Reliability of Genus 2 Curve Data (reviewed)

As described in Section 3.4 of [MR:3540958, arXiv:1602.03715], the completeness of the genus 2 curve database for curves of absolute discriminant $|\Delta|\le 10^6$ has been tested against other tables of genus 2 curves, including those of [Stoll], and Merriman and Smart [10.1017/S030500410007153X]. However, as explained on the completeness page, it is only complete within the boxes that were searched, and it is likely that there are genus 2 curves of minimal discriminant $|D|\le 10^6$ that are not included (even though no such curves are currently known).

The reliability of specific data associated to genus 2 curves is discussed below.

• In cases where the set of rational points has not been provably determined, this is indicated by the label known rational points. In cases where the set of rational points has been provably determined (via some variant of Chabauty's method implemented in Magma), this is indicated by the label Rational points; in all other cases lists of rational points are preceded by the caption Known rational points.

• The power of 2 in the conductor of the Jacobian (originally bounded analytically) has been rigorously verified by Tim Dokchitser and Christopher Doris [arXiv:1706.06162] using algebraic methods (the odd part of the conductor has also been rigorously computed).

• All L-function computations are conditional on the assumption that the L-function satisfies the Hasse-Weil conjecture (which implies that it lies in the Selberg class), and in particular, that it has a meromorphic continuation to $\C$ with no poles in the critical strip and satisfies a functional equation). This assumption is necessary in order for the analytic rank to be well-defined. In some cases (curves whose Jacobians are of $\GL_2$-type for example) the Hasse-Weil conejecture is known to hold, and this is indicated on curve's home page. The assumption of the Hasse-Weil conjecture is also used when computing bad Euler factors at 2.

• In addition to the assumption that the Hasse-Weil conjecture holds, analytic ranks greater than 1 are generally only upper bounds that are believed to be equal to the true analytic rank (this can fail to be true only if there is a zero very close to but not at the critical point). This is explicitly indicated on the home page of each curve by a parenthetical "(upper bound"), but these upper bounds are treated as exact for the purpose of searching. In cases where the Hasse-Weil conjecture is known the analytic rank is provably correct if it is equal to 0 or 1 (our analytic rank bounds are all compatible with the sign in the functional equation). There are some cases where analytic ranks greater than 1 are provably correct because the Jacobian is isogenous to a product of elliptic curves whose analytic ranks are known (in such cases the "(upper bound)" notation is removed).

• The data on the geometric endomorphism ring that was initially computed heuristically has now been rigorously certified by Davide Lombardo [arXiv:1610.09674] and by Edgar Costa, Nicolas Mascot, Jeroen Sijsling, and John Voight [arXiv:1705.09248], independently, by different methods. This rigorously confirms the Sato-Tate group computations (the fact that these independent computation agree is a consistency check).

• Isogeny class identifications are based on a comparison of Euler factors at good primes up to $2^{20}$. Jacobians that are not identified as members of the same isogeny class are provably non-isogenous, but the identification of membership within a particular isogeny class is heuristic (except for Richelot isogenies, no explicit isogenies have been computed). In principle it could be made rigorous in any particular case via a Faltings-Serre argument, as described in [arXiv:1805.10873], but this has only been done for a handful of cases such as the isogeny class of conductor 277.

• Mordell-Weil generators and regulators are conditional on BSD in cases where the Mordell-Weil rank is marked as a lower bound (in all such cases it is equal to our computed upper bound on the analytic rank).

• All values of the analytic order of sha are accurate to at least six decimal places, and under BSD exactly equal to the rounded integer displayed.

All invariants not specifically mentioned above were computed using rigorous algorithms that do not depend on any unproved hypotheses.