Abstract: Let $p$ and $\ell$ be primes $\geq 5$ such that $\ell$ divides $p-1$. Mazur introduced a certain quotient of the modular Jacobian $J_0(p)$ called the $\ell$-Eisenstein quotient. 

In a joint work with Jun Wang, we proved a congruence formula at the Eisenstein ideal for the central critical value of the $L$-function of the $\ell$-Eisenstein quotient twisted by even Dirichlet characters $\chi$ such that $\chi(p)=1$. Our proof relies on a conjecture of Sharifi relating modular symbols and $K$-theory. If the character is quadratic, inspired by previous work of Ono and others using the theory of half-integral weight modular forms, we use our congruence formula in a work in progress with Christian Maire to prove that if $p=2\ell+1$ (i.e. $\frac{p-1}{2}$ is a Sophie Germain prime), then there are infinitely many real quadratic fields in which $p$ splits and such that neither the class number is divisible by $\ell$ nor the fundamental unit is a $\ell$th power modulo a prime above $p$.