Élisabeth Lutz

Who was Élisabeth Lutz?

Élisabeth Lutz was a French mathematician. She was a student of André Weil at the Université de Strasbourg, from 1934 to 1938. In 1935 she began working on aspects of elliptic curves over p-adic fields.

An elliptic curve over Q can be put in the form y² = x³ − Ax − B with A and B integers. Recall that the abelian group of rational points on an elliptic curve over Q is finitely generated. In her published paper on the subject, Lutz makes two observations as a consequence of her analysis:

first, that any Q-rational point of finite order on such a curve has integer coordinates, and,

second, that either y equals 0 or y² divides 4A³ - 27B².

This result is now called the Nagell–Lutz theorem. It implies that the torsion subgroup of Q-rational points is effectively computable. It remains unknown whether the whole group of Q-rational points is effectively computable.

Weil describes Lutz's work, and its relationship to his own research, in his Collected Papers, vol. I, pp. 538–539. Perhaps as evidence of Weil's high standards, Lutz was granted only the lower-level French thesis for this work. She wrote a doctoral thesis after World War II on a different p-adic topic with a different advisor.