Birch-Swinnerton-Dyer Conjecture

In mathematics there are equations for which one only seeks solutions in whole numbers or fractions. A particularly important family of such equations has the form y² = x³ + ax + b, where a and b are fixed numbers. Such curves are called elliptic curves. For some of them one finds infinitely many solutions in fractions, for others only very few or none at all. The Birch-Swinnerton-Dyer conjecture states how this can be recognized in advance, without searching for all solutions. It was formulated in the 1960s by the British mathematicians Bryan Birch and Peter Swinnerton-Dyer and remains unproven to this day.

A Millennium Problem with a Million-Dollar Prize

In the year 2000, the Clay Mathematics Institute in the USA declared seven particularly difficult open problems to be the so-called Millennium Problems. A prize of one million US dollars is offered for each solution. The Birch-Swinnerton-Dyer conjecture is one of them. So far, only a single one of these seven problems has been solved.

The reason for this status lies not in the prize money but in the role the conjecture plays. It connects two areas of mathematics that at first glance have nothing to do with one another. One area counts solutions of equations. The other studies functions built up from infinitely long sums. Such bridges are considered especially valuable in mathematics, because tools from one side can be transferred to the other.

Elliptic curves are, moreover, no mere plaything. They are embedded in the encryption that protects messengers, bank cards, and websites. And they were the central tool in Andrew Wiles’s 1994 proof of Fermat’s Last Theorem. Whoever understands elliptic curves better also understands these applications better.

What the L-Function Reveals About the Solutions

For every elliptic curve one can form a so-called L-function. This is a computational rule assembled from many individual pieces of information about the curve. For each prime number, one counts how many solutions the curve has modulo that prime. Modulo here means: one calculates only with remainders, as on a clock face, where after 12 comes 1 again. All these counts are combined into a single function.

The conjecture now looks at exactly one point of this function, namely the point s = 1. If the value of the function there is not zero, then the curve has only finitely many solutions in fractions. If it is zero, there are infinitely many. More precisely: the degree to which the function becomes zero at this point is supposed to indicate the number of independent basic solutions. This number is called the rank of the curve.

One can picture the L-function as a kind of fever chart arising from thousands of small measurements. The conjecture claims that a single reading from it reveals the structure of the entire curve. This has only been proven in special cases, for instance for curves of rank zero or one. For higher ranks, no approach exists to this day. Computers have confirmed the conjecture in millions of individual cases, but that is not a proof.

From Cryptography to AI Proof Assistants

One does not encounter the conjecture directly in everyday life. Its surroundings, however, one does: elliptic curves secure a large portion of internet traffic via ECDSA and similar methods. In the news, the conjecture usually surfaces when reports cover the Millennium Problems or when a well-known mathematician presents an alleged proof. Such announcements have so far never withstood scrutiny.

Increasingly, the term also appears in reports about artificial intelligence. Companies such as Google DeepMind and OpenAI like to test their systems on difficult mathematical problems. Proof assistants like Lean formally verify each step for correctness in the process. Major open conjectures serve as a distant benchmark in this community, one against which progress is measured.

A common misconception is that the conjecture is merely a technical footnote for specialists. In fact, an entire program hinges on it: the question of how the number of solutions to equations can generally be read off from analytic functions. A proof would give this program a secure foundation. Until then, the conjecture remains exactly that: a very well-founded but unproven claim.