Further Computations on Maeda’s Conjecture

Xiaoyu Niu : Mathematics

Mentor: Sug Woo Shin, Mathematics

Yoshitaka Maeda made the conjecture in 1997: Let m be an integer greater than 1 and let F be the characteristic polynomial of the Hecke operator T_m acting on the space S_k of cusp forms of weight k and level one, then the polynomial F is irreducible over the field of rational numbers; the Galois group of the splitting field of F is the full symmetric group ₢_d, where d is the dimension of S_k. Most recent computations via Sage have verified the conjecture for k ≤ 14000. Xiaoyu’s project will focus on computing via Sage for k > 14000 and/or considering bigger n than what’s currently available.  She hopes to either provide more evidence or to find results that disprove the conjecture. She may make theoretical attempts at the conjecture.