10 problems

Hecke–Fricke converse theorem. If the Dirichlet series satisfies the stated functional equation and Euler product, then it equals $L(s,f)$ for some

Cogdell–Piatetski-Shapiro conjecture. There are a partition $n=n_1+\cdots+n_r$ and cuspidal automorphic representations $\pi_i$ of $GL_{n_i}(\mathbb{A})$ such that

Let $\ell$ be a prime and let $q$ be a prime power. The naive converse theorem for mod $\ell$ representations of $\operatorname{GL}_2(\mathbb{F}_q)$ asks whether the relevant mod…

Very weak edge removal–strong converse equivalence conjecture. An equivalence holds between very weak edge removal and the ordinary strong converse for discrete memoryless networks…

Let $N$ be the level and, for a prime $q$ coprime to $N$, consider the single-modulus converse theorem referred to as Theorem. Large-prime extension conjecture. The proof of that t…

Let $\xi$ be a Dirichlet character modulo $N$, let $k$ be a positive integer satisfying $\xi(-1)=(-1)^k$, and let $\{f_n\}_{n=1}^\infty$ and $\{g_n\}_{n=1}^\infty$ be sequences of…

Let $\mathcal S$ be the Selberg class, and let $d_F$ denote the degree of $F\in\mathcal S$. An $L$-function is called automorphic when it arises from an automorphic representation…

Cogdell–Piatetski-Shapiro's converse theorem conjecture. If the conditions $\mathcal{A}(n,m)$ hold for every $m$ with $0leq mleq

Let $\pi_1$ and $\pi_2$ be non-isomorphic unitary cuspidal automorphic representations of $\operatorname{GL}_{d_1}(\mathbb{A}_{\mathbb{Q}})$ and…

Consider degree-$2$ $L$-functions with Euler products, and a converse theorem intended to characterize them without using twists. Twist-free converse-theorem conjecture. It has bee…