We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields of characteristic p. We prove the prime-to-p part conditionally on the finiteness of the p-primary part of the Tate-Shafarevich group or the equality of the analytic and the algebraic rank. If the base is a product of curves, Abelian varieties and K3 surfaces, we prove the prime-to-p part of the conjecture for constant or isoconstant Abelian schemes, in particular the prime-to-p part for (1) relative elliptic curves with good reduction or (2) Abelian schemes with constant isomorphism type of \Acal[p] or (3) Abelian schemes with supersingular generic fibre, and the full conjecture for relative elliptic curves with good reduction over curves and for constant Abelian schemes over arbitrary bases. We also reduce the conjecture to the case of surfaces as the basis.
In this note, we prove a duality theorem for the Tate-Shafarevich group of a finite discrete Galois module over the function field $K$ of a curve over an algebraically closed field: There is a perfect duality of finite groups $Sha^1(K,F) \times Sha^1(K,F') \to Q/Z$ for $F$ a finite étale Galois module on $K$ of order invertible in $K$ and with $F' = \Hom(F,Q/Z(1))$.
Furthermore, we prove that $H^1(K,G) = 0$ for $G$ a simply connected, quasisplit semisimple group over $K$ not of type $E_8$.
Let $X$ be one of the $28$ Atkin-Lehner quotients of a curve $X_0(N)$ such that $X$ has genus $2$ and its Jacobian variety $J$ is absolutely simple. We show that the Shafarevich-Tate group $\Sha(J/\Q)$ is trivial. This verifies the strong BSD~conjecture for $J$.
We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the $p$-part. Along the way, we generalize previous results on the Tate-Shafarevich and the Brauer group in this situation.
We complete the computation of all Q-rational points on all the 64 maximal Atkin-Lehner quotients $X_0(N)^*$ such that the quotient is
hyperelliptic. To achieve this, we use a combination of various methods, namely the classical Chabauty–Coleman, elliptic curve Chabauty, quadratic Chabauty, and the bielliptic quadratic Chabauty method combined with the Mordell-Weil sieve. Additionally, for square-free levels N, we classify all Q-rational points as cusps, CM points (including their CM field and
j-invariants) and exceptional ones. We further indicate how to use this to compute the Q-rational points on all of their modular coverings.
Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over finitely generated fields to the the specialization theorem for N\'eron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after an alteration of the base surface $S$, for almost all curves $C$ on $S$ the Mordell-Rank of $A$ over $S$ stays the same when restricting to $C$.
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.