Beppo Levi - Mathematical Contributions

Mathematical Contributions

His early work studied singularities on algebraic curves and surfaces. In particular, he supplied a proof (questioned by some) that a procedure for resolution of singularities on algebraic surfaces terminates in finitely many steps. Later he proved some foundational results concerning Lebesgue integration, including a statement that even today appears in many measure theory textbooks as "Beppo Levi's lemma".

He also studied the arithmetic of elliptic curves. He classified them up to isomorphism, not only over C, but also over Q. Next he studied what in modern terminology would be the subgroup of rational torsion points on an elliptic curve over Q: he proved that certain groups were realizable and that others were not. He essentially formulated a conjecture as to what the complete list of possibilities should be, a conjecture that was to be made independently by Andrew Ogg about 60 years later, and finally proved by Barry Mazur.