29 problems

Holonomic rank conjecture. The solution rank of $\tau$ at the point $a\in V^\vee$ is given by

Let $k$ be a global field with ring of adeles ${\boldsymbol A}$, and let $D$ range over quaternion algebras over $k$. For an irreducible cuspidal automorphic representation $\pi$ o…

Let $G_n$ and $G_{n+1}$ be the relevant unitary groups, with automorphic representations $\pi_n$ and $\pi_{n+1}$, global period $\mathcal{P}$, and normalized local periods…

Let $\cM_{\rm cs}$ be the complex-structure moduli space of a Calabi–Yau manifold, and let its period integrals be the functions obtained by integrating the relevant Calabi–Yau for…

Let $Q=LU$ be a parabolic subgroup of $G$, let $N_L$ be a maximal unipotent subgroup of $L$, set $N=N_LU$, and let $\xi_L$ be a generic character of $[N_L]$, extended trivially acr…

Let $G$ be a reductive group, let $H$ be a subgroup equipped with a homomorphism $\boldsymbol{\rho}_H$ as in the BZSV setting, and let $\iota$ determine the dual homomorphism … For…

BZSV's period-sheaf conjecture. The period sheaf $\cP_X^r$ after spectral projection should correspond to the $L$-sheaf defined in the cited work.

Semi-local period-integral commutator relation. Under $\thref{genassum}$,

Period-integral commutator relation. Under $\thref{commsetup}$,

Let $\Delta=(G,H,\rho_H,\iota)$ be a strongly tempered BZSV quadruple, meaning that $\hat G=\hat H'Z_{\hat G}$, and let $\Pi$ be a global tempered $L$-packet of $G(\mathbb A)$. Whe…

Let $\pi$ be a cuspidal automorphic representation of $\mathbf{G}_0(\mathbb{A})$ with the archimedean and unramified hypotheses described above, and let $\chi$ be an anticyclotomic…

Let $F$ be a $p$-adic field and let $H\subset G$ be a pair of $F$-reductive groups such that $Y:=H\backslash G$ is spherical. Let $\sigma$ be a complex irreducible unitary $G(F)$-r…

General period bounds conjecture. These generic bounds should in fact hold for every eigenfunction in the sequence:

Let $E$ be a field, let $G$ be a reductive group over a local field $F$, and let $Y=H\backslash G$ be a spherical variety. For a smooth admissible $G(F)$-representation $\pi$ over…

Let $\Pi_{P}$ be the canonical form of local solutions of the Picard–Fuchs differential equations at a special boundary point $P$, and suppose mirror symmetry arises from $P$. Inte…

Let $\Pi_\phi=\bigcup_D\Pi_\phi(G_D)$ be the global tempered cuspidal $L$-packet and retain the notation for the models $(G_D,H_D)$, the embeddings $\nu$, and the period integrals…

Consider the first model in Table 1, $(G,H)=({\mathrm{GL}}_4\times{\mathrm{GL}}_2,{\mathrm{GL}}_2\times{\mathrm{GL}}_2)$, for which $Z_H/Z_{G,H}\cong {\mathrm{GL}}_1$. Under the no…

Let $(G,H)$ be one of the models in Table 1 other than the first, and let $D/k$ be a quaternion algebra, possibly split (or $D\in H^1(H/Z_{G,H},k)$ for Model 2). Let…

Let $\f$ and $\f'$ be factorizable vectors in representations $\c0$ and $\c0'$, respectively. Let $\a$ be the period pairing, ${\overline{\frak b}}$ the global arithmetic factor,…

Let $\pi$ be the cuspidal automorphic representation of $\operatorname{GU}_6(\mathbb A)$ considered in the paper, let $\chi$ be the character in the exterior-cubic $L$-function, an…

Getz–Wambach conjecture. The non-vanishing of the period integrals of the model $(H,H^{\sigma})$ for $\pi$ should be, roughly, equivalent to the non-vanishing of the period integra…

Let $X=G/P$ be a homogeneous variety, let $E=L_1\oplus\cdots\oplus L_r$ be a direct sum of homogeneous line bundles, and let $Y_i$ be the zero locus of the $L_i$-component of a sec…

Let $f$ be a homogeneous polynomial defining a smooth hypersurface family, and let $\alpha_s$ be the matrices associated with the Laurent polynomial construction, with $\sigma$ the…

Let $\mathop{\pi}$, $\mathop{\tau}$, and $\mathop{\sigma}\in \operatorname{Wd}_{\psi}(\tau)$ be as in the setup of the local triple product period. Assume that the local nonvanishi…

Let $X$ be a semi-Fano toric variety with mirror toric variety $X^\vee$, and let $B_X$ be the cohomology-valued function whose pairings with classes in $H_*(X^\vee,\mathbb Z)$ gene…