Euler systems and the symmetric square of a Hida family

Let $p\geq7$ be a prime number.

We build a non-trivial Euler system for the symmetric square of a $p$-adic Hida family of modular forms interpolating the Euler system constructed by Loeffler-Zerbes for the symmetric square of a $p$-ordinary newform.

As a second contribution, we prove an algebraic functional equation for dual Selmer groups in this setting.

Finally, building on recent work by Büyükboduk-Ganguly on functional equations of algebraic (Rankin-Selberg) $p$-adic $L$-functions, we prove a divisibility result towards the Iwasawa main conjecture for the symmetric square of a Hida family.