  This paper concerns a special class of $n$-point correlation functions of
operators in the stress tensor supermultiplet of $\mathcal{N}=4$ supersymmetric
$SU(N)$ Yang-Mills theory. These are "maximal $U(1)_Y$-violating" correlators
that violate the bonus $U(1)_Y$ charge by a maximum of $2(n-4)$ units. We will
demonstrate that such correlators satisfy $SL(2,\mathbb{Z})$-covariant
recursion relations that relate $n$-point correlators to $(n-1)$-point
correlators in a manner analogous to the soft dilaton relations that relate the
corresponding amplitudes in flat-space type IIB superstring theory. These
recursion relations are used to determine terms in the large-$N$ expansion of
$n$-point maximal $U(1)_Y$-violating correlators in the chiral sector,
including correlators with four superconformal stress tensor primaries and
$(n-4)$ chiral Lagrangian operators, starting from known properties of the
$n=4$ case. We concentrate on the first three orders in $1/N$ beyond the
supergravity limit. The Mellin representations of the correlators are
polynomials in Mellin variables, which correspond to higher derivative contact
terms in the low-energy expansion of type IIB superstring theory in $AdS_5
\times S^5$ at the same orders as $R^4, d^4R^4$ and $d^6R^4$. The coupling
constant dependence of these terms is found to be described by non-holomorphic
modular forms with holomorphic and anti-holomorphic weights $(n-4,4-n)$ that
are $SL(2, \mathbb{Z})$-covariant derivatives of Eisenstein series and certain
generalisations. This determines a number of non-leading contributions to
$U(1)_Y$-violating $n$-particle interactions ($n>4$) in the low-energy
expansion of type IIB superstring amplitudes in $AdS_5\times S^5$.
