Counting level-raising congruences using modular representation theory
Counting level-raising congruences using modular representation theory
Given distinct primes l and p and a newform f of level N, the level-raising problem asks whether or not there exists a newform g of level Np or Np^2 such that g is congruent to f modulo l. If l>3 and f is non-Eisenstein modulo l, then this problem and variants of it have been tackled by many authors, first by Ribet, and most generally by Diamond and Taylor, and the answer can be summarized roughly by saying that if it looks possible just by looking locally at p that such a g exists, then it does. If f is Eisenstein modulo l, then the situation is more complicated, but there are some results in special cases, starting with Mazur's famous Eisenstein-ideal paper. In this talk, we will explore a quantitative version of the level-raising problem: instead of asking if such a newform g exists, we ask how many such forms g there are. We give a partial answer using mod-l representation theory of PGL(2,Fp). The answer applies equally well in the Eisenstein and non-Eisenstein settings and gives new insight into the original problem in the Eisenstein case. This is joint work with Jaclyn Lang and Robert Pollack.