Elliptic Curves

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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 full ring of integers \(\mathcal{O}_{K}\) of an imaginary quadratic field \(K = \mathbb{Q}(\sqrt{-d})\). When the field is \(Q(\sqrt{-3})\), the ring of integers is the Eisenstein integers, resulting in curves with special 3-fold symmetric properties.
  • Relation to Eisenstein Series: The modular invariants \(j\) (and \(g_2, g_3\) in Weierstrass equations) of these curves are directly expressed using Eisenstein series (\(E_4, E_6\)) and theta functions, which are specialized modular forms.
  • Arithmetic Structure: Eisenstein curves, particularly those over imaginary quadratic fields, allow for explicit constructions of elliptic units, which are valuable in understanding class groups.
  • p-adic Properties: They are fundamental in the construction of p-adic analogues of Eisenstein-Kronecker series, which are utilized to prove p-adic Kronecker limit formulas.
  • Applications in Number Theory: These curves often act as "strong Weil curves" when studying the Fourier expansion of modular forms, aiding in the computation of the number of points on curves over finite fields, as seen with theta functions. [1, 2, 3, 4, 5, 6, 7]
Examples and Forms
  • Curves with 3-fold Symmetry: The curve defined by \(y^2 = x^3 - d\) for a non-zero \(d\) has CM by \(\mathbb{Z}[\omega]\).
  • Modular Relation: The theta function \(\theta(z, 4)\) can be expressed via Eisenstein series \(E_{2,4}(z)\), connecting them to the representation of integers as a sum of four squares. [1, 2]
Note: Eisenstein Elliptic Curves should not be confused with [Eisenstein's Irreducibility Criterion], which is a completely different topic. [1]

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