Exact verification of the strong BSD conjecture for some absolutely simple abelian surfaces

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 of $J/\mathbb{Q}$ is trivial. This verifies the strong BSD conjecture for $J$.


Introduction
Let A be an abelian variety over Q and assume that its L-series L(A, s) admits an analytic continuation to the whole complex plane.The weak BSD conjecture (or BSD rank conjecture) predicts that the Mordell-Weil rank r = rk A(Q) of A equals the analytic rank r an = ord s=1 L(A, s).The strong BSD conjecture asserts that the Shafarevich-Tate group X(A/Q) is finite and that its order equals the "analytic order of Sha", (1) #X(A/Q) an := Here A ∨ is the dual abelian variety, A(Q) tors denotes the torsion subgroup of A(Q), the product v c v runs over all finite places of Q and c v is the Tamagawa number of A at v, L * (A, 1) is the leading coefficient of the Taylor expansion of L(A, s) at s = 1, and Ω A and Reg A/Q denote the volume of A(R) and the regulator of A(Q), respectively.
If A is modular in the sense that A is an isogeny factor of the Jacobian J 0 (N ) of the modular curve X 0 (N ) for some N , then the analytic continuation of L(A, s) is known.If A is in addition absolutely simple, then A is associated (up to isogeny) to a Galois orbit of size dim(A) of newforms of weight 2 and level N , such that L(A, s) is the product of L(f, s) with f running through these newforms.Such an abelian variety has real multiplication: its endomorphism ring (over Q and over Q) is an order in a totally real number field of degree dim(A).If, furthermore, ord s=1 L(f, s) ∈ {0, 1} for one (equivalently, all) such f , then the weak BSD conjecture holds for A; see [12].
All elliptic curves over Q arise as one-dimensional modular abelian varieties [25,20,2] such that N is the conductor of A. For all elliptic curves of (analytic) rank ≤ 1 and N < 5000, the strong BSD conjecture has been verified [9,14,6].
In this note, we consider certain absolutely simple abelian surfaces and show that strong BSD holds for them.One class of such surfaces arises as the Jacobians of quotients X of X 0 (N ) by a group of Atkin-Lehner operators.Hasegawa [11] has determined the complete list of such X of genus 2; 28 of them have absolutely simple Jacobian J.For most of these Jacobians (and those of further curves taken from [24]), it has been numerically verified in [8,22] that #X(J/Q) an is very close to an integer, which equals #X(J/Q) [2] (= 1 in the cases considered here).We complete the verification of strong BSD for these Jacobians by showing that #X(J/Q) an is indeed an integer and X(J/Q) is trivial.

Methods and algorithms
In the following, we denote the abelian surface under consideration by A; it is an absolutely simple isogeny quotient of J 0 (N ), defined over Q.We frequently use the fact that A can be obtained as the Jacobian variety of a curve X of genus 2. The algorithms described below have been implemented in Magma [1].
Recall that a Heegner discriminant for A is a fundamental discriminant D < 0 such that for K = Q( √ D), the analytic rank of A/K equals dim A = 2 and all prime divisors of N split in K. Heegner discriminants exist by [3,23].Since Magma can determine whether ord s=1 L(f, s) is 0, 1, or larger (for a newform f as considered here), we can easily find one or several Heegner discriminants for A.
Associated to each Heegner discriminant D is a Heegner point y D ∈ A(K), unique up to sign and adding a torsion point.In particular, the Heegner index is an order in a real quadratic field.In all cases considered here, O is a maximal order and a principal ideal domain.For each prime ideal p of O, we have the residual Galois representation ρ p : Gal(Q|Q) → Aut(A[p]) ≃ GL 2 (F p ), where F p = O/p denotes the residue class field.
We can use Magma's functionality for 2-descent on hyperelliptic Jacobians based on [19] to determine X(A/Q) [2].In all cases considered here, this group is trivial, which implies that X(A/Q)[2 ∞ ] = 0. (In fact, this had already been done in [8] for most of the curves.)It is therefore sufficient to consider the p-primary parts of X(A/Q) for odd p. Theorem 1.Let A be an abelian variety of GL 2 -type over Q. Assume that ord s=1 L(f, s) ∈ {0, 1} for one (equivalently, all) newform associated to A.
(1) If the level N of A is square-free, ord p (#X(A/Q) an ) = ord p (#X(A/Q)) for all rational primes p = 2 such that ρ p is irreducible for all p | p.  2) is an explicit version of [12].
We have implemented the following algorithms.
(1) Image of the residual Galois representations.Extending the algorithm described in [7], which determines a finite small superset of the primes p with ρ p reducible in the case that End Q (A) = Z, we obtain a finite small superset of the prime ideals p of O such that ρ p is reducible.Building upon this and [5], we can also check whether ρ p has maximal possible image The irreducibility of ρ p for all p | p is the crucial hypothesis in [4, Theorems C and D], and in [12].
(2) Computation of the Heegner index.We can compute the height of a Heegner point using the main theorem of [10].By enumerating all points of that approximate height using [15], we can identify the Heegner point y D ∈ A(K) as a Q-point on A, or on the quadratic twist A K , depending on the analytic rank of A/Q.An alternative implementation uses the j-invariant morphism X 0 (N ) → X 0 (1) and takes the preimages of the j-invariants belonging to elliptic curves with CM by the order of discriminant D. A variant of this is based on approximating q-expansions of cusp forms analytically and finding the Heegner point as an algebraic approximation.(3) Determination of the (geometric) endomorphism ring of A/Q and its action on the Mordell-Weil group A(Q).Given the Heegner point y D , this can be used to compute the Heegner index I D .We can also compute the kernel of a given endomorphism as an abstract Gal(Q|Q)-module together with explicit generators in A(Q).We apply this to find the characters corresponding to the constituents of ρ p when the representation is reducible.(4) Analytic order of X.If the L-rank ord s=1 L(f, s) of A/Q is zero, then we can compute #X(A/Q) an exactly as a rational number using modular symbols via Magma's LRatio function, which gives L(A, 1)/Ω −1 A ∈ Q >0 , together with (1), since #A(Q) tors = #A ∨ (Q) tors and the Tamagawa numbers c v are known.
When the L-rank is 1, we can compute the analytic order of X from #X(A/K) an = #X(A/Q) an • #X(A K /Q) an • 2 (bounded exponent) and the formula deduced from [10]; here, the last two factors are integral.In the computation of #X(A K /Q) an , we use van Bommel's code to compute the Tamagawa numbers of A/Q and A K /Q and the real period of A K /Q.In the cases where his code did not succeed, we used another Heegner discriminant.(5) Isogeny descent.In the cases when p is odd and ρ p is reducible, we determined characters χ 1 and χ 2 such that ) above.We then compute upper bounds for the F p -dimensions of the two Selmer groups associated to the corresponding two isogenies of degree p whose composition is multiplication by a generator π of p on A; see [16].From this, we deduce an upper bound for the dimension of the π-Selmer group of A, which, in the cases considered here, is always ≤ 1.
Using the known finiteness of X(A/Q), which implies that X(A/Q)[p] has even dimension, this shows that X(A/Q)[p] = 0. (6) Computation of the p-adic L-function.We can also compute the p-adic Lfunctions of newforms of weight 2, trivial character and arbitrary coefficient ring for p 2 ∤ N .Computing ord p L * p (f, 0) and using the known results [18,17] about the GL 2 Iwasawa Main Conjecture (IMC) with the hypotheses

( 2 )
If there exists a polarization λ : A → A ∨ , X(A/Q)[p] = 0 for all prime ideals p | p = 2 such that ρ p is irreducible and p does not divide deg λ, and, for some Heegner field K with Heegner discriminant D, I D and the order of the groups H 1 (K nr v |K v , A) with v running through the places of K. Proof.(1) is [4, Theorems C and D]. (