Elliptic Curves
Elliptic Curves Elliptic Curves Elliptic Curves and Abelian Varieties Elliptic Curves, Eisenstein Series, and Bernoulli Numbers Modular Curves and the Eisenstein Ideal a la Barry Mazur Eisenstein Elliptic Curves AI on Eisenstein Elliptic Curves Eisenstein elliptic curves are elliptic curves associated with the lattice of Eisenstein integers \(\mathbb{Z}[\omega]\) (where \(\omega = e^{2\pi i/3}\)) or, more broadly, curves with complex multiplication (CM) by the ring of integers of an imaginary quadratic field, often used in constructing modular forms and p-adic L-functions. They often appear as curves with 3-fold or 6-fold symmetry, playing a key role in the study of Ramanujan-type congruence properties and generating functions, such as theta functions, defined through Eisenstein series. [ 1 , 2 , 3 , 4 , 5 ] Key Aspects of Eisenstein Elliptic Curves Complex Multiplication (CM): These curves have an endomorphism ring strictly larger than \(\mathbb{Z}\), specifically the ful...