Elliptic Curves and Modular Forms

Mordell–Weil, modularity, and BSD conjecture.

foundation tier

Elliptic Curves and Modular Forms. Mordell–Weil, modularity, and BSD conjecture.

Foundations and canonical references

The standard treatments of elliptic curves and modular forms approach the subject from complementary angles. Silverman, The Arithmetic of Elliptic Curves (2009) is the anchor reference for the subject and lays out the core definitions, theorems, and worked examples that practitioners return to. Silverman, Rational Points on Elliptic Curves (1992) offers an alternative presentation that complements the primary references and is useful for triangulating definitions and proof techniques.

Open methodological questions for elliptic curves and modular forms include sharpening the bridges between foundational theory and computational practice, extending classical results to broader or more structured settings, and integrating the techniques surveyed above with adjacent mathematical disciplines. The references listed in this page are the entry points that current work builds on.

Prerequisites

Sources

  • textbook · primary · 2009
    The Arithmetic of Elliptic Curves
    silverman-2009
  • textbook · supporting · 1992
    Rational Points on Elliptic Curves
    silverman-1992, tate-1992

In context

Where this topic sits in the prerequisite graph. Click any node to jump.

Open in full atlas →

Explore

  1. 01

    Modular Forms

    Cusp forms, Hecke operators, and the Eichler–Shimura correspondence.
  2. 02

    Birch and Swinnerton-Dyer Conjecture

    Rank conjectures, Selmer groups, and Heegner points.
  3. 03

    Abelian Varieties

    Higher-dimensional analogs of elliptic curves and their endomorphism rings.

Review this topic

This page was drafted by an agent and is waiting on expert review. Spotted a wrong prerequisite, a missing concept, a misattributed source, or a factual slip? Tell us — your review opens a tracked issue maintainers act on.