0 ABC予想　　　ABC予想　conjecture　　ABC定理

ABC予想証明に新理論？　望月氏「著者は無知」と一蹴、混迷深まる：朝日新聞デジタル (asahi.com)

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ABC予想（エービーシーよそう、英語: abc conjecture）は、1985年にジョゼフ・オステルレとデイヴィッド・マッサーによって提起された数論の予想です ¹ ²。この予想は、自然数の組(a, b, c)(a,b,c)で、互いに素でありかつa + b = ca+b=cを満たすような3つの自然数a, b, ca,b,cの和と積の関係について述べています ³ ⁴。

具体的には、自然数nnに対して、nnの互いに異なる素因数の積をnnの根基（radical）と呼び、\text{rad}(n)rad(n)と書きます。例えば、ppが素数ならば、\text{rad}(p) = prad(p)=pです。また、\text{rad}(8) = \text{rad}(23) = 2rad(8)=rad(23)=2、\text{rad}(45) = 15rad(45)=15などです。

ABC予想は、次のように述べられます：

任意の\varepsilon > 0ε>0に対して、次を満たすような自然数の組(a, b, c)(a,b,c)は高々有限個しか存在しないであろうという主張です：c > \text{rad}(abc)^{1 + \varepsilon}c>rad(abc)1+ε

また、同等の定式化として次のものもあります：

任意の\varepsilon > 0ε>0に対して、ある定数K(\varepsilon) > 0K(ε)>0が存在し、全ての互いに素な自然数の組(a, b, c)(a,b,c)について次が成り立つというものです：c > K(\varepsilon) \cdot \text{rad}(abc)c>K(ε)⋅rad(abc)

ABC予想は、数論における多くの有名な予想や定理が直ちに導かれることから有名です。現在も未解決の問題として、ディオファントス解析において最も重要な問題の一つとされています。

証明はまだ得られていないものの、数々の数学者によりABC予想の証明が提案されてきましたが、広く受け入れられたものはありません。望月新一教授による提案もありますが、現在も主流の数学コミュニティにおいては未証明とされています。¹ ² ³ ⁴

ABC予想は、数学の興味深い問題の一つであり、その証明が見つかることを期待しています。¹ ² ³ ⁴

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ABC予想は、次のように述べられます：

任意の\varepsilon > 0ε>0に対して、次を満たすような自然数の組(a, b, c)(a,b,c)は高々有限個しか存在しないであろうという主張です：c > \text{rad}(abc)^{1 + \varepsilon}c>rad(abc)1+ε

また、同等の定式化として次のものもあります：

任意の\varepsilon > 0ε>0に対して、ある定数K(\varepsilon) > 0K(ε)>0が存在し、全ての互いに素な自然数の組(a, b, c)(a,b,c)について次が成り立つというものです：c > K(\varepsilon) \cdot \text{rad}(abc)c>K(ε)⋅rad(abc)

ABC予想は、数学の興味深い問題の一つであり、その証明が見つかることを期待しています。¹ ² ³ ⁴

The theorem states that any elliptic curve over Q can be obtained via a rational map with integer coefficients from the classical modular curve
X 0 ( N ) {\displaystyle X_{0}(N)}
for some integer N; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level N. If N is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the conductor), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level N, a normalized newform with integer q-expansion, followed if need be by an isogeny.
The modularity theorem implies a closely related analytic statement: to an elliptic curve E over Q we may attach a corresponding L-series. The L-series is a Dirichlet series, commonly written
L ( E , s ) = ∑ n = 1 ∞ a n n s . {\displaystyle L(E,s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}

The generating function of the coefficients
a n {\displaystyle a_{n}}
is then
f ( E , q ) = ∑ n = 1 ∞ a n q n . {\displaystyle f(E,q)=\sum _{n=1}^{\infty }a_{n}q^{n}.}

If we make the substitution
q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau }}

we see that we have written the Fourier expansion of a function
f ( E , τ ) {\displaystyle f(E,\tau )}
of the complex variable τ, so the coefficients of the q-series are also thought of as the Fourier coefficients of
f {\displaystyle f}
. The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem.
Some modular forms of weight two, in turn, correspond to holomorphic differentials for an elliptic curve. The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher-dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it).
History[edit]
Yutaka Taniyama (1956) stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō. Goro Shimura and Taniyama worked on improving its rigor until 1957. André Weil (1967) rediscovered the conjecture, and showed that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the Langlands program.
The conjecture attracted considerable interest when Gerhard Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat’s Last Theorem. He did this by attempting to show that any counterexample to Fermat’s Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed when Jean-Pierre Serre (1987) identified a missing link (now known as the epsilon conjecture or Ribet’s theorem) in Frey’s original work, followed two years later by Ken Ribet (1990)’s completion of a proof of the epsilon conjecture.
Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to proof (Singh 1997, pp. 203–205, 223, 226). For example, Wiles’ ex-supervisor John Coates states that it seemed “impossible to actually prove”, and Ken Ribet considered himself “one of the vast majority of people who believed [it] was completely inaccessible”.
Wiles (1995), with some help from Richard Taylor, proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves, which he used to prove Fermat’s Last Theorem, and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996), Conrad, Diamond & Taylor (1999), and Breuil et al. (2001) who, building on Wiles’ work, incrementally chipped away at the remaining cases until the full result was proved.
Further information: Fermat’s Last Theorem and Wiles’ proof of Fermat’s Last Theorem
Once fully proven, the conjecture became known as the modularity theorem.
Several theorems in number theory similar to Fermat’s Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two coprime n-th powers, n ≥ 3. (The case n = 3 was already known by Euler.)

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