On the $abc$ conjecture

Abstract: Using adelic intersection theory and following the method envisioned by Parshin and Moret-Bailly, we prove a conjecture of Vojta for curves of genus two and a weak form of the $abc$ conjecture

• The paper derives an effective discriminante dependent height bound for rational points on genus 2 curves over number fields.
• It establishes a uniform exponent $24+ε$ for the $abc$ inequality under specific conditions, aligning with a weaker form of Vojta's conjecture.
• The work also uses a genus-six Kodaira fibration and Arakelov-intersection theory to control heights in genus-two curves and their applications.

Main objective and claimed results

The paper studies the arithmetic of rational points on genus-two curves through Arakelov intersection theory, with the explicit aim of deriving quantitative consequences for the $abc$ conjecture. Its principal geometric statement is an effective discriminant-dependent height bound for rational points on any fixed smooth, projective, geometrically irreducible curve of genus $2$ over a number field $K_0$. For every $\varepsilon>0$, the paper claims the existence of a constant $\beta(\varepsilon)$ such that, for every finite extension $F/K_0$ and every $P\in\mathcal C(F)$,

$$h(P)\leq (12+\varepsilon)\frac{\log|\Delta_F|}{[F:\mathbb Q]}+\beta(\varepsilon).$$

Here $h$ is a logarithmic height associated with the canonical bundle of $\mathcal C$. The result is presented as an effective form of Mordell’s conjecture for genus-two curves and as a weak form of a conjecture of Vojta. The coefficient $12$ is the key numerical output: the paper does not obtain the coefficient $1$ suggested by the corresponding Vojta prediction, but it does obtain an explicit uniform dependence on the normalized discriminant.

The paper then applies this estimate to obtain a weak uniform form of the $abc$ inequality. If $a,b,c$ are coprime positive integers satisfying $a+b=c$, and $N$ is the product of the primes dividing $abc$, it claims

$c\leq A(\varepsilon)N^{24+\varepsilon}.$

Equivalently, the resulting exponent in the logarithmic height formulation is $24+\varepsilon$. This is substantially weaker than the conjectural exponent $1+\varepsilon$, but it is claimed to follow unconditionally from the genus-two height estimate. More generally, the paper establishes a number-field version involving the height $H(a,b,c)$, the bad-prime term $N(a,b,c)$, and the normalized discriminant of the ground field:

$$\log H(a,b,c) \leq (24+\varepsilon)\log N(a,b,c) +(30+\varepsilon)\frac{\log|\Delta_K|}{[K:\mathbb Q]} +\beta_3(\varepsilon).$$

The paper also derives a discriminant-conductor inequality for semistable elliptic curves over $\mathbb Q$ and uses the resulting weak uniform $abc$ estimate to claim a zero-free region for quadratic Dirichlet $L$-functions.

Arakelov-theoretic framework

The argument is formulated in the language of adelic line bundles and arithmetic intersection theory. For an adelically metrized line bundle $\overline L$ on a smooth projective curve, the paper uses the arithmetic intersection pairing

$$\overline L\cdot \overline M = \widehat{\deg}_K\langle\overline L,\overline M\rangle.$$

Several foundational ingredients are established or recalled. First, the paper proves that sup norms on spaces of global sections of adelic line bundles define adelic vector-space structures. At almost all finite places this is obtained from a coherent integral model, and the key point is that the supremum norm agrees with the model norm. This permits the use of successive-minima estimates for adelic vector spaces.

Second, the paper proves a comparison principle for metrized line bundles of degree zero on a curve. If $\widehat x$ is an adelic Picard class whose underlying algebraic class has degree zero, then for an ample metrized line bundle $\overline M$ and every $\varepsilon>0$,

$$\left| \widehat{\deg}_F(u^*\widehat x) \right| \leq \varepsilon\widehat{\deg}_F(u^*\overline M)+O([F:\mathbb Q]).$$

This estimate is used repeatedly to replace one choice of metric by another while controlling the error by an ample height. Its role is essential because the main construction naturally produces several adelic metrics, whereas the height estimate must ultimately be independent of those choices.

A further ingredient is an intersection inequality for two ample line bundles with admissible metrics. If $s_i$ are global sections with disjoint divisors, then

$$-\overline L_1\cdot\overline L_2 \leq \deg(L_2)h(s_1)+\deg(L_1)h(s_2).$$

The proof uses the Deligne pairing and the positivity of admissible metrics. Locally, the arithmetic intersection pairing is expressed through the divisor of one section and the curvature measure of the other. Summing the resulting inequalities over all places gives the global bound. This is the mechanism that converts small adelic sections into an upper bound for an arithmetic self-intersection number.

The genus-six Kodaira fibration

The central geometric construction begins with a genus-two curve $\mathcal C$. After a finite extension $K/K_0$, the paper constructs a finite étale cover

$Y\longrightarrow \mathcal C_K$

and a non-isotrivial smooth proper morphism

$\phi:X\longrightarrow Y$

whose fibers are curves of genus $6$. The construction is a Kodaira–Parshin-type fibration. It proceeds through a degree-two étale cover $Y_1\to Y_0$, where $Y_0=\mathcal C_K$, followed by a degree-two cover of $Y_1$ branched at two points. Since $Y_1$ has genus $3$, the Riemann–Hurwitz formula gives genus $6$ for the resulting fibers.

The paper analyzes the geometry of these fibers in detail. The genus-three curve $Y_1$ is shown to be hyperelliptic. For a Weierstrass point $W$ on $Y_1$, its canonical class satisfies

$4\mathcal O_{Y_1}(W)=\omega_{Y_1}.$

The paper distinguishes four exceptional points on $Y_1$. These are characterized by the condition that the two branch points of the genus-six double cover are exchanged by the hyperelliptic involution. This exceptional locus is finite and becomes a source of torsion contributions in the direct-image calculations later in the proof.

The stable pointed-curve formalism is used to extend the family over an arithmetic base after an alteration. Specifically, a stable two-pointed model exists after replacing a regular projective model of $Y$ by an alteration that is finite étale over the generic curve. The construction uses level structures on the moduli space of stable pointed curves. This extension is necessary because the later application of successive minima requires integral models and adelic metrics defined at every finite place.

Intersection computations

Let $B_1$ and $B_2$ denote the two sections of the genus-six fibration arising from the branch divisors, and let $\omega=\omega_{X/Y}$ be the relative dualizing sheaf. If $k=\deg(Y/Y_1)$, the paper computes

$B_1^2=B_2^2=-2k, \qquad \omega^2=12k.$

The last equality is particularly important. It follows from the relation

$\omega\simeq B_1+B_2+\phi_1^*(\omega_{Y_1})$

and the identity $\omega_{Y_1}\simeq 4W$ for a Weierstrass divisor $W$. The positivity of $\omega^2$ implies that the Deligne pairing $\langle\overline\omega,\overline\omega\rangle$ is algebraically ample on the base. This allows it to serve as the ample comparison class in the degree-zero metric estimates.

The paper then introduces

$B=2B_1+2B_2, \qquad L=\omega-B.$

The main numerical identities are

$$\deg(L)=6, \qquad B^2=-16k, \qquad \omega\cdot B=8k, \qquad L^2=-20k.$$

The fiberwise dimensions of sections are also sharply controlled:

• $h^0(B_{\overline y})=3$ and $h^0(L_{\overline y})=4$ at exceptional points;
• $h^0(B_{\overline y})=1$ and $h^0(L_{\overline y})=2$ elsewhere.

The jump in dimension at the finite exceptional locus produces torsion in $R^1\phi_*(B)$. Since there are $4k$ exceptional points and the torsion length is at least $2$ at each one, the paper obtains

$$\deg\bigl(R^1\phi_*(B)_{\mathrm{tor}}\bigr)\geq 8k.$$

Combining this with Riemann–Roch and the calculation

$\deg R\phi_*(B)=-11k,$

the paper obtains

$\deg\phi_*(L)\geq -3k.$

This lower bound is one of the decisive numerical inputs. It compensates for the negative classical self-intersection $L^2=-20k$ and permits the arithmetic self-intersection of the restricted bundle to be controlled by the discriminant of the field of definition.

The discriminant estimate for the relative dualizing sheaf

The main technical theorem asserts that, for every $\varepsilon>0$ and every $P\in Y(F)$,

$$\overline\omega_P^2 \leq (36+\varepsilon)\log|\Delta_F|+O([F:\mathbb Q]).$$

The metric on $\omega$ is any integrable adelic metric. The paper observes that if the estimate holds for one such metric, it holds for every other one: the difference of the two self-pairings has algebraic degree zero, so the preceding degree-zero comparison estimate controls the change by an arbitrarily small multiple of $\overline\omega_P^2$ plus a linear error in the field degree.

The proof begins by lifting a point $P\in Y(F)$ to a point $P'\in Y'(F')$ on an alteration used to obtain an integral stable model. The extension $F'/F$ has bounded degree and is unramified outside a fixed finite set of places. Consequently,

$$\frac{\log|\Delta_{F'}|}{[F':\mathbb Q]} \leq \frac{\log|\Delta_F|}{[F:\mathbb Q]}+O(1).$$

On the fiber $X_{P'}$, the bundle $L$ has degree $6$ and, outside a finite exceptional set, two sections can be chosen with no common zero. The intersection inequality therefore gives

$$-\overline L_{P'}^2 \leq 6\bigl(h(s_1)+h(s_2)\bigr).$$

Successive minima applied to the two-dimensional adelic space of sections yields

$$h(s_1)+h(s_2) \leq -\widehat{\deg}(H_{\sup}) +\log|\Delta_{F'}| +O([F':\mathbb Q]),$$

where $H$ is the fiber of $\phi_*(L)$. The comparison between the degree of $\phi_*(L)$ and $\overline\omega^2$ gives

$$-\overline L_{P'}^2 \leq \frac{3+\varepsilon}{2}\overline\omega_{P'}^2 +6\log|\Delta_{F'}| +O([F':\mathbb Q]).$$

On the other hand, since $L^2=-20k$ and $\omega^2=12k$,

$$\overline L_{P'}^2 \leq \left(-\frac53+\varepsilon\right)\overline\omega_{P'}^2 +O([F':\mathbb Q]).$$

Combining the two inequalities yields

$$\overline\omega_{P'}^2 \leq (36+\varepsilon)\log|\Delta_{F'}| +O([F':\mathbb Q]).$$

The exceptional fibers do not satisfy the generic base-point-free argument, but they form a finite set. Their contribution is absorbed into the linear error term. The projection formula then descends the estimate from $F'$ and $Y'$ to the original field and curve.

Effective Mordell for genus two

The discriminant estimate is transferred from $Y$ to the original genus-two curve $\mathcal C$. Let $M$ be the canonical bundle on $\mathcal C$, and let $L=\pi^*M$ on $Y$. The paper constructs the degree-zero adelic class

$$\widehat x = \overline L-\frac13\langle\overline\omega,\overline\omega\rangle.$$

The coefficient $1/3$ is forced by the intersection computation: the degree of $L$ equals one third of $\omega^2$. Applying the degree-zero comparison estimate gives

$$\widehat{\deg}(u^*\overline L) \leq \frac{1+\varepsilon}{3}\overline\omega_P^2 +O([F:\mathbb Q]).$$

Substitution of the $(36+\varepsilon)$ discriminant estimate produces the coefficient $12+\varepsilon$:

$$\widehat{\deg}(u^*\overline L) \leq (12+\varepsilon)\log|\Delta_F| +O([F:\mathbb Q]).$$

After comparison with a Weil height and descent through the finite étale cover, the paper obtains

$$h(P) \leq (12+\varepsilon) \frac{\log|\Delta_F|}{[F:\mathbb Q]} +\beta_1(\varepsilon).$$

The paper explicitly contrasts this with the coefficient $1+\varepsilon$ predicted by the cited form of Vojta’s conjecture. Thus the result is effective in its dependence on the discriminant but is not numerically optimal relative to the conjectural bound.

Consequences for discriminants and the $abc$ inequality

The effective genus-two estimate is applied to the modular curve $X_0(23)$, which has genus $2$ and whose $j$-map has degree $24$. This yields a discriminant-conductor estimate for semistable elliptic curves:

$$\log|\Delta(E)| \leq \bigl(144(1+\log 24)+\varepsilon\bigr)\log N(E) +\beta_2(\varepsilon).$$

The large coefficient reflects the degree of the modular parametrization and the coefficient $12$ in the genus-two height estimate. The result is explicitly weaker than the expected optimal discriminant-conductor relation, but it is uniform once $\varepsilon$ is fixed.

For the $abc$ application, the paper considers the genus-two curve

$y^2+y=x^5$

and the degree-five map to $\mathbb P^1$ given by the $y$-coordinate. Given $a+b=c$, it sets $y=b/a$ and introduces a degree-one-or-five extension by adjoining a fifth root of $bc/a^2$. The resulting point on the genus-two curve allows the height estimate to be applied. The discriminant of this extension is bounded in terms of the radical of $abc$:

$$\frac{\log|\Delta_F|}{[F:\mathbb Q]} \leq \frac45\log N(a,b,c) + \frac{\log|\Delta_K|}{[K:\mathbb Q]} +\log 5.$$

Because the genus-two estimate contributes a factor $30$ to the height of the degree-five coordinate, multiplication by $4/5$ yields the coefficient $24$ in the radical term. The resulting number-field inequality is

$$\log H(a,b,c) \leq (24+\varepsilon)\log N(a,b,c) + (30+\varepsilon)\frac{\log|\Delta_K|}{[K:\mathbb Q]} +\beta_3(\varepsilon).$$

For coprime positive integers over $\mathbb Q$, this implies

$c\leq A(\varepsilon)N^{24+\varepsilon}.$

The argument therefore establishes a genuine uniform radical bound, but it does not approach the conjectural exponent $1+\varepsilon$. The paper’s claim is best understood as a weak $abc$ theorem obtained from a quantitative genus-two Mordell estimate.

Consequence for quadratic $L$-functions

The final application uses the weak uniform $abc$ estimate to control the heights of parameters in CM elliptic curves. Applying the inequality to a suitable triple involving $\lambda$, $1-\lambda$, and $1$ gives a bound of the form

$H(\lambda,1-\lambda,1)\ll d^{16},$

where $-d$ is a negative fundamental discriminant. A degree-six rational map from the corresponding Legendre parameter to the $j$-invariant then gives

$H(j,1)\ll d^{96}.$

Combining this with the standard relation between the height of a CM $j$-invariant and the class number yields a lower bound

$h'\gg \frac{\sqrt d}{\log d}$

for a weighted class-number quantity. The cited analytic argument then implies that there is an absolute constant $c>0$ such that the quadratic Dirichlet $L$-function satisfies

$$L(s,\chi_d)\neq 0 \qquad\text{for}\qquad 1-\frac{c}{\log d}<s\leq 1.$$

The implication is conditional only on the paper’s preceding weak uniform $abc$ estimate and the external analytic results it invokes. The zero-free region has the standard Landau–Siegel shape, but the paper does not provide an explicit numerical value for $c$.

Limitations and open questions

The principal limitation is quantitative. The paper’s effective Mordell estimate has coefficient $12+\varepsilon$, whereas the cited Vojta prediction would give $1+\varepsilon$. This loss propagates to the exponent $24+\varepsilon$ in the $abc$ consequence. The paper does not identify which part of the construction is responsible for the full numerical loss, although the constants arise transparently from the genus-six fibration, the relation $\omega^2=12k$, the estimate on $\deg\phi_*(L)$, and the degree-five cover used in the final $abc$ argument.

The construction also requires several auxiliary finite extensions, alterations, integral models, and adelic metrics. The discriminant of the auxiliary field is controlled only up to an additive constant after normalization. This is sufficient for the asserted asymptotic inequalities, but it prevents the results from being sharp at the level of field-by-field constants.

Finally, the supplied text contains substantial typographical corruption and anonymized or missing attributions, including incomplete formulas and citation metadata. The mathematical exposition nevertheless makes the claimed chain of implications clear: arithmetic intersection estimates imply the relative dualizing-sheaf bound, which implies the genus-two height estimate, which in turn implies the weak $abc$ theorem and its stated analytic consequences. Whether every cited external theorem applies with precisely the stated normalization would require verification against the original sources.

Conclusion

The paper develops an Arakelov-theoretic proof strategy based on a genus-six Kodaira fibration over a finite étale cover of a genus-two curve. Its central claimed estimate is

$$\overline\omega_P^2 \leq (36+\varepsilon)\log|\Delta_F|+O([F:\mathbb Q]),$$

which yields the effective genus-two height bound with coefficient $12+\varepsilon$. From this, the paper derives a weak uniform $abc$ inequality with exponent $24+\varepsilon$, a discriminant-conductor estimate for elliptic curves, and a Landau–Siegel zero-free region for quadratic characters. The principal contribution is therefore a coherent geometric-arithmetic mechanism connecting Arakelov intersection theory on Kodaira fibrations with explicit, though quantitatively nonoptimal, Diophantine and analytic consequences (2611.05217).