6 problems

The p-adic Artin formalism conjecture. In full generalities, there should be a factorization

Suppose that $\varepsilon(\textup{\bf f})=-1$ and the stated hypothesis holds. Let ${\rm Log}_{\omega_\textup{\bf f}}({\rm BK}_\textup{\bf f}^\dagger)$ denote the logarithm of the…

Let $r$ be a positive integer and let $(\kappa,\lambda)\in\mathcal A_r^{(2)}$. Write $X_r=X_1(Np^r)$, $f=\mathbf f_\kappa$, and $g=\mathbf g_\lambda$. Assume that $f$ and $g$ are n…

Let $N$ and $p\geq 5$ be primes with $p\mid N-1$. Let $g\in S_1(\Gamma_1(d))$ be a weight-one cuspidal newform of conductor $d$ with $\gcd(Np,d)=1$, let $g^*$ be its dual, and let…

Rank-zero Bloch–Kato conjecture. Then

Let $(f,g,h)$ have weights $(k,\ell,m)$ with $k\geq\ell+m$ and $k,\ell,m\geq2$, and let $M(f\otimes g\otimes h)$ be the associated triple-product motive. Put $c=(k+\ell+m-2)/2$ and…