718 problems

Let $p\equiv 4,7\, \mathrm{mod}\, 9$, let $i=1,2$, and let $E_{p^i}$ be the elliptic curve $y^2=x^3+\frac{p^{2i}}{4}$. Write $\Omega_{p^i}$ for its real period, let $\delta(p^i)=0$…

Let $L(s)$ range over the several families of $L$-functions considered by Keating and Snaith, evaluated at the central point $s=1/2$. Let $\mathrm{SO}(2N)$ and $\mathrm{USp}(2N)$ d…

Let $\mathfrak{L}_f(s)=\sum_{n=1}^{\infty}f(n)n^{-s}$ be a member of the Selberg class $\mathcal{S}$, with coefficients $f(n)$. Ramanujan's conjecture. One has…

Density conjecture. The Katz–Sarnak philosophy predicts

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…

Hasse–Weil conjecture. The function $L(s,C/K)$ has an analytic continuation, and the completed function satisfies

Conrey–Farmer–Keating–Rubinstein–Snaith conjecture. The asymptotic formula

Generalized Riemann hypothesis. All non-trivial zeros of $L$-functions lie on the critical line.

Let $f_d$ be a quadratic twist in the family and define … where $s=1/2+it$. For $-1/4<\operatorname{Re}(\alpha),\operatorname{Re}(\beta)<1/4$,…

High-moment asymptotic conjecture. Under these conditions,

Generalized Sato–Tate conjecture. The moment sequence of the distributions of coefficients of the normalized L-polynomials converges to the moment sequence of the Sato–Tate group.

Harder's conjecture. If $p_{\mathfrak{p}}>2k+j-2$ and $\mathfrak{p}\mid L^{\operatorname{alg}}(k+j,f)$, then there exist a Hecke cusp eigenform

Let $\pi$ and $\pi'$ be irreducible cuspidal automorphic representations of $\operatorname{GL}(m)$ and $\operatorname{GL}(m')$ over $\mathbb{Q}$, respectively, both with unitary ce…

Let $piinmathfrak{F}m$ be an automorphic representation, and let $alpha{r,pi}(p)$ denote its Satake parameters at a prime $p$. Generalized Ramanujan–Petersson conjecture. The Sat…

Let $q=p^a$ be a prime power, let $\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$, and let $\mathfrak{M}_2(2g)$ denote the set of primitive quadratic…

Let $L(f,s)$ be a “nice” $L$-function, and suppose its reciprocal has a Dirichlet-series expansion … valid in some right half-plane. The non-trivial zeros of $L(f,s)$ are the zeros…

Let $\Lambda_i(V;s)$ be the completed $L$-function associated with a degree-$d_i$, motivic-weight-$i$ $L$-function, including its gamma factors. Generalized Riemann hypothesis. All…

The Riemann hypothesis asserts that the nontrivial zeros of the Riemann zeta-function lie on the line $\Re(s)=1/2$. Generalized Riemann hypothesis. The same property holds for a mu…

Böcherer's conjecture. For every imaginary quadratic field $E$,

Let $\sigma$ be an automorphic representation and let $r$ be a finite-dimensional representation defining the associated $L$-function. Langlands functoriality conjecture. The $L$-f…

Let a family of $L$-functions be given, and consider the logarithms of their central values. Keating–Snaith conjecture. The logarithm of the central values of $L$-functions in vari…

Let the Stickelberger–Brumer elements be the special values of the $L$-function evaluators associated with an abelian extension of global fields, and let the corresponding ideal cl…

Let $E$ be an elliptic curve, let $\mathcal{L}(E)$ denote its algebraic $L$-value, let $\mathrm{BSD}(E)$ denote its Birch–Swinnerton-Dyer quotient, and let … be the $q$-adic valuat…

Deninger's conjecture.

Ivić's moment conjecture. In 2002, Ivić conjectured that