#### Milestone Year

### 1976

#### Contributions to the Langlands Program in Number Theory

In 1976, a Weizmann Institute mathematician (together with a scientist
at Columbia University) proved that every "automorphic form on GL(2)
lifted to GL(3) via the symmetric square lifting". From a certain point
of view, this result is rather technical. But from another, it is one
of the first results predicted by the vast and beautiful collection
of conjectures known as the "Langlands Program".

This program, which was laid out by R.P. Langlands in 1967, introduces
the notion of an "automorphic representation of a reductive group G",
a deep notion which excited and captured the attention of generations
of mathematicians. First of all, it provided a spectacular background
for several important themes in number theory, algebraic geometry,
and representation theory - subjects that were previously studied
completely separately. Secondly - and more significantly - Langlands
laid the groundwork for a rich program of future research.
He predicted that many objects, like representations of a Galois group
as studied by Artin, and elliptic curves as studied by Weil,
were really "automorphic representations of some GL(N)";
and he conjectured that "automorphic forms of GL(2) with
a symmetric n-th power representation" -
and even "automorphic representations of any reductive group G" -
should be "automorphic representations of some GL(N)".

The result about GL(2) and GL(3) alluded to above made it possible
for Langlands to later prove (using many other results) that certain
representations of a Galois group were of "automorphic" GL(2) type.
This type of result made it possible for Wiles in 1994 to
establish Fermat's Last Theorem.
(Wiles showed that Shimura-Taniyama's Conjecture held for elliptic curves,
i.e., that Langlands's special vision for elliptic curves was true.)

From the 1990's until today, the number of mathematicians devoted to the
Langlands Program has grown exponentially, especially in Israel.
At the Weizmann Institute, we now have several mathematicians generalizing
the tools used to attack Langlands Conjectures, principally,
the algebraic geometry behind GL(N), the Arthur-Selberg trace formula,
and the objects resembling general Rankin-Selberg zeta-integrals.
Combined with the work of others in Israel, the recent Israeli contributions
put this country at the forefront of the Langlands Program of
the twenty-first century, along with the USA and France.