Stark–Heegner theorem

Key takeaways

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

Reference excerpt

In number theory, the Heegner theorem or Stark-Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number. Let Q denote the set of rational numbers, and let d be a square-free integer. The field Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} is a quadratic extension of Q. The class number of Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} is one if and only if the ring of integers of Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} is a principal ideal domain. The Baker–Heegner–Stark theorem can then be stated as follows:

If d < 0, then the class number of Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} is one if and only if d ∈ { − 1 , − 2 , − 3 , − 7 , − 11 , − 19 , − 43 , − 67 , − 163 } . {\displaystyle d\in \{\,-1,-2,-3,-7,-11,-19,-43,-67,-163\,\}.}

These are known as the Heegner numbers. By replacing d with the discriminant D of Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} this list is often written as:

D ∈ { − 3 , − 4 , − 7 , − 8 , − 11 , − 19 , − 43 , − 67 , − 163 } . {\displaystyle D\in \{-3,-4,-7,-8,-11,-19,-43,-67,-163\}.}

History This result was first conjectured by Gauss in Section 303 of his Disquisitiones Arithmeticae (1798). It was essentially proven by Kurt Heegner in 1952, but Heegner's proof was not accepted until an academic mathematician Harold Stark published a proof in 1967 which had many commonalities to Heegner's work, though Stark considers the proofs to be different. Heegner "died before anyone really understood what he had done". In (Stark 1969a) Stark works through Heegner's proof to highlight what the gap in Heegner’s proof consisted of; other contemporary papers produced various similar proofs using modular functions. (Heegner's paper dealt mainly with the congruent number problem, also using modular functions.) Alan Baker's slightly earlier 1966 proof used completely different principles which reduced the result to a finite amount of computation, with Stark's 1963/4 thesis already providing this computation; Baker won the Fields Medal for his methods. Stark later pointed out that Baker's proof, involving linear forms in 3 logarithms, could be reduced to a statement about only 2 logarithms which was already known from 1949 by Gelfond and Linnik. Stark's 1969 paper (Stark 1969a) also cited the 1895 text by Weber and noted that if Weber had "only made the observation that the reducibility of [a certain equation] would lead to a Diophantine equation, the class-number one problem would have been solved 60 years ago". Bryan Birch notes that Weber's book, and essentially the whole field of modular functions, dropped out of interest for half a century: "Unhappily, in 1952 there was no one left who was sufficiently expert in Weber's Algebra to appreciate Heegner's achievement." Stark's 1969 paper can be seen as a good argument for calling the result Heegner's Theorem. In the immediate years after Stark, Deuring, Siegel, and Chowla all gave slightly variant proofs by modular functions. Other versions in this genre have also cropped up over the years. For instance, in 1985, Monsur Kenku gave a proof using the Klein quartic (though again utilizing modular functions). And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline). The work of Gross and Zagier (1986) (Gross & Zagier 1986) combined with that of Goldfeld (1976) also gives an alternative proof.

Real case On the other hand, it is unknown whether there are infinitely many d > 0 for which Q ( d ) {\displaystyle \mathbf {Q} ({\sqrt {d}})} has class number 1. Computational results indicate that there are many such fields. Number Fields with class number one provides a list of some of these.

Notes

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Worked examples

Example 1 — a first encounter with Stark–Heegner theorem

Start with the simplest possible case. Write down what Stark–Heegner theorem 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 Stark–Heegner theorem 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 Stark–Heegner theorem 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 Stark–Heegner theorem

In research: Stark–Heegner theorem 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 Stark–Heegner theorem 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: Stark–Heegner theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in algebraic number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Stark–Heegner theorem 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.

How to study Stark–Heegner theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Stark–Heegner theorem 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 Stark–Heegner theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Stark–Heegner theorem in simple terms?

In number theory, the Heegner theorem or Stark-Heegner theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are principal ideal domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields…

Why does Stark–Heegner theorem 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 Stark–Heegner theorem?

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 Stark–Heegner theorem.