Inspired by the invariant of a number field given by its zeta function, we define the notion of weak arithmetic equivalence and show that under certain ramification hypotheses this equivalence determines the local root numbers of the number field. This is analogous to a result of Rohrlich on the local root numbers of a rational elliptic curve. Additionally, we prove that for tame non-totally real number fields, the integral trace form is invariant under arithmetic equivalence

We study the global root number of the complex $L$-function of twists of elliptic curves over $\mathbb{Q}$ by real Artin representations. We obtain examples of elliptic curves over $\mathbb{Q}$ which, while not having any rational points of infinite order, conjecturally must have points of infinite order over the fields $\mathbb{Q}( \sqrt[3] {m} )$ for every cube-free $m > 1$. We describe analogous phenomena for elliptic curves over the fields $\mathbb{Q}( \sqrt[r] {m} )$, and in the towers $(\mathbb{Q}( \sqrt[r^n] {m})_{n \ge 1} )$ and $(\mathbb{Q}( \sqrt[r^n] {m}, \mu_{r^n})_{n \ge 1})$, where $r \ge 3$ is prime.

Let $\pi$ be a generic irreducible representation of either a symplectic group or a split special even orthogonal group over a local field of characteristic zero. We prove that $\varepsilon(\frac12,\pi,\psi)=\pi(-1)$.

For the reductive dual pair (U(n, n), U(n, n)) over a p-adic field, we study the local theta correspondence for certain tempered representations in terms of R-groups. In the case we consider, the Langlands parameter is preserved, but, a twist occurs in the L-packet. Moreover this twist is determined by root numbers.

We show that 17.9% of all elliptic curves over Q, ordered by their exponential height, are semistable, and that there is a positive density subset of elliptic curves for which the root numbers are uniformly distributed. Moreover, for any α > 1/6 (resp. α > 1/12) the set of Frey curves (resp. all elliptic curves) for which the generalized Szpiro Conjecture |Δ(E)| [Lt]α NE12α is false has density zero. This implies that the ABC Conjecture holds for almost all Frey triples. These results remain true if we use the logarithmic or the Faltings height. The proofs make use of the fibering argument in the square-free sieve of Gouvêa and Mazur. We also obtain conditional as well as unconditional lower bounds for the number of curves with Mordell–Weil rank 0 and [ges ]2, respectively.