33 problems

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension, let ${\rm U}(n)$ be the corresponding unitary group, and let $\pi$ be a representation of ${\rm GL}_n(E)$. A repre…

Let $F$ be a $p$-adic field, let $E/F$ be a degree-two extension, set $G={\rm GL}_n(E)$ and $H={\rm GL}_n(F)$, and let $\pi$ be an irreducible admissible representation of $G$. A r…

Let $K/k$ be a purely inseparable extension of degree $p$, let $X$ be a connected scheme proper over $k$, and let $x_K\in X_K(K)$ lie over $x\in X(k)$. Let $\mathcal{C}_X$ and…

Let $F$ be a totally real number field, let $f$ be a Hilbert newform of parallel even weight over $F$, let $F'$ be a totally real quadratic extension of $F$, and let $f'$ be the ba…

Let $F_0$ be a local field, let $G$ be a reductive group over $F_0$, let $\pi_0\in\Pi(G/F_0)$ be any supercuspidal representation, and let $\Pi_0\subset\Pi(G/F_0)$ be the packet co…

Let $F$ be a local field, let $G$ be a reductive group over $F$, and call a supercuspidal representation pure incorrigible when it is pure and remains supercuspidal in the base-cha…

Let $F' / F$ be an extension, let $F”/F'$ be cyclic, and let $BC_{F”/F'}$ be the conjectural cyclic base-change map on representations of $G(F')$. Let ${\mathcal{L}}^{ss}_{F'}$ and…

Let $E/F$ be a solvable extension, let $\mathfrak{q}$ be a prime of $E$ above $\mathfrak{p}$, and let $\pi_E$ be the base-change lift of $\pi$ to $PGL_{n,E}$. Let…

Let $f$ be a normalized newform, $F$ the number field in the paper, and $f_F$ its base change. Let $\widetilde{u}_2(f_F)$ be the normalized degree-two period quantity, $\Omega_f^+$…

Let $f$ be a normalized newform and $f_F$ its base change to $F$. Let $\eta_f^\sharp$ be the complementary factor in the factorization of the base-change congruence ideal, let…

Let $F$ be the totally real field of degree $d$, let $f$ be a normalized newform and $f_F$ its normalized holomorphic base change. For $\epsilon\in\widehat{W_F}$, let $d^+$ and…

Let $\mathcal E_{\mathrm{par}}$ be the parallel-weight eigenvariety, let $\mathcal C_M$ denote the relevant Coleman–Mazur eigencurve at level parameter $M$, and let $[\varphi]$ den…

Let $E/F$ be a quadratic extension of $p$-adic fields, let $U_{2n}(E/F)$ be the quasi-split unitary group in $2n$ variables, and let…

Let $F$ be the function field of a smooth projective curve over a finite field, let $G$ be a split semisimple group over $F$, and let $d70$ be an automorphic representation of…

Let $F/E$ be a Galois extension with group $G=\operatorname{Gal}(F/E)$, and let $f$ be a Hilbert newform of $G$-invariant level $\mathfrak{N}$, weight $k$, and character $\chi$. Fo…

Let $E/F$ be the finite extension and let $S_p(cpi)$ denote the relevant set of primes above $p$. For a prime $\tilde{\mathfrak{p}}\in S_p(\pi)$, let $e_{\tilde{\mathfrak{p}}}$ and…

Let $E/F'$ be a solvable extension of number fields, and let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_E)$ that is $\mathrm{Gal}(E/F')$-invariant.…

Let $E/F$ be a Galois extension of number fields and let $n$ be an integer such that … and, for every divisor $m\mid n$, there are no nontrivial irreducible representations…

Let $E \geq F' \geq F$ be a tower of number fields, let $S$ be a finite set of places of $F$ containing the infinite places, and let $S'$ and $S_0$ be the places of $F'$ and $E$ ab…

Let $F$ and $E$ be number fields as in the paper, let $n$ be a positive integer, and let $_{E/F}(\Sigma_{\phi}^{S_0}(X))$ denote the associated base-change test function. For a cus…

Let $G$ be a real reductive group and let $\pi$ be an irreducible unitary representation of $G(\mathbb{C})$. Discrete-spectrum base-change conjecture. There is at most one pure inn…

Let $G$ be a reductive algebraic group over $\mathbb{R}$, and let $\pi$ be an irreducible representation of $G(\mathbb{C})$. Complex-real distinction conjecture. If…

Let $F\supset E$ be a degree $l$ extension of characteristic-zero non-archimedean local fields with residue characteristic $p$, where $p$ and $l$ are distinct primes. Let $\pi$ be…

Hecke equivariance conjecture. One has

Galois structure conjecture. The graded $\mathbb{Z}[\sigma]$-modules