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64-397: The theorem states that any elliptic curve over Q {\displaystyle \mathbb {Q} } can be obtained via a rational map with integer coefficients from the classical modular curve 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

128-438: A , b , c , d with ad − bc = 1 and 37 | c . Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate. The most spectacular application of

192-472: A closely related analytic statement: To each elliptic curve E over Q {\displaystyle \mathbb {Q} } we may attach a corresponding L -series . The L -series is a Dirichlet series , commonly written The generating function of the coefficients a n is then If we make the substitution we see that we have written the Fourier expansion of a function f ( E , τ ) of

256-400: A completely symbolic form (e.g., as propositions in propositional calculus ), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which

320-467: A completely symbolic form—with the presumption that a formal statement can be derived from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and declare that the theory consists of all statements provable from these hypotheses. These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and

384-405: A formal symbolic proof can in principle be constructed. In addition to the better readability, informal arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, one might even be able to substantiate

448-410: A large family of elliptic curves. There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve E of conductor N can be expressed also by saying that there is a non-constant rational map defined over ℚ , from the modular curve X 0 ( N ) to E . In particular,

512-412: A layman. In mathematical logic , a formal theory is a set of sentences within a formal language . A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of logical consequence . Some accounts define a theory to be closed under

576-600: A lectureship. Here he supervised the PhD of Andrew Wiles , and together they proved a partial case of the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication . In 1977, Coates moved back to Australia, becoming a professor at the Australian National University , where he had been an undergraduate. The following year, he moved back to France, taking up a professorship at

640-411: A modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field . Most cases of these extended conjectures have not yet been proved. In 2013, Freitas, Le Hung, and Siksek proved that elliptic curves defined over real quadratic fields are modular. For example, the elliptic curve y − y = x − x , with discriminant (and conductor) 37,

704-456: A natural number n for which the Mertens function M ( n ) equals or exceeds the square root of n ) is known: all numbers less than 10 have the Mertens property, and the smallest number that does not have this property is only known to be less than the exponential of 1.59 × 10 , which is approximately 10 to the power 4.3 × 10 . Since the number of particles in the universe

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768-420: A semantics for them through interpretation . Although theorems may be uninterpreted sentences, in practice mathematicians are more interested in the meanings of the sentences, i.e. in the propositions they express. What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation

832-647: A single counter-example and so establish the impossibility of a proof for the proposition as-stated, and possibly suggest restricted forms of the original proposition that might have feasible proofs. For example, both the Collatz conjecture and the Riemann hypothesis are well-known unsolved problems; they have been extensively studied through empirical checks, but remain unproven. The Collatz conjecture has been verified for start values up to about 2.88 × 10 . The Riemann hypothesis has been verified to hold for

896-564: A summer working for BHP in Newcastle, New South Wales , though he was not successful in gaining a university scholarship with the company. Coates attended Australian National University on scholarship as one of the first undergraduates, from which he gained a BSc degree. He then moved to France, doing further study at the École Normale Supérieure in Paris, before moving again, this time to England. In England he did postgraduate research at

960-399: A theorem by using a picture as its proof. Because theorems lie at the core of mathematics, they are also central to its aesthetics . Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood,

1024-601: A theorem if proven true. Until the end of the 19th century and the foundational crisis of mathematics , all mathematical theories were built from a few basic properties that were considered as self-evident; for example, the facts that every natural number has a successor, and that there is exactly one line that passes through two given distinct points. These basic properties that were considered as absolutely evident were called postulates or axioms ; for example Euclid's postulates . All theorems were proved by using implicitly or explicitly these basic properties, and, because of

1088-589: A theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma , proposition and corollary for less important theorems. In mathematical logic , the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements become well-formed formulas of some formal language . A theory consists of some basis statements called axioms , and some deducing rules (sometimes included in

1152-564: A theorem of the ambient theory, although they can be proved in a wider theory. An example is Goodstein's theorem , which can be stated in Peano arithmetic , but is proved to be not provable in Peano arithmetic. However, it is provable in some more general theories, such as Zermelo–Fraenkel set theory . Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises . In light of

1216-434: A theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. Logically , many theorems are of the form of an indicative conditional : If A, then B . Such a theorem does not assert B — only that B

1280-442: A theorem. It comprises tens of thousands of pages in 500 journal articles by some 100 authors. These papers are together believed to give a complete proof, and several ongoing projects hope to shorten and simplify this proof. Another theorem of this type is the four color theorem whose computer generated proof is too long for a human to read. It is among the longest known proofs of a theorem whose statement can be easily understood by

1344-454: Is Fermat's Last Theorem , and there are many other examples of simple yet deep theorems in number theory and combinatorics , among other areas. Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture . Both of these theorems are only known to be true by reducing them to a computational search that

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1408-460: Is isogenous to the original curve (but not, in general, isomorphic to it). Yutaka Taniyama 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 rediscovered the conjecture, and showed in 1967 that it would follow from

1472-399: Is a device for turning coffee into theorems" , is probably due to Alfréd Rényi , although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking. The classification of finite simple groups is regarded by some to be the longest proof of

1536-419: Is a necessary consequence of A . In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture ), and B the conclusion of the theorem. The two together (without the proof) are called the proposition or statement of the theorem (e.g. " If A, then B " is the proposition ). Alternatively, A and B can be also termed the antecedent and

1600-403: Is a true statement about a formal system (as opposed to within a formal system) is called a metatheorem . Some important theorems in mathematical logic are: The concept of a formal theorem is fundamentally syntactic, in contrast to the notion of a true proposition, which introduces semantics . Different deductive systems can yield other interpretations, depending on the presumptions of

1664-405: Is associated to the form For prime numbers l not equal to 37, one can verify the property about the coefficients. Thus, for l = 3 , there are 6 solutions of the equation modulo 3: (0, 0) , (0, 1) , (1, 0) , (1, 1) , (2, 0) , (2, 1) ; thus a (3) = 3 − 6 = −3 . The conjecture, going back to the 1950s, was completely proven by 1999 using ideas of Andrew Wiles , who proved it in 1994 for

1728-512: Is generally considered less than 10 to the power 100 (a googol ), there is no hope to find an explicit counterexample by exhaustive search . The word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory (see mathematical theory ). There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role;

1792-443: Is that it is falsifiable , that is, it makes predictions about the natural world that are testable by experiments . Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its accuracy or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in

1856-636: Is then verified by a computer program. Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities. Theorems in mathematics and theories in science are fundamentally different in their epistemology . A scientific theory cannot be proved; its key attribute

1920-434: The consequent , respectively. The theorem "If n is an even natural number , then n /2 is a natural number" is a typical example in which the hypothesis is " n is an even natural number", and the conclusion is " n /2 is also a natural number". In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in

1984-635: The University of Paris XI at Orsay . In 1985, he returned to the École Normale Supérieure, this time as professor and director of mathematics. From 1986 until his death, Coates worked in the Department of Pure Mathematics and Mathematical Statistics (DPMMS) of the University of Cambridge. He was head of DPMMS from 1991 to 1997. His research interests included Iwasawa theory , number theory and arithmetical algebraic geometry. He served on

Modularity theorem - Misplaced Pages Continue

2048-486: The division algorithm , Euler's formula , and the Banach–Tarski paradox . A theorem and its proof are typically laid out as follows: The end of the proof may be signaled by the letters Q.E.D. ( quod erat demonstrandum ) or by one of the tombstone marks, such as "□" or "∎", meaning "end of proof", introduced by Paul Halmos following their use in magazines to mark the end of an article. The exact style depends on

2112-491: The semantic consequence relation ( ⊨ {\displaystyle \models } ), while others define it to be closed under the syntactic consequence , or derivability relation ( ⊢ {\displaystyle \vdash } ). For a theory to be closed under a derivability relation, it must be associated with a deductive system that specifies how the theorems are derived. The deductive system may be stated explicitly, or it may be clear from

2176-406: The set of all sets cannot be expressed with a well-formed formula. More precisely, if the set of all sets can be expressed with a well-formed formula, this implies that the theory is inconsistent , and every well-formed assertion, as well as its negation, is a theorem. In this context, the validity of a theorem depends only on the correctness of its proof. It is independent from the truth, or even

2240-631: 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 suggested in 1986 that it implies Fermat's Last Theorem . He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply

2304-842: The Mathematical Sciences jury for the Infosys Prize in 2009. Coates was elected a fellow of the Royal Society of London in 1985, and was President of the London Mathematical Society from 1988 to 1990. The latter organisation awarded him the Senior Whitehead Prize in 1997, for "his fundamental research in number theory and for his many contributions to mathematical life both in the UK and internationally". His nomination for

2368-473: The University of Cambridge, his doctoral dissertation being on p -adic analogues of Baker's method . In 1969, Coates was appointed assistant professor of mathematics at Harvard University in the United States, before moving again in 1972 to Stanford University where he became an associate professor. In 1975, he returned to England, where he was made a fellow of Emmanuel College , and took up

2432-400: The author or publication. Many publications provide instructions or macros for typesetting in the house style . It is common for a theorem to be preceded by definitions describing the exact meaning of the terms used in the theorem. It is also common for a theorem to be preceded by a number of propositions or lemmas which are then used in the proof. However, lemmas are sometimes embedded in

2496-411: The axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to proof theory , which allows proving general theorems about theorems and proofs. In particular, Gödel's incompleteness theorems show that every consistent theory containing the natural numbers has true statements on natural numbers that are not theorems of

2560-594: The complex variable τ , so the coefficients of the q -series are also thought of as the Fourier coefficients of 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

2624-591: The conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5 , the Fermat equation has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch  [ fr ] was the first to notice, the elliptic curve of discriminant cannot be modular. Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of

Modularity theorem - Misplaced Pages Continue

2688-407: The context. The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative to a given semantics, or relative to the standard interpretation of

2752-526: The derivation rules (i.e. belief , justification or other modalities ). The soundness of a formal system depends on whether or not all of its theorems are also validities . A validity is a formula that is true under any possible interpretation (for example, in classical propositional logic, validities are tautologies ). A formal system is considered semantically complete when all of its theorems are also tautologies. John H. Coates John Henry Coates FRS (26 January 1945 – 9 May 2022)

2816-429: The evidence of these basic properties, a proved theorem was considered as a definitive truth, unless there was an error in the proof. For example, the sum of the interior angles of a triangle equals 180°, and this was considered as an undoubtable fact. One aspect of the foundational crisis of mathematics was the discovery of non-Euclidean geometries that do not lead to any contradiction, although, in such geometries,

2880-707: The existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre 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'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. For example, Wiles's Ph.D. supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of

2944-463: The first 10 trillion non-trivial zeroes of the zeta function . Although most mathematicians can tolerate supposing that the conjecture and the hypothesis are true, neither of these propositions is considered proved. Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e.,

3008-495: The foundations of mathematics to make them more rigorous . In these new foundations, a theorem is a well-formed formula of a mathematical theory that can be proved from the axioms and inference rules of the theory. So, the above theorem on the sum of the angles of a triangle becomes: Under the axioms and inference rules of Euclidean geometry , the sum of the interior angles of a triangle equals 180° . Similarly, Russell's paradox disappears because, in an axiomatized set theory,

3072-484: The interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in case the hypotheses are true—without any further assumptions. However, the conditional could also be interpreted differently in certain deductive systems , depending on the meanings assigned to the derivation rules and the conditional symbol (e.g., non-classical logic ). Although theorems can be written in

3136-408: The link between these two statements, based on an idea of Gerhard Frey (1985), is difficult and technical. It was established by Kenneth Ribet in 1987. Theorem In mathematics and formal logic , a theorem is a statement that has been proven , or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that

3200-404: 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,

3264-498: The physical axioms on which such "theorems" are based are themselves falsifiable. A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time. Other terms may also be used for historical or customary reasons, for example: A few well-known theorems have even more idiosyncratic names, for example,

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3328-411: The points of E can be parametrized by modular functions . For example, a modular parametrization of the curve y − y = x − x is given by where, as above, q = e . The functions x ( z ) and y ( z ) are modular of weight 0 and level 37; in other words they are meromorphic , defined on the upper half-plane Im( z ) > 0 and satisfy and likewise for y ( z ) , for all integers

3392-457: The proof of a theorem, either with nested proofs, or with their proofs presented after the proof of the theorem. Corollaries to a theorem are either presented between the theorem and the proof, or directly after the proof. Sometimes, corollaries have proofs of their own that explain why they follow from the theorem. It has been estimated that over a quarter of a million theorems are proved every year. The well-known aphorism , "A mathematician

3456-549: The remaining cases until the full result was proved in 1999. 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 modularity theorem is a special case of more general conjectures due to Robert Langlands . The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of

3520-409: The same way such evidence is used to support scientific theories. Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. It is also possible to find

3584-635: The significance of the axioms. This does not mean that the significance of the axioms is uninteresting, but only that the validity of a theorem is independent from the significance of the axioms. This independence may be useful by allowing the use of results of some area of mathematics in apparently unrelated areas. An important consequence of this way of thinking about mathematics is that it allows defining mathematical theories and theorems as mathematical objects , and to prove theorems about them. Examples are Gödel's incompleteness theorems . In particular, there are well-formed assertions than can be proved to not be

3648-420: 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

3712-526: The structure of proofs. Some theorems are " trivial ", in the sense that they follow from definitions, axioms, and other theorems in obvious ways and do not contain any surprising insights. Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excellent example

3776-462: The sum of the angles of a triangle is different from 180°. So, the property "the sum of the angles of a triangle equals 180°" is either true or false, depending whether Euclid's fifth postulate is assumed or denied. Similarly, the use of "evident" basic properties of sets leads to the contradiction of Russell's paradox . This has been resolved by elaborating the rules that are allowed for manipulating sets. This crisis has been resolved by revisiting

3840-402: The theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic . Generally, an assertion that is explicitly called

3904-411: The theory (that is they cannot be proved inside the theory). As the axioms are often abstractions of properties of the physical world , theorems may be considered as expressing some truth, but in contrast to the notion of a scientific law , which is experimental , the justification of the truth of a theorem is purely deductive . A conjecture is a tentative proposition that may evolve to become

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3968-445: The underlying language. A theory that is inconsistent has all sentences as theorems. The definition of theorems as sentences of a formal language is useful within proof theory , which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory , which is concerned with the relationship between formal theories and structures that are able to provide

4032-478: The vast majority of people who believed [it] was completely inaccessible". In 1995, Andrew Wiles, with some help from Richard Taylor , proved the Taniyama–Shimura–Weil conjecture for all semistable elliptic curves . Wiles used this to prove Fermat's Last Theorem, and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond, Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at

4096-687: Was an Australian mathematician who was the Sadleirian Professor of Pure Mathematics at the University of Cambridge in the United Kingdom from 1986 to 2012. Coates was born the son of J. H. Coates and B. L. Lee on 26 January 1945 and grew up in Possum Brush (near Taree ) in New South Wales , Australia. Coates Road in Possum Brush is named after the family farm on which he grew up. Before university he spent

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