27 problems

$p$-parity conjecture. The $p^\infty$-Selmer rank $\operatorname{rk}_p E/K$ is even if and only if $w(E/K)=1$.

Let $E$ be an elliptic curve over a number field $K$. Let $w(E/K)\in\{\pm 1\}$ be the root number, defined as the conjectural sign in the functional equation for $L(E/K,s)$ under…

Parity conjecture.

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…

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

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.

Equidistribution of root numbers conjecture. Both $S_{-1}$ and $S_1$ have natural density $1/2$ in $\mathbb{N}$. This is the equivalent sign-equidistribution formulation of the con…

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…

Local conjecture. Under these hypotheses,

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…

Average-rank conjecture. One has

Root-number bias conjecture. For any $M>1$,

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

Chinburg's $\Omega(3)$ conjecture.

Let $L/\mathbb Q$ be a number-field extension and let $L_0/\mathbb Q$ be its Galois closure. Let $W(\chi)$ be the root number of the Artin $L$-function attached to $\chi$, and let…

Local root-number formula. The local root number of the Jacobian satisfies

Let $\mathcal{F}/\mathcal{K}$ be a finite Galois extension, let $A$ be an abelian variety over the global field $\mathcal{K}$, and let $A(\mathcal{F})_{\mathbb{C}}$ be the…

Let $A$ be an abelian variety over a global field $\mathcal{K}$. Write $\operatorname{rk} A/\mathcal{K}$ for the rank of its Mordell--Weil group, and let…

Let $K$ be a local field of characteristic zero, let $L/K$ be a quadratic extension, and let $C/K$ be a hyperelliptic curve with Jacobian $J$. Write $C^L$ and $J^L$ for the quadrat…

Let $A/K$ be an abelian variety over a number field, and let $p$ be a prime. Define the $p$-Selmer vector space … If $F/K$ is a Galois extension and $\tau$ is a self-dual represent…

Let $\mathcal K$ be a local field of characteristic zero, and let $E/\mathcal K$ be an elliptic curve with a $p$-isogeny $\phi:E\to E'$ over $\mathcal K$. Put…

Let $F$ be a local field of characteristic zero, and let $E/F$ be an elliptic curve with a 2-isogeny $\phi:E\to E'$ over $F$. In the notation of the source, let $a,b\in F$ be the p…