Rigid Local Systems and Sporadic Simple Groups

Nicholas Michael Katz, Princeton University, New Jersey, Antonio Rojas-Leon, Universidad de Sevilla, Spain, and Pham Huu Tiep, Rutgers University, Piscataway, New Jersey

• Chapters1. Introduction2. Almost quasisimple groups containing elements with simple spectra3. Preliminary results on condition $\mathrm {({\bf S })}$4. $G_\mathrm {geom}$ and $G_\mathrm {arith}$5. Structure of $G_\mathrm {geom}$6. Rationality, moments, and reduction mod $\ell $ of hypergeometric sheaves7. Descents of hypergeometric sheaves8. The notational scheme for descents9. Proving finiteness of $G_\mathrm {geom}$10. The alternating group $\mathsf {A}_6$11. The alternating group $\mathsf {A}_7$12. The Mathieu group $\mathrm {M}_{11}$13. The Mathieu group ${\mathrm M}_{22}$14. The Mathieu group $\mathrm {M}_{23}$15. The Mathieu group $\mathrm {M}_{24}$16. The MacLaughlin group $\mathrm {McL}$17. The Janko group $\mathrm {J}_2$18. The Janko group $\mathrm {J}_3$19. The Rudvalis group $\mathrm {Ru}$20. The special linear group $\mathrm {PSL}_3(4)$21. The special unitary group $\mathrm {PSU}_4(3)$22. The symplectic group $\mathrm {Sp}_6(2)$23. The orthogonal group $\Omega ^ _8(2)$24. The exceptional group $G_2(3)$25. The exceptional group $G_2(4)$ and its subgroup $\mathrm {SU}_3(4)$26. The ""exceptional"" group $\mathrm {SU}_3(3) \cdot 2 \cong G_2(2)$27. The Suzuki group ${}^2\! B_2(8)$28. The ""exceptional"" group $\mathrm {SL}_2(8) \cdot 3 \cong {}^2\! G_2(3)$29. The Conway group $\mathrm {Co}_1$ and the Suzuki group $\mathrm {Suz}$30. Complex reflection groups31. Further local systems for $\mathrm {Sp}_6(2)$, $\mathrm {SU}_3(3)$, ${}^2\! G_2(3)$, and $2\mathsf {A}_7$32. Further multi-parameter local systems