We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over Q. We apply our methods to all 28 Atkin–Lehner quotients of X_0(N) of genus $2$, all 97 genus 2 curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least 2. We also give an example where we verify that the order of the Tate–Shafarevich group is 7^2 and agrees with the order predicted by the BSD Conjecture.
Let f be a newform of weight k and level N with trivial nebentypus. Let p∤2N be a maximal prime ideal of the coefficient ring of f such that the self-dual twist of the mod-p Galois representation of f is reducible with constituents ϕ,ψ. Denote a decomposition group over the rational prime p below p by Gp. We remove the condition ϕ|Gp≠1,ω from [CGLS22], and generalize their results to newforms of arbitrary weights. As a consequence, we prove some Iwasawa main conjectures and get the p-part of the strong BSD conjecture for elliptic curves of analytic rank 0 or 1 over Q in this setting. In particular, non-trivial p-torsion is allowed in the Mordell–Weil group. Using Hida families, we prove a Iwasawa main conjecture for newforms of weight 2 of multiplicative reduction at Eisenstein primes. In the above situations, we also get p-converse theorems to the theorems of Gross–Zagier–Kolyvagin. The p-converse theorems have applications to Goldfeld's conjecture in certain quadratic twist families of elliptic curves having a 3-isogeny.
For an abelian variety A over a finitely generated field K of characteristic p > 0, we prove that the algebraic rank of A is at most a suitably defined analytic rank. Moreover, we prove that equality, i.e., the BSD rank conjecture, holds for A/K if and only if a suitably defined Tate--Shafarevich group of A/K (1) has finite ell$primary component for some/all ell neq p, or (2) finite prime-to-p part, or (3) has p-primary part of finite exponent, or (4) is of finite exponent. There is an algorithm to verify those conditions for concretely given A/K.
We develop an algorithm to test whether a non-CM elliptic curve $E/\Q$ gives rise to an isolated point of any degree on any modular curve of the form $X_1(N)$. This builds on prior work of Zywina which gives a method for computing the image of the adelic Galois representation associated to $E$. Running this algorithm on all elliptic curves presently in the $L$-functions and Modular Forms Database and the Stein--Watkins Database gives strong evidence for the conjecture that $E$ gives rise to an isolated point on $X_1(N)$ if and only if $j(E)=-140625/8, -9317,$ $351/4$, or $-162677523113838677$.
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 Néron-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 $C$.
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 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 $E_8$.
In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves X_0(N) of genus up to 8, and genus up to 10 with N prime, for which they were previously unknown. The values of N we consider are contained in the set
L = {58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127}.
We obtain that all the non-cuspidal quadratic points on X_0(N) for N in L are CM points, except for one pair of Galois conjugate points on X_0(103) defined over Q(sqrt{2885}). We also compute the j-invariants of the elliptic curves parametrised by these points, and for the CM points determine their geometric endomorphism rings.
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 $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
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.