- Title Pages
- Let 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 X1(5l) -
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 Vf over the complex numbers -
Chapter Thirteen Computing Vf modulo p -
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 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 X1(5l) -
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 Vf over the complex numbers -
Chapter Thirteen Computing Vf modulo p -
Chapter Fourteen Computing the residual Galois representations -
Chapter Fifteen Computing coefficients of modular forms - Epilogue
- Bibliography
- Index
