Book contents
- Frontmatter
- Contents
- Preface
- List of contributors
- 1 Modular Forms
- 2 On the basis problem for Siegel modular forms with level
- 3 Mock theta functions, weak Maass forms, and applications
- 4 Sign changes of coefficients of half integral weight modular forms
- 5 Gauss map on the theta divisor and Green's functions
- 6 A control theorem for the images of Galois actions on certain infinite families of modular forms
- 7 Galois realizations of families of Projective Linear Groups via cusp forms
- 8 A strong symmetry property of Eisenstein series
- 9 A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences
- 10 Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms
- 11 Modular shadows and the Lévy—Mellin ∞—adic transform
- 12 Jacobi forms of critical weight and Weil representations
- 13 Tannakian Categories attached to abelian varieties
- 14 Torelli's theorem from the topological point of view
- 15 Existence of Whittaker models related to four dimensional symplectic Galois representations
- 16 Multiplying Modular Forms
- 17 On projective linear groups over finite fields as Galois groups over the rational numbers
7 - Galois realizations of families of Projective Linear Groups via cusp forms
Published online by Cambridge University Press: 08 October 2009
- Frontmatter
- Contents
- Preface
- List of contributors
- 1 Modular Forms
- 2 On the basis problem for Siegel modular forms with level
- 3 Mock theta functions, weak Maass forms, and applications
- 4 Sign changes of coefficients of half integral weight modular forms
- 5 Gauss map on the theta divisor and Green's functions
- 6 A control theorem for the images of Galois actions on certain infinite families of modular forms
- 7 Galois realizations of families of Projective Linear Groups via cusp forms
- 8 A strong symmetry property of Eisenstein series
- 9 A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder's conjecture on congruences
- 10 Petersson's trace formula and the Hecke eigenvalues of Hilbert modular forms
- 11 Modular shadows and the Lévy—Mellin ∞—adic transform
- 12 Jacobi forms of critical weight and Weil representations
- 13 Tannakian Categories attached to abelian varieties
- 14 Torelli's theorem from the topological point of view
- 15 Existence of Whittaker models related to four dimensional symplectic Galois representations
- 16 Multiplying Modular Forms
- 17 On projective linear groups over finite fields as Galois groups over the rational numbers
Summary
Introduction
Let Sk be the space of cusp forms of weight k for SL2(ℤ) and write S2(N) for the space of cusp forms of weight 2 for Γ0(N).
We are going to consider the Galois representations attached to eigenforms in these spaces, whose images have been determined by Ribet and Momose (see [Ri 75] for Sk and [Mo 81], [Ri 85] for S2(N)).
Our purpose is to use these representations to realize as Galois groups over ℚ some linear groups of the following form: PSL2(pr) if r is even and PGL2(pr) if r is odd. In order to ease the notation, we will call both these families of linear groups PXL2(pr), so that PXL stands for PSL if r is even and PGL if r is odd.
Extending the results in [Re-Vi], where it is shown that for r ≤ 10 these groups are Galois groups over ℚ for infinitely many primes p, we will cover the cases r = 11, 13, 17 and 19, using the representations attached to eigenforms in Sk and again the cases 11 and 17 using the ones coming from S2(N).
We will give the explicit criterion for the case r = 3: for every prime p > 3 such that p ≡ 2, 3, 4, 5 mod 7 the group PGL2(p3) is a Galois group over ℚ.
- Type
- Chapter
- Information
- Modular Forms on Schiermonnikoog , pp. 85 - 92Publisher: Cambridge University PressPrint publication year: 2008
- 1
- Cited by