  We introduce a sequence of isolated curve singularities, the elliptic m-fold
points, and an associated sequence of stability conditions, generalizing the
usual definition of Deligne-Mumford stability. For every pair of integers
0<m<n, we prove that the moduli problem of n-pointed m-stable curves of
arithmetic genus one is representable by a proper irreducible Deligne-Mumford
stack. We also consider weighted variants of these stability conditions, and
construct the corresponding moduli stacks. In forthcoming work, we will prove
that these stacks have projective coarse moduli and use the resulting spaces to
give a complete description of the log minimal model program for M_{1,n}.
