189 problems

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…

A period is a complex number represented by an integral of an algebraic differential form over a domain defined by algebraic equations and inequalities. Kontsevich–Zagier period re…

Grothendieck's period conjecture. The period map

Integral homology basis conjecture. The set $\{v_0,v_1',v_2,v_3\}$ is the image under $\iota_{\omega}$ of a $\mathbb Z$-basis of $H_3(X_{\phi}(\mathbb C),\mathbb Z)$.

A period is a number admitting an integral representation whose functions and domains of integration are algebraic with coefficients in $\bar{\mathbb{Q}}$. The relevant transformat…

Periods in this setting have algebraic relations, and motivic relations provide the corresponding algebraic framework. Grothendieck's period conjecture. The relations between perio…

Let $g_{50}$ be the modular form whose associated periods are well-defined modulo contributions from its residues. These boundary terms are considered modulo magnetic modular forms…

Let $X$ be a smooth projective irreducible variety over $\overline{\mathbb{Q}}$, let $\beta$ be the de Rham--Betti comparison matrix in $\mathrm{GL}_N(\mathbb{C})$ under suitable b…

Grothendieck Period Conjecture. The transcendence degree over $\mathbb{Q}$ of the field generated by its period matrix equals the dimension of the Mumford–Tate group of the Hodge s…

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…

Mixed-Tate motives are motives belonging to the tannakian subcategory generated by Tate motives, and a period is a number arising from the comparison between algebraic de Rham and…

Let $F$ be a non-Archimedean locally compact field, let $D$ be a non-split quaternion algebra with centre $F$, and fix an integer $n>1$. Set $G={\rm GL}_n(D)$, equipped with an inv…

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 $(Y,(-\log t)\omega)$ and $(Z_t,\Omega_t)$ be a Calabi-Yau mirror pair, and let $C_t\subset Z_t$ be a certain family of $n$-cycles. Let $F$ and…

Let $M$ be a 1-motive defined over $\overline{\mathbb{Q}}$, let $\Pi_M$ be its period matrix, and let $\mathcal{G}_{\mathrm{mot}}(M)$ be its motivic Galois group. Grothendieck's pe…

Let $K$ be a CM field that is Galois over $\mathbb{Q}$, and let $\Phi$ be a CM type of $K$. The quantities $p_K(\operatorname{id},\sigma)$ are Shimura period symbols, and…

Let periods be obtained by integrating algebraic differential forms over algebraic cycles, and consider the formal properties of integration, including additivity, change of variab…

Guo–Jacquet conjecture. If $\pi_D$ is $H_D$-distinguished and $\pi$ is deduced from $\pi_D$ by the Jacquet–Langlands correspondence, then $\pi$ is $H$-distinguished.

Let $\tilde{\mathcal{P}}$ be the algebra of formal periods generated by symbols $(X,Y,i,\omega,\sigma)$, modulo bilinearity, functoriality, and boundary relations, and let…

Let $X$ be a smooth projective variety over $\overline{\mathbb Q}$, let $Z_X$ denote the comparison image associated with the periods of $X$, and let $\Omega_X^{\rm mot}$ denote it…

Let the formal algebra of pairs of polygons be denoted by $d53cd53a$, and let the formal multizeta value algebra be the algebra corresponding to formal multizeta values. The algebr…

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 $P_+$ be the rational vector space generated by symbols representing equivalence classes of algebraic de Rham forms and relative Betti homology cycles, modulo linearity, change…

Let $n=\ell+3\geq 4$, let $\mathfrak{M}_{0,n}$ be the moduli space of genus-$0$ curves with $n$ marked points, and let $\overline{\mathfrak{M}}_{0,n}$ be its smooth compactificatio…

A mixed Tate motive unramified over $\mathbb Z$ is a mixed Tate motive whose arithmetic ramification is absent over the integers. Its periods are complex numbers obtained by integr…