Szpiro's conjecture

Szpiro's conjecture

Definitions and mathematical statement

For an elliptic curve $E/{\mathbb{Q}}$, $\Delta_E\in{\mathbb{Z}}\setminus\{0\}$ is the discriminant of a global minimal Weierstrass equation. Its conductor $N_E=\prod_p p^{f_p}$ records the local conductor exponents of its Tate-module representation. The conjecture is \[ \forall\varepsilon>0\ \exists C_\varepsilon>0\ \forall E/{\mathbb{Q}}:\qquad |\Delta_E|\le C_\varepsilon N_E^{6+\varepsilon}. \] The constant is independent of the curve. The selected exponent is the classical $6+\varepsilon$ conductor--discriminant bound, not a differently normalized modified Szpiro assertion.

Formulation and concept references: [S1].

Short English statement

Can every elliptic curve's minimal discriminant be bounded uniformly by essentially the sixth power of its conductor?

Research attempt

Current frontier.

Here $\operatorname{num}(q)$ and $\operatorname{den}(q)$ denote the absolute numerator and positive denominator in reduced form, following [S1]. Conditions involving $\log\operatorname{num}(j_E)$ below apply only where that expression is defined; $j_E=0$ is covered separately by the integral-$j$ case. Pasten's small-denominator theorem gives $|\Delta_E|\leq256A N_E^6\log N_E$ when $\operatorname{den}(j_E)\leq A\log\operatorname{num}(j_E)$, with $A$ fixed. Its local input also yields $|\Delta_E|\leq N_E^5$ for integral $j_E$. These are uniform results within their stated subclasses, not a bound for all rational $j$-invariants. [S1]

A newer manuscript, posted on 15 September 2026, states the general estimate $\log|\Delta_E|\ll N_E\log\log N_E$ and the estimate $\log|\Delta_E|\ll_S N_E$ for curves semistable outside a fixed finite set $S$. These are recent preprint results, not independently audited here. Even if accepted, they remain far from the selected bound: its logarithmic form is $\log|\Delta_E|\leq(6+\varepsilon)\log N_E+O_\varepsilon(1)$. In particular, a bound with an unspecified exponent on $N_E$ would not settle this exact entry. [E1]

Attack route.

Three routes are tested. One can try to control the denominator of $j$ using its radical, but the numerator and other bad primes may be essential. One can seek counterexamples among quadratic twists, where minimal discriminants and conductors are tractable. Finally, local reduction data may isolate the part of the denominator that must be paid for by other bad primes. I use the third route to formulate a uniform sufficient estimate, and use the first two as adversarial tests. All constants introduced below are independent of the curve unless a subscript explicitly says otherwise.

Attempt.

1. Keeping the three local types separate. Write $\delta_p=v_p(\Delta_E)$, $f_p=v_p(N_E)$ and $B=\operatorname{den}(j_E)$. At $j_E=0$, take $B=1$ and interpret $v_p(j_E)=+\infty$. Partition the bad primes into the following disjoint sets: \[ \mathcal A=\{p:v_p(j_E)\geq0\},\quad \mathcal M=\{p:v_p(j_E)<0,\ E\text{ multiplicative at }p\}, \] \[ \mathcal V=\{p:v_p(j_E)<0,\ E\text{ additive at }p\}. \] Define the conductor factors \begin{equation}\label{wsref-0-39-34-30-30-30-39-30-3a-66-61-63-74-6f-72-73} K=\prod_{p\in\mathcal A}p^{f_p},\qquad U=\prod_{p\in\mathcal M}p,\qquad V=\prod_{p\in\mathcal V}p^{f_p}. \end{equation} Thus $N_E=KUV$, and the factors have disjoint prime support. Good primes cannot occur in $B$. The local estimates proved in Section 3 of [E2] are \begin{align} p\in\mathcal A &: \quad\delta_p\leq5f_p,\nonumber\\ p\in\mathcal M &: \quad f_p=1,\quad\delta_p=-v_p(j_E),\label{wsref-0-39-34-30-30-30-39-30-3a-6c-6f-63-61-6c}\\ p\in\mathcal V &: \quad \delta_p\leq3f_p-v_p(j_E)+8\,\mathbf1_{p=2}.\nonumber \end{align} The last term is retained: treating the prime $2$ as a tame prime would be an unjustified simplification.

Multiplying (9400090:local), without replacing each conductor factor by $N_E$, proves \begin{equation}\label{wsref-0-39-34-30-30-30-39-30-3a-77-65-69-67-68-74-65-64} |\Delta_E|\leq 256\,B K^5V^3, \qquad \frac{|\Delta_E|}{N_E^{6+\varepsilon}} \leq256\frac{B}{U^6V^3K\,N_E^\varepsilon}. \end{equation} This is the useful bookkeeping lemma of the attempt. In the tame potentially multiplicative case $I_n^*$, one has $\delta_p=n+6$, $f_p=2$, and $-v_p(j_E)=n$; hence \[ p^{\delta_p-6f_p}=p^{n-6}. \] For multiplicative type $I_n$ the same expression is $p^{n-6}$. The extra discriminant power from the star is exactly cancelled by the extra conductor power at exponent six. This explains why counting only the number of multiplicative primes loses important structure. [S1]

2. A sufficient global arithmetic estimate. Consider the following statement, which is not proved here:

For every $\varepsilon>0$ there is $A_\varepsilon$ such that, for every elliptic curve over ${\mathbb{Q}}$, \begin{equation}\label{wsref-0-39-34-30-30-30-39-30-3a-6d-69-73-73-69-6e-67} \operatorname{den}(j_E)\leq A_\varepsilon U^6V^3K\,N_E^\varepsilon, \end{equation} with $K,U,V$ defined by (9400090:factors).

Combining it with (9400090:weighted) would give the original conjecture, with $C_\varepsilon=256A_\varepsilon$. This is a stronger sufficient arithmetic statement, not an asserted equivalent formulation: the first line of (9400090:local) can have substantial slack. Any proposed counterexample to (9400090:missing) must still be checked against the actual minimal discriminant before being called a counterexample to Szpiro.

Conversely, an unbounded sequence of the original Szpiro ratios necessarily has unbounded right-hand ratios in (9400090:weighted). The latter are therefore a legitimate filter for a counterexample search, although not a certificate of success. This distinction prevents an upper-bound estimate from being used backwards.

3. Why a denominator-only radical bound is false. A tempting simplification would be \begin{equation}\label{wsref-0-39-34-30-30-30-39-30-3a-66-61-6c-73-65} B\leq C_\varepsilon\operatorname{rad}(B)^{6+\varepsilon} \quad\text{for every realized rational $j$-invariant.} \end{equation} It fails decisively. Here is an explicit realization, rather than an appeal to a moduli-space existence statement. For coprime integers $u,v$ with $v>0$, $u\neq0$ and $w=u-1728v\neq0$, set \[ E_{u,v}:\quad y^2=x^3-3uwx-2uw^2. \] A direct invariant calculation gives \begin{align} c_4&=144uw, &c_6&=1728uw^2,\ \Delta&=1728^2vu^2w^3, &j&=\frac uv. \label{wsref-0-39-34-30-30-30-39-30-3a-6a-6d-6f-64-65-6c} \end{align} Indeed $c_4^3-c_6^2=1728\Delta$, and the displayed nonzero discriminant ensures nonsingularity. This is an integral model, not necessarily a minimal model; only its $j$-invariant is used in the present argument.

Take $u=1$, $v=q^m$, with a fixed prime $q\geq5$. Then $B=q^m$ while $\operatorname{rad}(B)=q$. Letting $m\to\infty$ disproves (9400090:false) for every proposed fixed constant. It does not disprove Szpiro: primes dividing $1-1728q^m$ also enter the displayed model, and their contributions to the minimal conductor cannot simply be erased. This family pinpoints the missing interaction between the denominator and other bad primes in (9400090:missing).

4. Counterexample search through quadratic twisting. Fix an elliptic curve $E/{\mathbb{Q}}$. Let $d$ be a positive squarefree integer with \[ d\equiv1\pmod{24N_E}. \] In particular $d$ is coprime to $6N_E$ and is a local square at $2$, $3$ and all bad primes of $E$: use the square criterion modulo $8$ at $2$ and Hensel lifting at odd primes. Let $E^{(d)}$ be the quadratic twist, with short-model coefficients multiplied by $d^2,d^3$. Then \begin{equation}\label{wsref-0-39-34-30-30-30-39-30-3a-74-77-69-73-74} |\Delta_{E^{(d)}}|=d^6|\Delta_E|, \qquad N_{E^{(d)}}=d^2N_E. \end{equation}

Local verification. At the old bad primes the curves are locally isomorphic, so their local minimal discriminants and conductors agree. The same holds at $2$ and $3$. At a prime $p\mid d$, necessarily $p\geq5$, start with a minimal short equation for $E$ with unit discriminant. The twist has discriminant valuation $6$. It is minimal because lowering that valuation would require subtracting a positive multiple of $12$ from it. The original Tate-module representation is unramified; after the ramified quadratic twist, tame inertia acts by $-I$. Its invariant space has dimension zero and its Swan conductor is zero, so its conductor exponent is $2$. All remaining primes stay good. This proves both equalities in (9400090:twist). The discriminant-scaling and conductor conventions are those in the source's local discussion. [S1]

It follows exactly, not just asymptotically, that \begin{equation}\label{wsref-0-39-34-30-30-30-39-30-3a-74-77-69-73-74-2d-72-61-74-69-6f} \frac{|\Delta_{E^{(d)}}|}{N_{E^{(d)}}^{6+\varepsilon}} =\frac{|\Delta_E|}{N_E^{6+\varepsilon}}\,d^{-6-2\varepsilon}. \end{equation} Thus this natural unbounded twist search moves away from a counterexample. In (9400090:factors), these new primes instead multiply $K$ by $d^2$ while leaving $B,U,V$ unchanged. The sufficient-estimate ratio in (9400090:weighted) falls only as $d^{-2-2\varepsilon}$, displaying explicitly the slack between that criterion and the actual Szpiro ratio. A constant depending on the initially fixed curve cannot be promoted to a constant uniform over all curves.

5. Combining the small-denominator route with the recent height input. There is a modest, conditional sharpening that can be made without pretending to solve the denominator problem. The local divisibility bound $|\Delta_E|\leq16 B N_E^5$ and the comparison $h(j_E)\ll1+h(E)$ are given in [S1]. If one accepts the new preprint's estimate $h(E)\ll N_E\log\log(3N_E)$, then for fixed $A,\beta>0$, \[ B\leq A(\log\operatorname{num}(j_E))^\beta \quad\Longrightarrow\quad |\Delta_E|\ll_\beta A\, N_E^{5+\beta}\left(\log\log(3N_E)\right)^\beta. \] This follows simply by bounding $\log\operatorname{num}(j_E)$ by $h(j_E)$ and substituting. At $\beta=1$ the logarithmic loss in the older subclass bound becomes an iterated logarithm. The dependence on the recent input is explicit; no independent verification of that input, and no novelty claim for this combination, is made. [E1]

Stress test.

The argument has kept minimal discriminants distinct from discriminants of convenient equations; (9400090:jmodel) is used only to realize $j$. The wild-prime allowance in (9400090:local) is not absorbed into an unmentioned curve-dependent constant. The twisting calculation imposes a congruence that makes the local behavior at the old bad primes explicit. All powers of $N_E$ in (9400090:weighted) have been accounted for before formulating the missing estimate.

Neither a finite set of large Szpiro ratios nor the false estimate (9400090:false) provides a counterexample to the quantified conjecture: for a fixed $\varepsilon$ one needs ratios unbounded over curves. The symbolic checks in the companion script verify (9400090:jmodel) and the exponent cancellations. They do not compute minimal models or test (9400090:missing) over all elliptic curves. In particular, no invocation of an unproved $abc$ or modified Szpiro statement has been hidden in the local-to-global multiplication.

Outcome.

Outcome: Conditional reduction.

The proved result of this attempt is the weighted inequality (9400090:weighted), obtained by retaining the distinct local conductor factors, together with the explicit failure of the denominator-only radical lemma and the exact negative result for the specified twist search. Estimate (9400090:missing), uniformly for every curve and every $\varepsilon>0$, would imply the exact classical Szpiro statement. It is not established, and is stronger than the original conjecture at some local types. The recent-height combination concerns only a small-denominator subclass. No complete proof or counterexample candidate is claimed.

Sources

• S1 H. Pasten, Szpiro's conjecture when the denominator of the j-invariant is small, arXiv v2 (15 August 2023). https://arxiv.org/html/2308.07114. Catalogue formulation source.
• E1 H. Pasten, Improved bounds for Szpiro's conjecture, arXiv:2609.17390v1 (15 September 2026), Theorem 1.1, Corollaries 1.2--1.3, and equation $1.1$. https://arxiv.org/html/2609.17390v1. Consulted 27 September 2026; recent preprint, not independently proof-audited here.
• E2 H. Pasten, Szpiro's conjecture when the denominator of the $j$-invariant is small, arXiv:2308.07114v2 (15 August 2023), Theorem 1.2, Corollary 1.3, Section 2, Lemmas 3.1--3.3 and Corollary 3.4. https://arxiv.org/html/2308.07114v2. Same work as [S1]; this added locator records the exact results checked on 27 September 2026. The HTML's later generated date is not treated as a new arXiv revision.