Small non-Gorenstein residually Eisenstein Hecke algebras
In Mazur's work proving the torsion theorem for rational elliptic curves, he studied congruences between cusp forms and Eisenstein series in weight two and prime level. One of his innovations was to measure such congruences using a residually Eisenstein Hecke algebra. He asked for generalizations of his theory to squarefree levels. The speaker made progress toward such generalizations in joint work with Preston Wake; however, a crucial condition in their work was that the Hecke algebra be Gorenstein, which is often but by no means always true. We present joint work with Catherine Hsu and Preston Wake in which we study the smallest possible non-Gorenstein case and leverage this smallness to draw an explicit link between its size and an invariant from algebraic number theory.