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Modular Forms, Lattices, and Hecke Operators

Ben Merbaum

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Opening question: Ramanujan’s conjecture

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Section 1�What is a modular form?

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Lattices

 

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Functions on lattices

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Towards a definition of modular forms

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The modular group

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The modular group

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The modular group

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Modular forms

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Section 2�The space of modular forms

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An example: the Eisenstein series

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Modular discriminant

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Spaces of modular forms

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Section 3�Obtaining an eigenbasis

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Hecke operator: action on lattices

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Hecke operator: action on modular forms

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Simultaneous eigenforms

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An eigenbasis

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Section 4�Conclusion

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Returning to Ramanujan’s conjecture

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Applications in number theory

  • Sphere packing problem
  • Solutions of quintic equations
  • Fermat’s Last Theorem
  • Connections to elliptic curves

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References

[1] R.E. Greene and S.G. Krantz. Function Theory of One Complex Variable. Graduate studies in mathematics. American Mathematical Society, 2006.

[2] J.P. Serre. A Course in Arithmetic. Graduate texts in mathematics. Springer, 1973.

[3] J. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves. Graduate texts in mathematics. Springer, 1994.

[4] J Silverman. Arithmetic of Elliptic Curves. Graduate texts in mathematics. Springer, 2009.