54 problems

Let $T\subset G$ be a maximal torus split over $F$, with root datum $(X(T),\Phi(T),X^{\vee}(T),\Phi^{\vee}(T))$, and let $G^\sphat$ be the dual group. Define … and let…

Let $F$ be the base field, let $E/F$ be the quadratic extension, let $A_{/F}$ be the elliptic curve, and let $S$ be the set of $r$ primes used to define the plectic invariant…

Chinburg's $\Omega(3)$ conjecture modulo the kernel group.

Let $E_T$ be an elliptic curve of rank $r$ over $\mathbb{Q}(T)$, and let … Write $\mathcal F_4^+$ and $\mathcal F_4^-$ for the subfamilies according to the sign of the root number…

Parity conjecture. One expects

Let $P$ be a polynomial, and let … denote the Liouville function evaluated at $P(n)$. Polynomial Liouville average conjecture. The average of $\lambda(P(n))$ over the integers is z…

Parity conjecture.

Let $E$ be an elliptic curve over $\mathbb{Q}$. Denote by $\operatorname{rk}_{p^{\infty}}(E/\mathbb{Q})$ its $p^{\infty}$-Selmer rank, and by $w(E/\mathbb{Q})$ its global root numb…

Let $E$ be an elliptic curve over a number field $F$. Write \begin{displaystyle}\operatorname{rk}(E/F)\end{displaystyle} for its algebraic rank and $w(E/F)$ for its global root n…

Let $F / L$ be a Galois extension of number fields, let $G = \operatorname{Gal}(F / L)$, let $E / L$ be an elliptic curve, and let…

Let $E$ be an elliptic curve over a number field $L$, and let $w(E / L) \in \\{\pm1\\}$ be its global root number. Parity conjecture. The root number determines the parity of the M…

Let $r,s,t>0$ be integers such that $r+s+t=\ell^N$ and $\ell\nmid rst$. Define … When $\operatorname{ord}_{\ell}(\delta)=0$, define … Here $(\frac{\cdot}{\ell})$ denotes the Legend…

The conjecture. For $0\leq k\leq \left\lceil\frac{j-i_0}{2}\right\rceil$, the coefficients satisfy the stated congruences for $U_k^{(u)}$ and $U_k'$, and consequently

Let $E/\mathbb Q$ be the elliptic curve in the preceding construction, and let $W(E)\in\{\pm1\}$ be its global root number. Parity conjecture. The rank of $E$ is even if and only i…

Sign-dependent Iwasawa main conjecture. If $[?]$, then $[?]$ is torsion and

Let $E/F$ be the quadratic extension, let $mathfrak{n}{mathrm{ur}}$ denote the unramified part of the relevant conductor $mathfrak{n}$, and let $omega$ be a character of…

Let $(\mathcal N,\mathcal Q)$ be an arithmetically compatible sequence of pairs $(N,Q)$ of Type I. Let $\mathcal E$ be the set of rational newforms lying in $S_2(N)$ for some…

Parity conjecture for Artin twists.

Parity conjecture.

Averages of root numbers conjecture. The average root number in this family is zero. Helfgott gives strong support and proves the assertion for several elliptic surfaces under suit…

For an elliptic curve $E/\Q$, let $W(E)\in\{-1,1\}$ be the root number in the functional equation of its $L$-function, and let $\operatorname{rank} E(\Q)$ denote the rank of its Mo…

Local conjecture. Under these hypotheses,

Let $A/K$ be an abelian variety over a number field $K$. For each place $v$ of $K$, let $w_{A/K_v}$ be the local root number of $A$. Parity conjecture. The parity of the Mordell–We…

Let $G$ be a quasi-split connected reductive group over a non-archimedean local field $F$, with center $Z$, and let $Z^{\circ}$ denote the identity component of $Z$. Let $\pi$ be a…

Parity conjecture.