MATH 845 • SPRING 2027

Introduction to Elliptic Curves

Elliptic curves have played an important role in complex analysis since the 18th century and have been a foundational topic in number theory since the 20th century. They are complicated enough to illustrate many deep mathematical phenomena, yet simple enough to allow for explicit computations.

In this course, we will develop the theory of elliptic curves from elementary foundations. One of our main arithmetic goals will be to prove the Mordell–Weil theorem, which states that the rational points on an elliptic curve form a finitely generated abelian group. There will also be room to tailor the specific topics we cover to the interests of the participating students.

The course will be challenging, but it will be well worth the effort for anyone interested in pursuing further studies in number theory, algebraic geometry, or certain areas of complex analysis.

Curves with a group law

The real points of y² = x³ − 43x + 166, a single curve with a bulge on the left. Three tangent lines and three dashed vertical arrows form a closed cycle: the tangent at T = (3, 8) meets the curve again at (−5, 16), which reflects down to 2T = (−5, −16); the tangent at 2T meets the curve at (11, −32), which reflects up to 4T = (11, 32); the tangent at 4T meets the curve at (3, −8), which reflects back up to T. — The tangent at a point meets the curve once more; reflecting that point in the $x$ -axis gives the point’s double. Starting at $T$ : $T \to 2 T \to 4 T \to 8 T = T$ .

The points on an elliptic curve can be added, and they form a group.

On the curve $E : y^{2} = x^{3} - 43 x + 166$ , double the point $T = (3, 8)$ three times and you are back where you started: $8 T = T$ , so $T$ has order 7.

Those are all the rational points: $(3, \pm 8)$ , $(- 5, \pm 16)$ , $(11, \pm 32)$ , and the point at infinity. By Mazur’s theorem, no rational point on an elliptic curve over $ℚ$ has finite order larger than 12.

One curve, many fields

The same equation makes sense over any field, and each choice of field shows a different side of the curve.

Over $ℂ$ : the curve is a torus

ℂ / Λ

with

Λ = ω_{1} ℤ + ω_{2} ℤ

. Adding points is adding complex numbers. We see

8 T = T + ω_{1}

, so

T

has order 7.

A histogram of a_p/(2√p) over the interval from −1 to 1 that closely follows a red semicircle. — **Over the finite fields $𝔽_{p}$ :** $| E (𝔽_{p}) | = p + 1 - a_{p}$ . The histogram shows the distribution of $a_{p} / (2 \sqrt{p})$ for primes below a million. The Sato–Tate theorem predicts the shape. The $a_{p}$ are the coefficients of the $L$ -series of the elliptic curve.

What we will cover

Core

Elliptic curves through Weierstrass models
The group law, explained through linear equivalence of divisors and the Riemann–Roch theorem
The Tate module and the Weil pairing
Isogenies, endomorphism rings, and automorphism groups
The formal group of an elliptic curve
Elliptic curves over finite fields, over $ℂ$ , and over local fields
Elliptic curves over global fields: the Mordell–Weil theorem

Further topics, chosen with the class

Moduli of elliptic curves
Elliptic surfaces: reduction types and Kodaira symbols
Computing Mordell–Weil groups
$L$ -series of elliptic curves and the Birch and Swinnerton-Dyer conjecture
Modularity of elliptic curves over $ℚ$ (Wiles, Taylor–Wiles, Breuil–Conrad–Diamond–Taylor)

The precise content of the course can be tuned to the needs and interests of the students.

Who should take it

Graduate students interested in number theory, algebraic geometry, or arithmetic geometry. The course assumes:

undergraduate algebra (MATH 340/342, MATH 341)
some 400-level algebra (MATH 440 or 441)
complex analysis (MATH 322)

Students interested in taking the course should consult with the instructor.

Format and assessment

Lectures supplemented with examples and exercises; student participation is strongly encouraged. Assignments are biweekly, and the course ends with student presentations.

15%	Assignments (biweekly)
20%	Midterm
15%	Presentation
50%	Final examination

Texts

Joseph H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Springer GTM 106, 2009
(course text; access through SFU Library)
Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer GTM 151, 1994
(additional reading)

Course information

Course: MATH 845-4: Introduction to Elliptic Curves
Lectures: Wednesdays and Fridays, 2:30–4:20 pm
First class: Wednesday, January 6, 2027
Room: TBA
Prerequisites: MATH 340/342, MATH 341, MATH 440 or 441, MATH 322, or equivalent; consult the instructor
Instructor: Nils Bruin • nbruin@sfu.ca
SC K 10507 • (778) 782 3794