mathematics

Heegner point

Heegner point is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Heegner point rather than just read about it. In short: In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. Heegner points were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.

Key takeaways

Heegner point belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Heegner point to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Heegner point from memory before moving on to harder problems.

Reference excerpt

In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. Heegner points were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.

Gross–Zagier theorem The Gross–Zagier theorem (Gross & Zagier 1986) describes the height of Heegner points in terms of a derivative of the L-function of the elliptic curve at the point s = 1. In particular if the elliptic curve has (analytic) rank 1, then the Heegner points can be used to construct a rational point on the curve of infinite order (so the Mordell–Weil group has rank at least 1). More generally, Gross, Kohnen & Zagier (1987) showed that Heegner points could be used to construct rational points on the curve for each positive integer n, and the heights of these points were the coefficients of a modular form of weight 3/2. Shou-Wu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties (Zhang 2001, 2004, Yuan, Zhang & Zhang 2009).

Birch and Swinnerton-Dyer conjecture Kolyvagin later used Heegner points to construct Euler systems, and used this to prove much of the Birch–Swinnerton-Dyer conjecture for rank 1 elliptic curves. Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic (Brown 1994).

Computation Heegner points can be used to compute very large rational points on rank 1 elliptic curves (see (Watkins 2006) for a survey) that could not be found by naive methods. Implementations of the algorithm are available in Magma, PARI/GP, and Sage.

References Birch, B. (2004), "Heegner points: the beginnings", in Darmon, Henri; Zhang, Shou-Wu (eds.), Heegner Points and Rankin L-Series (PDF), Mathematical Sciences Research Institute Publications, vol. 49, Cambridge University Press, pp. 1–10, doi:10.1017/CBO9780511756375.002, ISBN 0-521-83659-X, MR 2083207, archived from the original (PDF) on 2023-09-01, retrieved 2023-09-01. Brown, M. L. (2004), Heegner modules and elliptic curves, Lecture Notes in Mathematics, vol. 1849, Springer-Verlag, doi:10.1007/b98488, ISBN 3-540-22290-1, MR 2082815. Darmon, Henri; Zhang, Shou-Wu, eds. (2004), Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications, vol. 49, Cambridge University Press, doi:10.1017/CBO9780511756375, ISBN 978-0-521-83659-3, MR 2083206 Gross, Benedict H.; Zagier, Don B. (1986), "Heegner points and derivatives of L-series", Inventiones Mathematicae, 84 (2): 225–320, Bibcode:1986InMat..84..225G, doi:10.1007/BF01388809, MR 0833192, S2CID 125716869. Gross, Benedict H.; Kohnen, Winfried; Zagier, Don (1987), "Heegner points and derivatives of L-series. II", Mathematische Annalen, 278 (1–4): 497–562, doi:10.1007/BF01458081, MR 0909238, S2CID 121652706. Heegner, Kurt (1952), "Diophantische Analysis und Modulfunktionen", Mathematische Zeitschrift, 56 (3): 227–253, doi:10.1007/BF01174749, MR 0053135, S2CID 120109035. Watkins, Mark (2006), Some remarks on Heegner point computations, arXiv:math.NT/0506325v2. Brown, Mark (1994), "On a conjecture of Tate for elliptic surfaces over finite fields", Proc. London Math. Soc., 69 (3): 489–514, doi:10.1112/plms/s3-69.3.489. Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei (2009), "The Gross–Kohnen–Zagier Theorem over Totally Real Fields", Compositio Mathematica, 145 (5): 1147–1162, doi:10.1112/S0010437X08003734, S2CID 17981061. Zhang, Shou-Wu (2001), "Gross-Zagier formula for GL2", Asian Journal of Mathematics, 5 (2): 183–290, doi:10.4310/AJM.2001.v5.n2.a1. Zhang, Shou-Wu (2004), "Gross–Zagier formula for GL(2) II", in Darmon, Henri; Zhang, Shou-Wu (eds.), Heegner points and Rankin L-series, Mathematical Sciences Research Institute Publications, vol. 49, Cambridge University Press, pp. 191–214, doi:10.1017/CBO9780511756375, ISBN 978-0-521-83659-3, MR 2083206.

Worked examples

Example 1 — a first encounter with Heegner point

Start with the simplest possible case. Write down what Heegner point claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Heegner point before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Heegner point ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Heegner point

In research: Heegner point appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Heegner point in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Heegner point is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic number theory, Elliptic curves, so understanding it makes those chapters shorter.
In everyday life: Look for Heegner point outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Heegner point in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Heegner point means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Heegner point out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Heegner point in simple terms?

Why does Heegner point matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Heegner point?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Heegner point.