Eichler–Shimura Correspondence

The relationship that associates weight-two modular forms (newforms) with algebraic geometric objects: it identifies certain newforms with factors of the Jacobian of modular curves (or with abelian varieties A_f) and relates Hecke eigenvalues to Frobenius traces in cohomology, linking analytic modular forms and algebraic geometry.

A relationship between classical modular forms (especially weight two eigenforms), two-dimensional l-adic Galois representations, and the cohomology (or Jacobians) of modular curves that connects complex-analytic objects and arithmetic representations.