Today the Weil and Tate pairings are used a lot in cryptography. I'm curious, what was the original motivation of Weil and Tate for defining them? (Especially curious about Weil.) I've understood Weil introduced them in his 1940 paper where he proved the Riemann hypothesis for curves over finite fields. I've seen a proof of the the hypothesis for elliptic curves using the Weil pairings, but doesn't seem they help for general curves. If someone knows where to find an English translation of that paper: Weil, André Sur les fonctions algébriques à corps de constantes fini. (French) C. R. Acad. Sci. Paris 210, (1940). 592–594.

That could also help.

Today the Weil and Tate pairings are used a lot in cryptography. I'm curious, what was the original motivation of Weil and Tate for defining them? (Especially curious about Weil.) I've understood Weil introduced them in his 1940 paper where he proved the Riemann hypothesis for curves over finite fields. I've seen a proof of the hypothesis for elliptic curves using the Weil pairings, but doesn't seem they help for general curves. If someone knows where to find an English translation of that paper: Weil, André Sur les fonctions algébriques à corps de constantes fini. (French) C. R. Acad. Sci. Paris 210, (1940). 592–594.

That could also help.