17 problems

Let $E_{/\mathbb{Q}}$ be an elliptic curve of conductor $N$, let $K$ be the quadratic imaginary field associated with a Heegner discriminant, and let $y_K\in E(K)$ be the correspon…

Let $F/\mathbb Q$ be a $p$-adic Lie extension with Galois group $G$, let $\Lambda=\Lambda(G)$ be its Iwasawa algebra, let $T_p$ be a Galois-stable lattice, and put…

Weil's conjecture. If $G$ is simply connected, then

Harder's conjecture. If ${\mathcal G}$ is split, the Tamagawa number of $G$ equals the number of connected components of the moduli space of ${\mathcal G}$-torsors on $X$.

Vanishing conjecture. For every $i>0$,

Let $p$ be an odd prime, let $K/k$ be a finite abelian extension of number fields with Galois group $G$, and let $S$ contain $S_p\cup S_\infty\cup S_{\operatorname{ram}}(K/k)$, wit…

Let $V$ be a crystalline representation of $G_\ell$ over $K_\lambda$, and let $T\subseteq V$ be a $G_\ell$-stable $\mathcal{O}_\lambda$-lattice. Let $\omega$ and $\omega^*$ be latt…

Let $M$ be a critical object of $\operatorname{{\bf PM}}_K$, let $b(M)$ be the basis supplied by Deligne's period conjecture, let $\lambda$ be a place of $K$ for which the stated v…

Let $M$ be a motive over $K$, let $F=\mathcal{F}_\lambda$, and let $T$ be a stable $\mathcal{O}_F$-lattice in its realization. Assume the local hypotheses and the motivic cohomolog…

Let $M$ be a critical motive over $K$ with coefficient field $\mathcal{F}$, let $F=\mathcal{F}_\lambda$, and let $T$ be a stable $\mathcal{O}_F$-lattice in the associated represent…

Let $f\in S_k(\Gamma_0(N))$ be a newform with $k\ge 2$ and $p\ge 3$. Assume that $\rho_f$ has large image, that $f$ is not congruent modulo $\pi$ to any anomalous ordinary CM modul…

Let $f\in S_k(\Gamma_0(N))$ be a newform with $k\ge 2$ and $p\ge 3$, and suppose that $\rho_f$ has large image. Let $N_0$ be the prime-to-$p$ part of $N$. The Kato-system Tamagawa…

Let $f\in S_k(\Gamma_0(N))$ be a newform with $k\ge 2$ and $p\ge 3$, with $p\nmid N$ and $\rho_f$ of large image. Suppose that $f$ is ordinary at $p$ and $p$-distinguished, or that…

Let $E / L$ be an elliptic curve, let $F / L$ be a Galois extension with Galois group $G$, and let $\rho$ be an irreducible representation of $G$. Suppose that … for some…

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $F / \mathbb{Q}$ be a finite Galois extension with $G = \operatorname{Gal}(F / \mathbb{Q})$, and let $\rho$ be an irreducible re…

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ be a smooth affine group scheme with semisimple simply connected generic fiber and connected fibers, and let…

Let $F=\mathbb C(X)$ be the function field associated with the Riemann surface $M$, let $\mathbb A$ be its adelic ring, and let $\mathbb K$ be the specified maximal compact subgrou…