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Szpiro's conjecture

Szpiro's conjecture is a statement in and arithmetic geometry that provides a uniform bound on the size of the minimal of an defined over the rational numbers in terms of its . Formulated by the French mathematician Lucien Szpiro in , the asserts that for every \epsilon > 0, there exists a constant \kappa(\epsilon) > 0 such that if E is an over \mathbb{Q} with N(E) and minimal \Delta(E), then |\Delta(E)| < \kappa(\epsilon) \, N(E)^{6 + \epsilon}. This inequality implies bounds on the exponents of primes dividing the in terms of the conductor, reflecting deep arithmetic properties of . In a slightly modified form—replacing N(E)^{6 + \epsilon} with N(E)^6 \mathrm{rad}(N(E))^{\epsilon}, where \mathrm{rad} denotes the radical—Szpiro's conjecture is equivalent to the , a broader Diophantine inequality proposed in 1985 by and that relates the prime factors of triples of coprime integers a, b, and c = a + b. The equivalence arises from associating elliptic curves to abc triples via , where the discriminant and conductor encode the radical and size of the triple. Szpiro's conjecture predates the abc conjecture and originated from studies of elliptic curves in the context of modular forms and arithmetic geometry. The conjecture has profound implications for Diophantine analysis, including uniform boundedness results for the torsion subgroups of elliptic curves over number fields and effective versions of the Mordell conjecture. It has been described as one of the most important unsolved problems in the field, with partial progress achieved through connections to the Langlands program and modular forms, though a full proof remains elusive as of November 2025. Recent efforts, including claimed proofs via inter-universal Teichmüller theory by Shinichi Mochizuki and by Kirti Joshi, have sparked debate but lack universal acceptance among mathematicians.

Background and Formulation

Historical Development

Lucien Szpiro proposed his conjecture during a talk at a meeting of the German Mathematical Society in Hannover in 1982, building on earlier discussions from 1978 with fellow mathematicians. This proposal arose in the context of arithmetic geometry, where elliptic curves serve as central objects of study. Szpiro's motivation stemmed from efforts to establish uniform boundedness for the torsion subgroups of elliptic curves over number fields, seeking bounds that would hold independently of the specific curve. A key early influence was Andrew Ogg's formula, which provides a relationship between the conductor N and the discriminant \Delta of elliptic curves defined over the rationals, highlighting structural connections in their arithmetic invariants. In a 1983 manuscript, Szpiro elaborated on these ideas, formulating a preliminary version of the conjecture in terms of discriminants and conductors for elliptic curves over number fields. During the 1980s and 1990s, the conjecture evolved through contributions from Szpiro and collaborators like David Masser, who explored its implications within arithmetic geometry and connected it to broader Diophantine problems. These developments emphasized applications to effective versions of classical conjectures, such as the . A formal printed statement of the conjecture appeared in the literature by the 1990s, notably in works referencing Oesterlé's 1988 formulation.

Original Statement

Szpiro's conjecture concerns elliptic curves defined over the rational numbers \mathbb{Q}. For such an elliptic curve E, the minimal discriminant \Delta_{\min}(E) is the discriminant of a minimal Weierstrass model of E over \mathbb{Z}, which is unique up to isomorphism and has the property that |\Delta_{\min}(E)| is minimized among all integral Weierstrass models of E. The conductor N(E) of E is the positive integer \prod_p p^{f_p}, where the product runs over all primes p and f_p is the exponent at p: f_p = 0 if E has good reduction at p, f_p = 1 if E has multiplicative bad reduction at p, and f_p = 2 + \delta_p (with \delta_p \geq 0) if E has additive bad reduction at p. The original statement of the conjecture, proposed by Lucien Szpiro in the early 1980s, asserts that for every \varepsilon > 0, there exists a constant C_\varepsilon > 0 such that for all elliptic curves E over \mathbb{Q}, |\Delta_{\min}(E)| \leq C_\varepsilon \, N(E)^{6 + \varepsilon}. This inequality relates the arithmetic complexity of E, as measured by its , to the size of its minimal . The exponent 6 is conjectured to be optimal, meaning that no smaller exponent works uniformly for all E/\mathbb{Q}, and the \varepsilon allows for the bound to hold with a power arbitrarily close to 6. The optimality of the exponent 6 is supported by explicit constructions of elliptic curves where the Szpiro ratio \sigma(E) = \frac{\log |\Delta_{\min}(E)|}{\log N(E)} exceeds 6 and can be made arbitrarily close to 6 in certain families. Such examples demonstrate that the constant in the exponent cannot be reduced below 6 without violating the inequality for infinitely many curves.

Modified Statement

The modified statement of Szpiro's conjecture extends the original formulation to elliptic curves defined over arbitrary number fields K, incorporating ideals in the \mathcal{O}_K. For an elliptic curve E over K, the minimal discriminant is an ideal \Delta_{\min}(E) in \mathcal{O}_K, and the conductor is an ideal \mathfrak{f}(E) whose norm is denoted N(E) = N_{K/\mathbb{Q}}(\mathfrak{f}(E)). The radical \mathrm{rad}(\Delta_{\min}(E)) is defined as the product of the distinct prime ideals of \mathcal{O}_K dividing \Delta_{\min}(E), generalizing the notion of the square-free part of the discriminant. The core inequality of the modified conjecture asserts that there exists a constant C > 0 (depending only on K) such that \mathrm{rad}(\Delta_{\min}(E)) \leq C \cdot N(E)^6 for every E over K. This version replaces the absolute norm of the used in with the , providing a more precise control that aligns closely with the structure of the . Over the rationals \mathbb{Q}, this reduces to a special case where the ideals are principal and norms are absolute values. To quantify the relationship, the quality of an E over K is defined as q(E) = \frac{\log |\Delta_{\min}(E)|}{\log N(E)}, where |\Delta_{\min}(E)| denotes the absolute norm N_{K/\mathbb{Q}}(\Delta_{\min}(E)). The modified implies that q(E) \leq 6 + \varepsilon for any \varepsilon > 0 and sufficiently large N(E), with the constant depending on \varepsilon and K. This measure highlights exceptional curves where the discriminant grows slower than expected relative to the conductor. Szpiro introduced this refinement in the , shifting from the absolute to the of the to enhance compatibility with the while preserving the exponent 6. The resulting form facilitates equivalences and generalizations in arithmetic geometry.

Relation to Other Conjectures

Equivalence to abc Conjecture

The , proposed by David Masser and Joseph Oesterlé in 1985, posits that for any \epsilon > 0, there exists a constant \kappa(\epsilon) > 0 such that for all coprime positive integers a, b, c with a + b = c, we have c < \kappa(\epsilon) \cdot \mathrm{rad}(abc)^{1 + \epsilon}, where \mathrm{rad}(n) denotes the of n, the product of its distinct prime factors. This formulation captures a profound relationship between addition and the prime factors of integers, with the constant \kappa(\epsilon) depending only on \epsilon. The modified Szpiro conjecture, which refines the original statement by incorporating the conductor of s, asserts that for any \epsilon > 0, there exists a constant C(\epsilon) > 0 such that for every E over \mathbb{Q} with discriminant [\Delta(E)](/page/Delta) and conductor N(E), |\Delta(E)| < C(\epsilon) \cdot N(E)^{6 + \epsilon}. The abc conjecture and the modified Szpiro conjecture are deeply intertwined, with their equivalence established in the late 1980s through work by Masser, Oesterlé, and Lucien Szpiro. The proof of equivalence hinges on a bidirectional mapping between the conjectures using Frey–Hellegouarch curves. Given a primitive abc-triple (a, b, c) with a + b = c and \gcd(a, b, c) = 1, associate the elliptic curve E_{a,b}: y^2 = x(x - a)(x + b); its discriminant is \Delta(E_{a,b}) = 16 (abc)^2, and its conductor N(E_{a,b}) divides \mathrm{rad}(abc) up to a bounded power of 2. If the abc conjecture fails for some triple, then |\Delta(E_{a,b})| \gg N(E_{a,b})^{6 + \epsilon} for arbitrarily small \epsilon, yielding a counterexample to the modified . Conversely, a violation of the modified Szpiro conjecture by some elliptic curve E over \mathbb{Q} implies the existence of abc-triples violating the abc bound, via the uniform boundedness of certain torsion points and the ability to construct Frey-like curves from the minimal model of E, ensuring the conductor aligns closely with the radical of the associated triple. This mutual implication demonstrates that the two conjectures are logically equivalent, bridging arithmetic geometry and Diophantine equations.

Connections to Height Conjectures

Szpiro's conjecture establishes a bound on the ratio of the minimal discriminant \Delta_E to the conductor N_E of an elliptic curve E over \mathbb{Q}, specifically |\Delta_E| \leq C N_E^{6+\epsilon} for some constant C depending on \epsilon > 0. This bound has implications for height conjectures in arithmetic geometry, particularly through its connection to Lang's height conjecture, a special case of the broader Bombieri-Lang conjecture on rational points on varieties of general type. The Bombieri-Lang conjecture posits that for a variety of general type over a number field, the rational points are not Zariski dense and lie in a proper subvariety union a finite set; for elliptic curves, this manifests as height bounds ensuring that nontorsion rational points have canonical heights bounded below relative to the curve's height. Szpiro's conjecture implies such height bounds, yielding a uniform lower bound \hat{h}(P) \geq C_1 h(E) - C_2 for nontorsion points P on E, where h(E) is the (stable) Faltings height of E, thereby providing Szpiro-type ratios via control over the growth of heights of rational points. The connection extends to the Manin conjecture, which predicts asymptotic formulas for the number of rational points of bounded on projective varieties, such as \# \{ P \in E(\mathbb{Q}) : \hat{h}(P) \leq B \} \sim c B^2 (\log B)^{r-1} for an E of r, where the constant c depends on the of E, including factors involving the . Szpiro's conjecture influences these asymptotics by bounding the in terms of the , which regulates the growth of points and the leading constants in Manin's predictions for families of s. In particular, for elliptic curves ordered by , Szpiro ensures that the does not grow excessively relative to the conductor, thereby stabilizing the asymptotic behavior and preventing pathological growth in point counts. This linkage highlights how Szpiro controls the arithmetic complexity of elliptic curves in -based enumerations. Unconditional progress toward these height-Szpiro links includes results bounding the Faltings height in terms of the . For instance, Murty and Pasten established h(E) \ll N_E \log N_E for all elliptic curves E over \mathbb{Q}, improving prior bounds and tying value prime factors to elliptic curve s via modular forms and techniques. This subexponential bound advances the height conjecture h(E) \ll \log N_E (implied by Szpiro) by controlling the prime factors in discriminants and conductors, with applications to bounding the greatest prime factor of values of quadratic and cubic . In arithmetic statistics, recent work examines average behavior of the Szpiro ratio \sigma(E) = \log |\Delta_E| / \log N_E over families of elliptic curves ordered by height or conductor. Heuristics and partial results indicate that the average \sigma(E) is significantly less than 6, with almost all curves (e.g., those with trivial torsion) having \sigma(E) close to 1, far below the conjectured uniform bound. For curves with prescribed torsion, the expected average ratio varies but remains bounded well under 6, supporting Szpiro via statistical evidence from large-scale computations and analytic methods. These findings, up to 2024, including a result by Chan showing that almost all elliptic curves with prescribed torsion have Szpiro ratio close to the expected value, reinforce partial height-Szpiro links without assuming the full conjecture.

Mathematical Consequences

Implications for Elliptic Curves

Szpiro's conjecture implies a uniform bound on the order of the torsion of the Mordell-Weil group E(K)_{\mathrm{tors}} for elliptic curves E over number fields K. Specifically, the order is bounded by a constant depending only on the Szpiro \sigma_E and the [K:\mathbb{Q}], and under the conjecture, \sigma_E \leq 6 + \epsilon for any \epsilon > 0 and sufficiently large conductors, yielding a bound depending on the . This follows from lower bounds on the heights of non-torsion points, which limit the possible torsion structures. The conjecture also provides bounds on the Mordell-Weil rank r = \mathrm{rank}(E(\mathbb{Q})) through its control on canonical heights and descent methods. In particular, Szpiro's conjecture implies that the canonical height \hat{h}(P) of any non-torsion point P \in E(\mathbb{Q}) satisfies \hat{h}(P) \geq c(\sigma_E) \log |\Delta_E| , where c > 0 is an effective constant, providing an effective version of Lang's conjecture on minimal heights. Since the conjecture bounds \sigma_E and relates \log |\Delta_E| to \log N_E (the conductor), this yields lower bounds on generator heights, and combined with the in the height pairing matrix, descent procedures (such as 2-descent) constrain the possible . Furthermore, the conjecture controls the minimal Weierstrass models of elliptic curves by bounding the minimal discriminant in terms of the conductor. For an elliptic curve E/\mathbb{Q} with conductor N_E, the modified form of the conjecture states that |\Delta_E| \leq C(\epsilon) N_E^{6 + \epsilon} for any \epsilon > 0 and some constant C(\epsilon) > 0. This restricts the possible integer coefficients in the minimal model y^2 = x^3 + ax + b (with \Delta_E = -16(4a^3 + 27b^2)), as bounded |\Delta_E| implies bounded |a| and |b|. Consequently, the valuations v_p(\Delta_E) at primes p of bad reduction are limited relative to v_p(N_E) \leq 2, constraining the possible Kodaira symbols and thus the reduction types (e.g., no arbitrarily high multiplicative or additive reduction exponents). As an illustrative example, there are only finitely many elliptic curves over \mathbb{Q} with a given fixed conductor N up to \mathbb{Q}-isomorphism; Szpiro's conjecture provides an explicit bound on their number. The possible minimal discriminants are finite, and for each such discriminant, only finitely many minimal Weierstrass models exist with integer coefficients yielding that discriminant value. This finiteness holds despite the infinite families of twists, as twists generally alter the conductor.

Applications in Number Theory

Szpiro's conjecture, being equivalent to the , has significant implications for bounding ideal class groups in number fields. Specifically, a uniform version of the over number fields implies effective lower bounds on the class numbers of imaginary quadratic fields, showing that the class number h(-d) grows at least like \sqrt{d} \log d for discriminants -d, which rules out Siegel zeros for associated L-functions and provides quantitative control over the arithmetic structure of these fields. This connection arises from applying the abc bound to solutions of Diophantine equations derived from modular functions, yielding explicit estimates on the distribution of class groups. The conjecture also applies to generalizations of and superelliptic equations. In particular, it implies the finiteness of nontrivial primitive solutions to generalized Fermat equations of the form Ax^p + By^q = Cz^r where p, q, r are fixed integers satisfying $1/p + 1/q + 1/r < 1, as the abc bound limits the size of solutions x, y, z effectively. This extends to broader superelliptic equations like z^m = F(x, y) for polynomials F of fixed degree, where the conjecture ensures only finitely many coprime integer solutions beyond a computable threshold, facilitating the resolution of cases in the Fermat-Catalan conjecture. Furthermore, Szpiro's conjecture influences arithmetic statistics concerning the distribution of discriminants and conductors in families of elliptic curves. By positing a uniform bound on the Szpiro ratio \beta_E = \log |\Delta_E| / \log N_E \leq 6 + \epsilon for any \epsilon > 0, it constrains the tail behavior of this ratio across all curves, complementing unconditional results that show the average ratio approaches 1 for most curves ordered by conductor. This has implications for counting elliptic curves with bounded invariants and understanding their modular properties in large families. Recent developments as of 2025 link Szpiro's conjecture to Zsigmondy-type theorems in abc-fields, providing bounds on prime divisors in dynamical sequences. In fields where the holds, for rational maps \phi \in K(x) of degree greater than 1, the Zsigmondy set of iterates \{\phi^n(\alpha)\} is finite under conditions, ensuring prime divisors appear with explicit multiplicity bounds for sufficiently large n. For unicritical polynomials over \mathbb{Q} or imaginary quadratic fields, such divisors exist for n > D_1 \max(\log v(\phi), 0) + D_2, where constants depend on the map's , tying to the conjecture's prime distribution control.

Progress and Partial Results

Unconditional Bounds

Early efforts to establish unconditional bounds on the size of the minimal \Delta_E in terms of the conductor N_E for elliptic curves E over \mathbb{Q} date back to the 1960s. Using Ogg's formula relating the conductor exponent to the discriminant valuation and the number of components in the special , one can derive a crude upper bound of the form |\Delta_E| \ll N_E^{12}. This exponent reflects the worst-case growth from local contributions at primes of bad reduction, though the constant is large and the bound is far from optimal. Significant progress came in the with improvements to exponents slightly better than 8. In their 2020 study of elliptic curves with good outside the first six primes, Best and Matschke provided explicit computations supporting bounds approaching |\Delta_E| \ll N_E^{8 + \epsilon} for small \epsilon > 0 in restricted families, though the global unconditional exponent remained higher. These results leverage modular methods and systematic searches to refine the growth rate. The strongest unconditional bound as of 2025 is due to Murty and Pasten (2013), who proved that for any E over \mathbb{Q}, \log |\Delta_E| < 1.2 \, N_E \log N_E + 93. This exponential bound arises from combining modular forms with effective Diophantine approximation techniques, marking the first explicit upper bound applicable to all elliptic curves over \mathbb{Q}. Recent work by Cuevas Barrientos and Pasten (2025) extends this to one-parameter families of elliptic curves (arising from elliptic surfaces), achieving subexponential growth: \log |\Delta_n| \ll_\epsilon N_n^\epsilon for every \epsilon > 0, where N_n is the conductor at parameter n. This improves upon the earlier exponential bound in average cases over families by incorporating linear forms in logarithms. On average, results from Bhargava and Shankar (2015) imply bounded average Selmer ranks, which indirectly support controlled growth of discriminants when ordering curves by height, though not directly yielding a global exponent below 6 + 1/3 + \epsilon. Theorems bounding the largest prime factor P(\Delta_E) of the minimal discriminant relative to N_E follow from the above size bounds, yielding P(\Delta_E) < \exp(1.2 N_E \log N_E + 93) unconditionally. In families, the 2025 results provide stronger control, ensuring P(\Delta_n) \gg \exp(c \sqrt{\log N_n \log^*_2 N_n}) for some c > 0 as a lower bound on the radical, implying the largest prime is substantial relative to N_n. Szpiro's conjecture would sharpen this to P(\Delta_E) \ll N_E^\epsilon for any \epsilon > 0. Computational evidence reinforces these bounds. Extensive searches in databases like LMFDB, covering elliptic curves up to conductors exceeding $10^{10}, reveal no over \mathbb{Q} with Szpiro ratio q(E) = \log |\Delta_E| / \log N_E > 8.76. The largest known ratios occur for curves with good reduction outside small primes, such as those tabulated by Best and Matschke, confirming that while ratios exceed the conjectured bound of 6, they remain below 9 even for large conductors.

Conditional Results and Local Versions

Under the Generalized Riemann Hypothesis (GRH), effective versions of the Shafarevich theorem for elliptic curves provide bounds on the size of classes with given , leading to improvements in the exponent of Szpiro's conjecture. These effective finiteness results allow the Szpiro ratio to be bounded by an exponent of 6 + ε for any ε > 0, approaching the conjectured value of 6, by controlling the heights and discriminants of minimal models via p-adic methods and class number bounds. Local versions of Szpiro's conjecture focus on bounding the p-adic valuation of the minimal discriminant v_p(Δ_E) in terms of the local contribution to the conductor at p. A key result in this direction is due to Bennett and Yazdani, who proved that if E is an elliptic curve over ℚ with conductor N(E) = M p, where p is a prime and M is a fixed integer, and E admits a nontrivial rational isogeny of degree n > 1, then v_p(Δ_E) ≤ 6 provided p is sufficiently large relative to M. This establishes the local conjecture at p under the strong hypothesis of a large prime in the conductor and the presence of an isogeny, using and bounds on rational points on modular curves. In contrast to the number field case, the analogue of Szpiro's conjecture holds fully over function fields of curves over s. Szpiro proved this geometric version, showing that for an E over the function field k(C) of a smooth projective curve C over a k, the degree of the is bounded by 6 times the degree of the plus a constant depending only on the of C. This result relies on the of the moduli of elliptic curves and properties of the minimal in characteristic p. Bogomolov extended this to stable curves over function fields with only irreducible fibers, confirming the bound using Arakelov and equidistribution techniques. Recent work has explored partial local-global principles linking local Szpiro-type inequalities to the in p-adic settings. For instance, discussions on the global nature of abc/Szpiro inequalities indicate that while direct summation of local p-adic bounds does not yield the full , modified local inequalities at each prime—incorporating p-adic valuations—can imply abc under certain compatibility conditions across places. These insights, from 2024 analyses, highlight the interplay between local p-adic data and global but remain partial without full resolution.

Claimed Proofs and Controversies

Mochizuki's Inter-universal Teichmüller Theory

Shinichi Mochizuki introduced Inter-universal Teichmüller Theory (IUT) as an arithmetic analogue of classical Teichmüller theory, aimed at deforming and reconstructing arithmetic structures for number fields. The theory builds on , which leverages the rigidity of arithmetic fundamental groups to recover scheme-theoretic data from étale fundamental groups, allowing for the reconstruction of arithmetic objects without relying on explicit embeddings into larger fields. Central to IUT are Frobenius profiles, which describe canonical liftings in p-adic settings and involve multiradial structures on domains like the Θ-link, facilitating comparisons across different "universes" of arithmetic data. This inter-universal reconstruction occurs through "changes of universe," where log-links and Θ-links enable the transfer of information between distinct arithmetic holomorphic structures, treating Galois representations abstractly to estimate distortions in data. Mochizuki claims that IUT proves the —and by equivalence, a modified form of Szpiro's conjecture—by establishing sharp bounds on the radical of via log-volume computations in deformation spaces. Specifically, the constructs log-theta-lattices, which serve as central objects for encoding information, and uses them to control the growth of conductors and discriminants for elliptic curves over number fields. These lattices, combined with mono-analytic spaces and coric structures, allow for the deformation of initial data into "pilot" objects that yield Diophantine inequalities bounding the size of rad() relative to the product . The proof culminates in 4.2 of the theory's fourth manuscript, asserting the validity of the abc/Szpiro inequalities. Mochizuki published the core of IUT in a series of four manuscripts in as RIMS preprints, totaling over 500 pages. A pivotal event occurred in during a at , where mathematicians and Jakob Stix presented a critique, arguing that 3.12 in the third fails to distinguish properly between pilot objects, rendering the key step in the proof invalid. As of 2025, the mainstream mathematical community has not accepted Mochizuki's proof, viewing the as unresolved due to persistent verification challenges. The controversy centers on difficulties in verifying the intricate commutative diagrams and links in IUT, particularly those involving the Θ-link and multiradial representations, which require a novel conceptual framework not widely adopted. In response, Kirti Joshi and Mochizuki issued a FAQ in November 2025 defending the proof's integrity, clarifying the role of distinct arithmetic structures and addressing alleged flaws in the Scholze-Stix analysis through additional expositions on anabelomorphy and p-adic Teichmüller theory. Despite these efforts, including Joshi's series of preprints attempting to bridge gaps, no broad consensus has emerged, with debates continuing over the theory's foundational assumptions and the feasibility of independent verification. Mochizuki has proposed formalizing parts of IUT in a computer-verifiable language to resolve the impasse, but this initiative remains in early stages as of late 2025.

Other Proposed Approaches

In the history of attempts to prove Szpiro's conjecture, early efforts included a 2007 sketch by Lucien Szpiro himself, who proposed a solution linking the conjecture to properties of elliptic curves and torsion points, but the argument was soon identified as containing a fatal flaw and withdrawn. Similarly, other exploratory approaches in the late , such as those exploring height functions and integral points on elliptic curves by Joseph H. Silverman, yielded important partial bounds but fell short of a full proof, often relying on unproven assumptions related to canonical heights. More recent alternative approaches have focused on local and p-adic methods to tackle aspects of the conjecture. For instance, in 2012, Michael A. Bennett and Soroosh Yazdani established a version of Szpiro's conjecture, bounding the minimal of the for elliptic curves over under strong hypotheses, including the being a large and the presence of a nontrivial rational ; however, this does not constitute a global proof and remains conditional. These efforts highlight the potential of local-global principles but underscore the challenges in extending such results unconditionally to the full conjecture. A notable 2025 claim emerged in the independently published book The Grand Unification: A Categorical Trace Theory and Szpiro's Conjecture by David R. Ely, which proposes a unified proof via a novel "Categorical Trace Theory" abstracting from the Arthur-Selberg trace formula to derive Szpiro's conjecture as a consequence within a categorical . Despite its ambitious scope, the work has received no or academic validation as of late 2025, positioning it as a contribution unlikely to gain traction without rigorous verification. As of November 2025, no alternative approaches to Szpiro's conjecture have achieved acceptance within the mathematical community, with attention primarily centered on resolving the controversies surrounding Shinichi Mochizuki's inter-universal Teichmüller theory. Researchers continue to pursue incremental advances, but a remains elusive.