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Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
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Conceptualizing Weil Pairing for elliptic curves ( and number fields)
There are two explanations in Silverman ( Arithmetic of Elliptic Curves), one in exercises developing the Weil reciprocity law ( for algebraic curves) and then generalizing, and then there is a differ …