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Introduction to Modular Symbols: Motivation, Examples, Applications

Explore the world of elliptic curves and abelian varieties through modular forms, which explain the relationship between modular symbols and elliptic curves. Learn about the Birch and Swinnerton-Dyer Conjecture and the BSD Conjecture, and how modular symbols are used to compute and enumerate elliptic curves. Discover the connection between modular symbols and modular forms, and their applications in number theory.

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Introduction to Modular Symbols: Motivation, Examples, Applications

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  1. William A. Stein Introduction to Modular Symbols Math 252 September 26, 2003

  2. Motivation Examples Applications

  3. Motivation Modular forms give order to the mysterious world of elliptic curves and abelian varieties. The modularity theorem of Wileset al. implies that modular formsof level N "explain" all of the elliptic curves of conductor N.

  4. Birch and Swinnerton-Dyer • In the 1960s, B. Birch and H.P.F. Swinnerton- Dyer computed amazing data about elliptic curves, which lead to a fundamental conjecture. • The conjecture is still very much open! For more details, see Wiles's paper at the Clay Math Institute Millenial Problems web page.

  5. The BSD Conjecture

  6. Birch first introducedmodular symbols • While gather data towards the conjecture, Birch introduced modular symbols. • Yuri Manin and Barry Mazur independently developed a systematic theory. • John Cremona later used modular symbols to enumerate the > 30000 elliptic curves of conductor up to 6000.

  7. How can we compute withobjects attached to subgroupsof the modular group?

  8. Modular Curves

  9. Modular curve for N=3: Helena Verrill

  10. Modular curve X (37): 0 Helena Verrill

  11. Modular Forms Ribet

  12. Examples of modular forms

  13. Modular Symbols N=11 A modular symbol {a,b} is the homology class (relative to cusps) of the image of a geodesic path from the cusp a to the cusp b. The three modular symbols to the right, denoted {-1,oo}, {0,1/5}, and {0,1/7}, are a basis for the space of modular symbols for Gamma_0(11). Compute some examples using MAGMA.

  14. Computing the space of modular symbols Assume for simplicity that N=p is prime.

  15. Explicit presentation of modular symbols

  16. Relations

  17. Example: N=11

  18. , Manins Trick

  19. Example

  20. The connection with modular forms

  21. Example

  22. Some Applications of Modular Symbols • Enumerate all elliptic curves of given conductor. • Compute basis of modular forms of given weight and level. • Proving theorems towards the BSD conjecture; e.g., that L(E,1)/Omega is a rational number.

  23. Some References • Manin:Parabolic points and zeta-functions of modular curves, 1972. • Mazur: Courbes elliptiques et symboles modulaires, 1972. • Cremona: Algorithms for modular elliptic curves, 1997. • My modular symbols package in MAGMA.

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