Cycle (Reciprocity)

Periodicity in Cycles

Trignometric and Elliptic functions as examples. Fundamental Parallelogram for Elliptci functions. Abel and Jacobi’s Elliptic Integrals. Weierstrass’s Simple Elliptic function. Poincaré and hypergeometric series.

Automorphic Forms

Ramanujan’s Conjecture. Periodic functions on Topological Groups. Modular forms are specific to special linear group or projective group. These may be seen through adeles.

Cusp Forms

Work of Langlands on Discrete and continuous (Eisenstein Series) spectra.

Reciprocity

Emil Artin. Quadratic Reciprocity.

Fundamental Lemma

Ngô Bảo Châu proved the Fundamental Lemma.