12 problems

Let $S_{N,r}$ be the index set used to define the coefficient space $\mathbb{P}_{N,r}$, let $\mathbf{W}_{N,r}\subseteq\mathbb{P}_{N,r}$ be the space of polynomials satisfying the s…

Let $\mathfrak{dg}_1$ be the depth-one component of the depth-graded motivic Lie algebra, let $\mathbb{P}$ be the space of restricted even period polynomials, and let…

Tasaka's isomorphism conjecture. The map $\widetilde{\xi}$ is an isomorphism. The conjecture concerns the depth-three case and is attributed to Tasaka. The paper does not report a…

Diamantis–Rolen odd-part containment conjecture. There is a real number $b=b(f,m,k)>0$ such that

Let $f\in S_k(\Gamma_0(16))$ be a newform of level $N=16$, and let $r_f^+(X)$ denote its even period polynomial. Endpoint-zero conjecture. The polynomial $r_f^+(X)$ has zeros at bo…

For $N=2$ and $4$, let $k\geq 4$ be even and let $1\leq j\leq (k-2)/2$. Define the Bernoulli polynomial variant … Define $\widetilde{S}_{N,k,j}(X)$ by … and … Write … Then the asso…

For $N=2$ or $4$, even integers $k\ge4$, and integers $j$ with $1\le j\le (k-2)/2$, define coefficients $a_i$ and $b_i$ by … and … Here $P_{N,k,j}^{(+)}$ and $P_{N,k,j}^{(-)}$ are…

Let $\mathbf W_{N,r}$ be the space of restricted even period polynomials, and let $\varphi^{(r)}_j$ be the maps on the polynomial spaces used in the preceding lemmas. For $r\geq3$,…

Kernel generating-series conjecture. The following identities are expected:

Generalized Ramanujan polynomial conjecture. The polynomial $R_k(z;\chi,\psi)$ is unimodular: all its roots lie on the unit circle.

Let $f\in S_k(\Gamma_0(N),\chi)$ be a newform of weight $k$, level $N$, and nebentypus $\chi$. Its associated period polynomial is … The functional equation pairs roots under…

Eichler–Shimura–Manin correspondence. There are isomorphisms defined over $\mathbb C$