News Release
A Major Milestone in Algebraic Number Theory
Langlands Conjecture Proved
For further information, contact:
Professor Jonathan Rogawski, UCLA
e-mail: jonr@math.ucla.edu
telephone: (310) 825-4048
fax: (310) 206-6673
December 9, 1999
PROVIDENCE, RI---Consider the sequence of numbers 4, 9, 16, 25, 36, 49, 64...
What is the pattern? All are perfect squares; for example, 25 = 5 x 5 =
52. Perfect squares arise in many fundamental questions in number
theory. For instance, one can find infinitely many examples of whole numbers
a, b, and c that satisfy the equation of the Pythagorean Theorem: a2
+ b2 = c2. Fermat's Last Theorem tells us that when the
exponent is a whole number bigger than 2, no solutions are possible (assuming
that one rules out trivial cases like 0n + 0n =
0n).
Another basic ingredient in number theory is modular arithmetic: An everyday
example is "clock arithmetic", whereby 5 o'clock plus 10 hours equals 3
o'clock. Mathematicians would put it this way: 5 + 10 = 3, modulo 12. In
fact, the "modulus" need not be 12, but can be any whole number. Suppose the
modulus is 23; then 20 + 7 = 4, modulo 23. Often mathematicians work with a
modulus that is a prime number (one that is not divisible by any number except
itself and 1).
Combining perfect squares with modular arithmetic, one can pose a fundamental
question in number theory: when is a number d a perfect square modulo p, where
p is prime? That is, when is there a solution x to the equation x2 = d,
modulo p? For example, if d = 2 and p = 23, then there is a solution, x = 5,
because 52 = 25 = 2, modulo 23.
Here arises a deep mystery in number theory, the "Law of Quadratic
Reciprocity", which reveals a special relationship between d and p when d is a
perfect square modulo p. The law is so surprising and definitive that
mathematicians in the 18th century tried to generalize it to cover other cases,
such as when the exponent is not 2 but 3 or even larger.
Since that time, the search for a general notion of a reciprocity law has been
a deep theme in number theory. It was not until the 1960s that mathematician
Robert Langlands formulated the "Local Langlands Correspondence", which
describes how reciprocity laws work in a very general number-theoretic context.
Mathematicians cheered when the Local Langlands Correspondence, which had
stubbornly resisted proof for more than 30 years, was finally proved in recent
work by Michael Harris and Richard Taylor, and independently by Guy Henniart.
This result represents a major milestone in algebraic number theory.
The article, "The Nonabelian
Reciprocity Law for Local Fields" by
Jonathan Rogawski, describes the proof of the Local Langlands Correspondence.
The article appears in the January 2000 issue of Notices of the AMS.
Founded in 1999 to further mathematical research and scholarship, the
30,000-member AMS fulfills its mission through programs and services that
promote mathematical research and its uses, strengthen mathematical education,
and foster awareness and appreciation of mathematics and its connections to
other disciplines and to everyday life.
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