- Title Pages
- Let τ:ℕ→ℤ be defined by
- Contents
- Preface
- Acknowledgments
- Author information
- Dependencies between the chapters
- Chapter One Introduction, main results, context
- Chapter Two Modular curves, modular forms, lattices, Galois representations
- Chapter Three First description of the algorithms
- Chapter Four Short introduction to heights and Arakelov theory
- Chapter Five Computing complex zeros of polynomials and power series
- Chapter Six Computations with modular forms and Galois representations
- Chapter Seven Polynomials for projective representations of level one forms
- Chapter Eight Description of <i>X</i><sub>1</sub>(5<i>l</i>)
- Chapter Nine Applying Arakelov theory
- Chapter Ten An upper bound for Green functions on Riemann surfaces
- Chapter Eleven Bounds for Arakelov invariants of modular curves
- Chapter Twelve Approximating <i>V<sub>f</sub></i> over the complex numbers
- Chapter Thirteen Computing <i>V<sub>f</sub></i> modulo <i>p</i>
- Chapter Fourteen Computing the residual Galois representations
- Chapter Fifteen Computing coefficients of modular forms
- Epilogue
- Bibliography
- Index

# Polynomials for projective representations of level one forms

# Polynomials for projective representations of level one forms

- Chapter:
- (p.159) Chapter Seven Polynomials for projective representations of level one forms
- Source:
- Computational Aspects of Modular Forms and Galois Representations
- Author(s):
### Johan Bosman

- Publisher:
- Princeton University Press

This chapter explicitly computes mod-ℓ Galois representations attached to modular forms. To be precise, it looks at cases with *l* ≤ 23, and the modular forms considered will be cusp forms of level 1 and weight up to 22. The chapter presents the result in terms of polynomials associated with the projectivized representations. As an application, it will improve a known result on Lehmer's nonvanishing conjecture for Ramanujan's tau function.

*Keywords:*
modular forms, Galois representations, cusp forms, polynomials, Ramanujan's tau function, Lehmer, nonvanishing conjecture

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- Title Pages
- Let τ:ℕ→ℤ be defined by
- Contents
- Preface
- Acknowledgments
- Author information
- Dependencies between the chapters
- Chapter One Introduction, main results, context
- Chapter Two Modular curves, modular forms, lattices, Galois representations
- Chapter Three First description of the algorithms
- Chapter Four Short introduction to heights and Arakelov theory
- Chapter Five Computing complex zeros of polynomials and power series
- Chapter Six Computations with modular forms and Galois representations
- Chapter Seven Polynomials for projective representations of level one forms
- Chapter Eight Description of <i>X</i><sub>1</sub>(5<i>l</i>)
- Chapter Nine Applying Arakelov theory
- Chapter Ten An upper bound for Green functions on Riemann surfaces
- Chapter Eleven Bounds for Arakelov invariants of modular curves
- Chapter Twelve Approximating <i>V<sub>f</sub></i> over the complex numbers
- Chapter Thirteen Computing <i>V<sub>f</sub></i> modulo <i>p</i>
- Chapter Fourteen Computing the residual Galois representations
- Chapter Fifteen Computing coefficients of modular forms
- Epilogue
- Bibliography
- Index