate–Shafarevich group of every $E/\Q$ is finite.
- E. $E(\Q)$ is finitely generated.
- F.The ranks of elliptic curves over $\Q$ are unbounded.

Answer key — Arithmetic geometry and elliptic curves

1.
$(0, 1) = 2P$ and $P$ has order $6$ , so the order of $2P$ is $6/\gcd(2, 6) = 3$ . Check with the table: $2 \cdot (0,1) = 4P = (0, -1)$ and $3 \cdot (0, 1) = 6P = O$ .
2.
$x^3 \bmod 7$ for $x = 0, \dots, 6$ is $0, 1, 1, 6, 1, 6, 6$ , so $x^3 + 2 \equiv 2, 3, 3, 1, 3, 1, 1$ . The values $2$ and $1$ are non-zero squares ( $3^2 \equiv 2$ , $1^2 = 1$ ) and $3$ is not. So $x = 0, 3, 5, 6$ give two points each: $8$ affine points, and $\#E(\mathbb{F}_7) = 9$ . Hasse: $|9 - 8| = 1 \le 2\sqrt 7 \approx 5.3$ .
3.
$\alpha^2 + \beta^2 = a^2 - 2q = 9 - 10 = -1$ , so $\#E(\mathbb{F}_{25}) = 25 + 1 - (-1) = 27$ . (A direct count in $\mathbb{F}_{25} = \mathbb{F}_5[t]/(t^2 - 2)$ confirms this.) Note that $9 \mid 27$ , as it must: $E(\mathbb{F}_5)$ is a subgroup of $E(\mathbb{F}_{25})$ .
4.
$y^2 \in \{1, 9\}$ or $y = 0$ , so $y \in \{-3, -1, 0, 1, 3\}$ . Each value does occur: $(2, \pm 3)$ , $(0, \pm 1)$ and $(-1, 0)$ are exactly the torsion points $P, \dots, 5P$ of the lesson's table. Nagell–Lutz gives a necessary condition, which makes torsion computable: test the finitely many candidates.
5.
If $m^2 - amn + qn^2 \ge 0$ for all integers, then by scaling for all rationals and by continuity for all reals; so $t^2 - at + q \ge 0$ for all real $t$ , which forces the discriminant $a^2 - 4q \le 0$ , i.e. $|a| \le 2\sqrt q$ .
6.
$E(\R)$ (uncountable) and $E(\overline{\Q})$ (divisible) are not finitely generated. $E(\Q)$ and $E(\Q(i))$ are, by Mordell–Weil, and $E(\mathbb{F}_7)$ is finite. The hypothesis that $K$ is a number field is essential: the proof uses heights with finitely many points of bounded height (false over $\R$ and $\overline{\Q}$ ) and the finiteness of class groups and unit groups modulo squares.
7.
With $x_1 = x_2$ and $y_1 = y_2$ , hℓ reads $0 = \ell \cdot 0$ , true for every $\ell$ . Take $y^2 = x^3 + 1$ , $P_1 = P_2 = (0, 1)$ , $\ell = 1$ : then $x_3 = 1$ , $y_3 = 1 \cdot (1 - 0) + 1 = 2$ , and $2^2 = 4 \neq 1^3 + 1 = 2$ . So the statement without hx is false; the hypothesis is exactly what makes the chord, and hence its slope, well defined. The tangent case needs the extra information $2y_1\ell = 3x_1^2 + a$ .
8.
```
def count_points(p, a, b):
    roots = [0] * p
    for y in range(p):
        roots[y * y % p] += 1
    return 1 + sum(roots[(x ** 3 + a * x + b) % p] for x in range(p))
```
Tabulating square roots makes the count $O(p)$ instead of $O(p^2)$ . The Hasse checks pass for every smooth curve over these fields — finite evidence for a theorem that holds for all $q$ (Theorem 6). Schoof's algorithm counts points in time polynomial in $\log p$ , which matters for cryptographic primes of $256$ bits.
9.
Theorems: (a) Gross–Zagier–Kolyvagin plus modularity; (c) the modularity theorem; (e) Mordell–Weil. Open: (b) BSD, (d) finiteness of Ш, (f) unboundedness of ranks (heuristics predict boundedness).