85 problems

Let $X$ be a pure or mixed motive defined over $\overline{\mathbb{Q}}$. Write $\mathrm{periods}(X)$ for its periods and $\mathcal{G}_{\mathrm{mot}}(X)$ for its motivic Galois group…

Let $P$ be the algebra of periods, let $P_{\mathbb{Z},\mathrm{Tate}}$ be the subalgebra generated by $(2\pi i)^{\pm1}$ and periods of mixed Tate motives unramified over…

Let $F$ be a number field, let ${\cal O}_{F,S}$ be a ring of $S$-integers, and let ${\cal A}_{\bullet}({\cal O}_{F,S})$ be its motivic Hopf algebra. Let…

Let $N$ be a positive integer. Let ${\cal Z}^{\cal H}_{\bullet}(\mu_N)$ be the commutative graded Hopf algebra generated by framed Hodge-Tate structures associated with multiple po…

Let $\mathcal P$ be a regular rational polypol, and let $F_{\mathcal P}(u,v)$ denote its associated dual period. Consider the holonomic system annihilating $F_{\mathcal P}$, and th…

Let $C/\mathbb{Q}(\zeta_m)$ be a superelliptic curve $y^m=f(x)$ as in the paper, let $\tau$ be a character of $C_m$, and fix an embedding…

Let $M$ be a $\mathbb{Q}$-motive with coefficients in a number field $F$ and odd weight. Let $H_B^+(M)$ be the $+1$-eigenspace of complex conjugation, let $\alpha_M^+$ be the compa…

Let $G$) be a connected quasi-split reductive group over $F$ with a fixed Whittaker datum. Suppose there is a categorical equivalence … For a dual pair…

Let $M$ be a critical motive over $K$ with coefficient field $\mathcal{F}$. Let $\gamma$ and $\delta$ be $\mathcal{F}$-bases of ${\det}_\mathcal{F}(H_B(M)^+)$ and…

Let $M=[u:\mathbb{Z}\to G^n]$ be the 1-motive with $u(1)=(R_1,\dots,R_n)$ defined by the points in the source. Let $k$ be the endomorphism field and let $\mathrm{LieBracket}$ and…

Let $f$ and $f'$ be the forms specified in Theorem BS, chosen so that for every $v\in S_\infty$ their local components lie in the indicated coefficient-field-rational subspaces of…

Let $K$ be the weight, $S_K$ the space of cusp forms under consideration, and $r_{\ell,\chi}$ the twisted period indexed by $\ell$ and a primitive Dirichlet character $\chi$. Let…

Let $K$ be the weight, $S_K$ the space of cusp forms under consideration, and $r_{\ell,\chi}$ the twisted period indexed by $\ell$ and a primitive Dirichlet character $\chi$. Let…

Prasanna--Venkatesh conjecture. One has

Let $\mathcal{E}$ be an elliptic curve with period lattice $\Omega$, invariants $g_2,g_3$, and endomorphism field $k$. Let $s,n\geq0$, let $\mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s}$…

Let $A$ be a CM abelian variety of dimension $g\geq 1$, let $[o]\in\mathbb{A}_g(\overline{\mathbb{Q}})$ parametrize $A$, and set $[\mathbf{o}]:=([o],\ldots,[o])$. An elementary qua…

Let $f$ be a Hilbert newform of parallel weight $2$ over the totally real field $F$, with Hecke field $K_f$ of degree $g$. Let $s=s_i$ be a sign vector corresponding to the real em…

Hoffman basis conjecture. The Hoffman elements $\zeta(k_1,\ldots,k_d)$ with $k_i\in\{2,3\}$ form a basis for $\mathcal{Z}$; in particular,

Let $\tilde{\mathbf G}$ be the twisted group in the Shalika-period setting, let $\omega$ be the specified central character, and let $\mathcal{C}(\tilde{G},\omega)$ be the strongly…

Let $G=\mathrm{GL}_m(D)$, let $H$ be the subgroup defining the linear period, let $\omega$ be a central character, and let $\mathcal{C}(G(F),\omega)$ be the corresponding Schwartz–…

Let $F$ be as in the linear-period setting, let $G=\mathrm{GL}_m(D)$, let $E/F$ be quadratic, let $H$ be the centralizer of $E^\times$ in $G$, and let $m_\mathrm{geom}(\pi,\chi)$ b…

Let $0pi$ be a cuspidal automorphic representation of $0\operatorname{GSp}4$ of weight $(k1,k2)$, and let $0\sigma$ be a cuspidal automorphic representation of…

Motivated Grothendieck period conjecture. The motive $M$ satisfies MGPC if $\Omega^{\mathrm{And}}_M$ is connected and

Let $f$ be a parallel weight-one form, let $U_f^\vee$ be the dual space appearing in the cited construction, and for $u\in U_f^\vee$ let $u_i$ denote its $\sigma_i$-component. Let…

Let $\Pi$ be a globally generic representation satisfying the paper's newvector assumption, and let $f=\bigotimes_p f_p\in\Pi$ be a newform. Let $W$ and $W_\infty$ be the Whittaker…