4 problems

Let $k$ and $j$ be non-negative integers with $j$ even and $k\geq 3$. Let $f=\sum a(n,f)q^n\in S_{2k+j-2}(SL_2(\mathbb Z))$ be a primitive form, and suppose that a sufficiently lar…

Let $\rho^\theta_\blambda$ be the $p$-adic realization of the motive associated with the Siegel modular form, let $\rho^thetablambda aturaltheta$ denote its twist by $\theta$, and le…

Kudla's conjecture. The formal generating series

Let $a\geq b\geq 0$ with $(a,b)\neq(0,0)$, let $V_{a,b}$ be the corresponding local system on $\A{2}$, let $T(p)$ be the Hecke operator at $p$, and let $S_{a-b,b+3}$ be the space o…