Mordell Surplus in Diophantine Geometry

• Mordell surplus is defined as the difference between the total number of integral or rational points on a curve and a natural benchmark based on absolute parameters or the Mordell–Weil rank.
• It connects classical Mordell curves and higher genus curves through explicit classifications using binary cubic forms, class number estimates, and computational verifications.
• Recent analyses provide explicit upper and lower bounds and extremal classifications, demonstrating the surplus's role in quantifying unexpected point abundances in Diophantine equations.

The Mordell surplus quantifies the excess of integral points on an elliptic or higher genus curve over a natural combinatorial benchmark, forming a central object in the modern arithmetic paper of Diophantine equations. For the classical one-parameter family of Mordell curves $E_k: y^2 = x^3 + k$, where $k \in \mathbb{Z}$, the Mordell surplus is defined as $S(k) = N(k) - |k|$, with $N(k)$ denoting the number of integral solutions (including the point at infinity). For curves of general genus $g \geq 2$ over number fields, an analogous notion considers the difference between the total count of rational points and the Mordell–Weil rank of the Jacobian. This framework provides both a precise arithmetic measure of “unexpected” integral or rational points and a driver for explicit bounds, extremal classification, and uniformity results in arithmetic geometry (Anand, 30 Jun 2024, Dimitrov et al., 2020).

1. Definition of the Mordell Surplus

For a Mordell curve $E_k : y^2 = x^3 + k$, with $(x, y) \in \mathbb{Z}^2$, the Mordell surplus is

$S(k) = N(k) - |k|,$

where $N(k)$ counts all integral solutions, including the point at infinity. The “naïve” benchmark $|k|$ reflects the intuition that most Mordell curves should not admit significantly more solutions than their absolute parameter. The Mordell surplus, therefore, captures any overabundance of integral points relative to this expectation (Anand, 30 Jun 2024).

For higher genus curves $C/K$ over a number field $K$, with Jacobian $\operatorname{Jac}(C)$ of Mordell–Weil rank $r$, the surplus is analogously defined by

$S(C, K) := \#C(K) - r,$

with $\#C(K)$ the number of rational points on $C$. This definition generalizes the Mordell surplus by benchmarking against algebraic group complexity rather than a simple parameter (Dimitrov et al., 2020).

2. Complete Classification of Extremal Cases

The extremal scenario $N(k) = |k|$ for Mordell curves admits a complete classification: the only integers $k$ (including the point at infinity) satisfying this identity are

$k = 3,\, 8,\, 17.$

Excluding the point at infinity, there are precisely four $k$ for which $N(k) = |k|$: $k = -1,\ -2,\ -4,\ 2.$ Among all possible quadratic twists, the unique case with $N(k) = 2|k|$ arises for $k = -1$ (including the point at infinity). The Diophantine argument yielding these classifications invokes the correspondence between integral Mordell curve points and binary cubic forms of discriminant $-108k$, together with explicit class number bounds and computational verification up to $|k| \leq 119$. This finiteness is a consequence of elementary class number estimates and a finite search (Anand, 30 Jun 2024).

3. Explicit Upper and Lower Bounds for Mordell Surplus

Upper and lower bounds for $N(k)$ and thus the Mordell surplus are fundamentally tied to the Mordell–Weil rank and local arithmetic invariants:

Bound Name	Upper Bound Expression	Dependence on Rank
Bhargava–Shankar–Taniguchi–Thorne–Tsimerman–Zhao	$$N(k) = O_\varepsilon(|\mathrm{Disc}(E_k)|^{0.1117 + \varepsilon})$$, $\mathrm{Disc}(E_k) = -432k^2$	Indirect via discriminant
Helfgott–Venkatesh	$$N(k) \le e^{O(\omega(k))} 1.33^{\operatorname{rank}(E_k)} (\ln |k|)^2$$	Explicit in the exponent
Alpöge–Ho (second moment)	$$N(k) \ll 2^{\operatorname{rank}(E_k)} \prod_{p^2\mid k^2} \min\left(4\left\lfloor\tfrac{\nu_p(k^2)}{2}\right\rfloor+1,\,7^{2^7}\right)$$	Explicit in the exponent

The dependence on the Mordell–Weil rank is central: higher rank admits potentially much larger Mordell surplus. The binary cubic forms correspondence guarantees an upper bound $N(k) \le 10 h_3(-108 k)$, with $h_3$ the 3-part of the class number of $\mathbb{Q}(\sqrt{-3k})$. Lower bounds are provided by results of Silverman: for suitable cubic forms $F$ of rank $r$,

$N_F(m) > c \log(m)^{r/(r+2)}$

for infinitely many $m$. Elkies exhibited 3-isogenous Mordell curves of rank 17 where

$N_F(m) > c (\ln m)^{17/19}$

for infinitely many $m$. This suggests that unbounded rank could, in principle, drive the Mordell surplus arbitrarily high for selected parameter values, although average-order results (such as Duke’s theorem) show that the mean surplus remains small over large families (Anand, 30 Jun 2024).

4. Mordell Surplus in Parametric Families and Twists

For a fixed elliptic curve $E : y^2 = x^3 + Ax + B$ over $\mathbb{Q}$ with real period $\Omega_E$ and discriminant $\Delta$, consider the quadratic twists

$E_n : y^2 = x^3 + n^2 A x + n^3 B.$

Define

$$\nu_E(n) = \#\{ (x, y) \in \mathbb{Z}^2 : y^2 = x^3 + n^2 A x + n^3 B,\, \gcd(x, n) = 1,\, e_1 \le x/n \le e_2 \}$$

where $e_1 < e_2 < e_3$ are the real roots of $x^3 + Ax + B$. Duke’s theorem states that the Dirichlet series

$\sum_{n \ge 1} \nu_E(n) n^{-s}$

has a simple pole at $s = \frac{3}{2}$, and

$$\lim_{N \to \infty} \frac{1}{\sqrt{N}} \sum_{n \le N} \nu_E(n) = \frac{3 \Delta \Omega_E}{2\pi^2 \psi(\Delta)} h_E,$$

where $$\psi(\Delta) = \Delta \prod_{p \mid \Delta} (1 + 1/p)$$, and $h_E$ is a Hurwitz class number. For Legendre-normal-form curves, this yields an explicit bound involving the logarithmic mean and known transcendental estimates (Anand, 30 Jun 2024).

5. Binary Cubic Forms and the Integral Point Correspondence

Integral points on $E_k$ are in bijection with the set of binary cubic forms of discriminant $-108k$, up to $SL_2(\mathbb{Z})$-equivalence. Precisely, solutions are triples $(F, x, y)$ where $F(x, y) = 1$ and $\operatorname{Disc}(F) = -108k$. This correspondence permits the transfer of class number bounds to integral point counting and underlies the sharp classification and bounding techniques for Mordell surplus (Anand, 30 Jun 2024).

6. Uniformity and Mordell Surplus in Higher Genus Curves

For smooth, projective curves $C/K$ of genus $g \geq 2$ over number fields $K$ of degree $d$, there exists an explicit three-parameter uniform bound: $\#C(K) \le c(g, d)^{1 + r},$ where $r = \operatorname{rank} \operatorname{Jac}(C)(K)$ and $c(g, d)$ depends only on $g$ and $d$ (Dimitrov et al., 2020). Consequently, for the surplus

$S(C, K) \le c(g, d)^{1 + r},$

which is exponential in rank, and polynomial or less in $g$ and $d$. This demonstrates that, even in the higher-genus setting, the Mordell surplus cannot be bounded purely in terms of geometric invariants without explicit control over the Mordell–Weil rank. Previous one-parameter family results, giving bounds of the form $C(g) 7^r$, are subsumed under this more general, uniform approach.

7. Significance and Open Directions

The Mordell surplus encapsulates both the exceptional and typical behavior of integral or rational points on algebraic curves and their families. Its explicit connections to class numbers, ranks, and moduli data bridge arithmetic geometry, analytic number theory, and arithmetic statistics. For Mordell curves and twists, record examples with large rank yield the greatest known surpluses, but for the majority of curves, the surplus is routinely small. The exponential dependence on the Mordell–Weil rank in all major upper bounds is a dominant theme, suggesting that potential progress on uniformity or effective Mordell–Weil rank bounds would immediately impact surplus estimates. Conversely, understanding the distribution and extremal behavior of Mordell surplus values in families remains an active area, fundamentally connected to the arithmetic of ranks and the finer structure of the moduli of curves (Anand, 30 Jun 2024, Dimitrov et al., 2020).