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December 24, 2023: Article on the verification of strong BSD for many modular abelian surfaces over $\mathbf{Q}$

In our recent preprint “Complete verification of strong BSD for many modular abelian surfaces over $\mathbf{Q}$” Michael Stoll and I verify the strong Birch–Swinnerton-Dyer (BSD) conjecture for the first time for abelian surfaces over $\mathbf{Q}$ in cases where it cannot immediately be reduced to elliptic curves, i.e., such that the surface is absolutely simple.

Why this matters. The Birch–Swinnerton-Dyer conjecture is one of the Clay Millennium Problems. Its strong form predicts not only the rank but also the order of the mysterious Tate–Shafarevich group, and it had been verified only for elliptic curves and for abelian varieties that reduce to them. Ours is the first verification of strong BSD for absolutely simple abelian surfaces, genuinely two-dimensional examples, including the computation of the order of their Tate–Shafarevich groups, which was previously out of reach.

Recall the conjecture for $A/\mathbf{Q}$ a principally polarized abelian variety: $$L^*(A/\mathbf{Q},1) :=\lim_{s \to 1} (s-1)^{-r} L(A/\mathbf{Q}, s) = \frac{\Omega_A \prod_p c_p(A) \cdot \mathrm{Reg}_{A/\mathbf{Q}} \#\mathrm{Ш}(A/\mathbf{Q})}{(\#A(\mathbf{Q})_\mathrm{tors})^2} \,.$$

Assume the abelian surface is the Jacobian $J$ of a genus $2$ curve, with real multiplication, absolutely simple, and that the analytic rank of one of the newforms associated with it is $0$ or $1$. Then our Magma code can do the following:

1. Computation of the images of all residual Galois representations.

2. Computation of Heegner points and Heegner indices.

3. Formulas for the “analytic order of Sha”.

4. Euler system computations.

5. Isogeny descents.

6. Computations of approximations of $p$-adic $L$-functions.

7. Compute a finite set of primes for which one has to compute $\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}^\infty]$ to verify strong BSD. In practice, this set is small and often empty.

Assuming that one can compute the Mordell–Weil group of the abelian variety and heights, most of the algorithms generalize to modular Jacobians of any dimension.

Let me illustrate our algorithm with an example, the Jacobian of the genus $2$ curve 4225.a.274625.1 of level $N = 65 = 5\cdot13$ and rank $0$.

1. All residual Galois representations at primes not above $2$ or $3$ are irreducible.

2. For the Heegner discriminant $D = -51$, one gets the Heegner index $2^2\cdot3$.

3. The analytic order of Sha is $1$, as one can compute using modular symbols since the rank is $0$.

4. The Euler system of Kolyvagin–Logachëv in an explicit version says that $\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}] = 0$ for $\mathfrak{p}\nmid 2,3$: One always has to exclude the primes above $2$, and additionally those dividing the Tamagawa product and the gcd of some Heegner indices, as well as those such that the residual Galois representation is reducible.

5. For the torsion prime $\mathfrak{p}$ above $3$, an isogeny descent shows that $\mathrm{Ш}(J/\mathbf{Q})[\mathfrak{p}] = 0$.

6. No $p$-adic $L$-function is needed. In fact, for the “LMFDB examples”, we only use it for the curve with $N = 188$.

7. One easily computes $\mathrm{Ш}(J/\mathbf{Q})[2] = 0$ using existing algorithms in Magma. Hence $\#\mathrm{Ш}(J/\mathbf{Q}) = 1 = \#\mathrm{Ш}(J/\mathbf{Q})_\mathrm{an}$.

I’m currently running our algorithms on all absolutely simple modular Jacobians associated with RM newforms with real quadratic coefficients up to level $N \leq 1000$.

Update (October 10, 2024). Our article has been accepted for publication in Forum of Mathematics, Sigma.