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Modularity theorem

The Modularity theorem, formerly known as the Taniyama–Shimura conjecture, asserts that every elliptic curve over the field of rational numbers \mathbb{Q} is modular, in the sense that its associated L-function arises from a weight-2 newform (a cuspidal Hecke eigenform) of corresponding level and character. This bijection between isomorphism classes of elliptic curves over \mathbb{Q} and such modular forms provides a deep link between the arithmetic of elliptic curves and the analytic theory of modular forms.^[1] The conjecture originated in the 1950s through independent insights by Yutaka Taniyama and Goro Shimura, who proposed connections between elliptic curves and modular forms as part of broader reciprocity laws in algebraic number theory.^[2] André Weil refined and popularized the statement in 1967, emphasizing its implications for the analytic continuation and functional equations of L-functions attached to elliptic curves. Numerical evidence supported the conjecture for many specific curves, but a general proof remained elusive until the 1990s. A major breakthrough came in 1995 when Andrew Wiles, building on the Langlands program and earlier work by Gerhard Frey, Jean-Pierre Serre, and Kenneth Ribet, proved the semistable case of the theorem using advanced techniques in Galois representations, deformation theory, and the structure of Hecke algebras. This partial result sufficed to establish Fermat's Last Theorem as a corollary, since semistable elliptic curves arising from putative counterexamples to Fermat's theorem were shown to violate modularity.^[2] Wiles' proof involved demonstrating that the residual Galois representation attached to an elliptic curve lifts to a modular representation, with key innovations in showing Hecke algebras are complete intersections.^[1] The full theorem for all elliptic curves over \mathbb{Q} was established in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, who extended Wiles' methods to handle the "wild" ramification cases at the prime 3 using base change to totally real fields and automorphic induction. Their work completed the proof by addressing potential non-modularity in curves with more general reduction types at primes of bad reduction.^[1] The theorem has profound implications beyond Fermat's Last Theorem, including the resolution of many cases of the Birch and Swinnerton-Dyer conjecture via the Gross–Zagier formula and Heegner points, as well as advancements in the Langlands program for GL(2).^[3] It also underpins the study of elliptic curves over number fields, with generalizations to higher dimensions and other fields now actively pursued.^[4]

Background Concepts

Elliptic Curves over the Rationals

An elliptic curve over the rational numbers \mathbb{Q} is defined as a smooth projective algebraic curve of genus 1 equipped with a specified base point, which serves as the identity for the group structure on its points. Such curves can be represented by a Weierstrass equation of the form

y^2 = x^3 + a x + b,

where a, b \in \mathbb{Q} and the discriminant \Delta = -16(4a^3 + 27b^2) \neq 0 ensures the curve is nonsingular.^[5]^[6] The point at infinity provides the base point, and this model embeds the curve in the projective plane over \mathbb{Q}.^[5] The set of points on the elliptic curve E forms an abelian group under a geometric addition law derived from the chord-and-tangent construction. For distinct points P_1 = (x_1, y_1) and P_2 = (x_2, y_2) with x_1 \neq x_2, the sum P_3 = P_1 + P_2 = (x_3, y_3) is the reflection across the x-axis of the third intersection point of the line through P_1 and P_2 with the curve, given by

x_3 = \left( \frac{y_2 - y_1}{x_2 - x_1} \right)^2 - x_1 - x_2, \quad y_3 = \frac{y_2 - y_1}{x_2 - x_1} (x_1 - x_3) - y_1.

Doubling a point P = (x, y) with y \neq 0 uses the tangent line:

x_3 = \left( \frac{3x^2 + a}{2y} \right)^2 - 2x, \quad y_3 = \frac{3x^2 + a}{2y} (x - x_3) - y.

The identity element is the point at infinity \mathcal{O}, and the inverse of P = (x, y) is -P = (x, -y).^[7] The rational points E(\mathbb{Q}) form a subgroup, and by the Mordell-Weil theorem, E(\mathbb{Q}) is a finitely generated abelian group, isomorphic to \mathbb{Z}^r \oplus T where r is the rank and T is the finite torsion subgroup.^[8] The j-invariant, j(E) = \frac{2^8 3^3 a^3}{4a^3 + 27b^2}, classifies elliptic curves up to isomorphism over \overline{\mathbb{Q}} and is independent of the choice of Weierstrass model.^[5]^[9] Associated to an elliptic curve E over \mathbb{Q} is its Hasse-Weil L-function L(E, s), defined for \Re(s) > 3/2 by the Dirichlet series

L(E, s) = \sum_{n=1}^\infty \frac{a_n}{n^s},

where the coefficients a_n arise from the Euler product over primes, with a_p = p + 1 - \#E(\mathbb{F}_p) for primes of good reduction.^[10]^[11] This L-function encodes arithmetic data about E and admits analytic continuation to the complex plane.^[11]

Modular Forms and Representations

A modular form of weight 2 and level N is a holomorphic function f: \mathbb{H} \to \mathbb{C} on the upper half-plane \mathbb{H} that satisfies the transformation property f\left(\frac{az + b}{cz + d}\right) = (cz + d)^2 f(z) for all matrices \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(N), where \Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) \mid c \equiv 0 \pmod{N} \right\}, and f is holomorphic at the cusps of \Gamma_0(N) \backslash \mathbb{H}^*.^[12] Such forms admit a Fourier expansion f(z) = \sum_{n=0}^\infty a_n q^n at the cusp \infty, where q = e^{2\pi i z}, with a_0 = 0 for cusp forms.^[12] The space of cusp forms of weight 2, level N, and nebentypus character \varepsilon: (\mathbb{Z}/N\mathbb{Z})^\times \to \mathbb{C}^\times with \varepsilon(-1) = 1 is denoted S_2(\Gamma_0(N), \varepsilon). Newforms are the normalized Hecke eigenforms in the new subspace S_2^{\mathrm{new}}(\Gamma_0(N), \varepsilon), meaning they are eigenfunctions of the Hecke operators T_\ell for all primes \ell with eigenvalues a_\ell, normalized so that a_1 = 1, and they generate the one-dimensional eigenspaces under the Hecke algebra action.^[12] The Hecke operators T_n act on the space of modular forms by summing over certain cosets, commuting with each other when coprime, and preserving the cusp forms subspace.^[12] To a newform f \in S_2^{\mathrm{new}}(\Gamma_0(N), \varepsilon) with rational Fourier coefficients, Deligne attached a continuous representation \rho_{f,\lambda}: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\overline{\mathbb{Q}}_\ell) for primes \ell \nmid N, which is unramified at primes outside \ell and the primes dividing N, and satisfies \mathrm{trace}(\rho_{f,\lambda}(\mathrm{Frob}_p)) = a_p for primes p \neq \ell. For a prime p \nmid N not dividing the level, the residual representation \overline{\rho}_{f,p}: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\mathbb{F}_p) is the reduction modulo p of \rho_{f,p}, which is unramified outside p and the primes dividing N, with \mathrm{trace}(\overline{\rho}_{f,p}(\mathrm{Frob}_\ell)) = a_\ell \pmod{p} for \ell \neq p.^[13] The dimension of S_2(\Gamma_0(N), \varepsilon) equals the genus g of the modular curve X_0(N), given explicitly by

g = 1 + \frac{\mu}{12} - \frac{\varepsilon_\infty}{4} - \frac{\nu_2}{3} - \frac{\nu_3}{2},

where \mu = [ \mathrm{SL}_2(\mathbb{Z}) : \Gamma_0(N) ] = N \prod_{p \mid N} (1 + 1/p) is the index, \varepsilon_\infty is the number of cusps, and \nu_i (for i=2,3) counts the elliptic points of order i.^[14] For trivial nebentypus \varepsilon = 1, this simplifies to approximately \mu/12 for large N, reflecting the growth of the space.^[15] The Eichler-Shimura isomorphism identifies the space S_2(\Gamma_0(N)) of cusp forms of weight 2 and trivial nebentypus with the \mathbb{C}-vector space of Hecke-invariant classes in H^1(X_0(N), \mathbb{C}), more precisely, S_2(\Gamma_0(N)) \cong H^1_c(X_0(N), V_2)^+ where V_2 is the standard 2-dimensional representation of \mathrm{SL}_2(\mathbb{R}), up to the action of complex conjugation.^[16] This links the analytic theory of modular forms to the étale cohomology of the modular curve, providing a geometric realization of the Hecke eigenvalues as traces on cohomology classes.^[16]

Formal Statement

The Theorem

The modularity theorem states that every elliptic curve E defined over the rational numbers \mathbb{Q} is modular. Specifically, for any such E, there exists a cuspidal newform f of weight 2 and level equal to the conductor N_E of E such that the L-function of E coincides with that of f, i.e.,

L(E, s) = L(f, s).

This equality implies that the Hecke eigenvalues of f match the Fourier coefficients of the L-series expansion of E. The theorem was first established in 1995 for the case of semistable elliptic curves by Andrew Wiles. The proof for the general case, covering all elliptic curves over \mathbb{Q}, was completed in 2001 by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. Modularity of E further implies that the p-adic Galois representation \rho_{E,p}: \Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \GL_2(\mathbb{F}_p) attached to E is modular, meaning it is isomorphic to the residual representation \rho_{f,p} attached to some weight-2 newform f of level dividing N_E. Conversely, the modularity criterion asserts that if an irreducible, odd, two-dimensional residual representation \rho: \Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \GL_2(\mathbb{F}_p) arises from a modular form (i.e., \rho \cong \rho_{f,p} for some newform f), then under suitable conditions on the determinant and image, \rho is the residual representation attached to some elliptic curve over \mathbb{Q}.

Equivalent Formulations

One equivalent formulation of the modularity theorem arises in the context of Galois representations. For an elliptic curve E over \mathbb{Q}, the p-adic Tate module T_p(E) yields a continuous representation \rho_{E,p}: \Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \GL_2(\mathbb{Z}_p). The modularity theorem is equivalent to the statement that, for some prime p, the reduction modulo p of \rho_{E,p} is modular, meaning it is isomorphic to the mod p Galois representation attached to a weight-two newform f of level equal to the conductor of E with rational coefficients.^[17] The modularity theorem is a precise realization of the broader Taniyama-Shimura-Weil conjecture, which posits that every elliptic curve over \mathbb{Q} is modular. This conjecture asserts that for any such elliptic curve E with conductor N, there exists a cuspidal newform f \in S_2(\Gamma_0(N)) with rational Fourier coefficients such that the L-function of E equals the L-function of f. The full conjecture, now a theorem, encompasses all elliptic curves over \mathbb{Q}, extending beyond the original semi-stable cases initially proved by Wiles and others.^[18] From the perspective of the Langlands program, the modularity theorem represents a special case of the Langlands correspondence for \GL_2/\mathbb{Q}. It establishes that the two-dimensional Galois representation \rho_{E,p} attached to an elliptic curve E corresponds to a cuspidal automorphic representation of \GL_2(\mathbb{A}_\mathbb{Q}) generated by a weight-two modular form, thereby realizing functoriality in this setting. This connection highlights modularity as an instance of the reciprocity conjecture linking Galois representations to automorphic forms.^[19]^[20]

Historical Development

Origins and Conjectures

The origins of the modularity theorem trace back to early observations in the 1930s by Erich Hecke, who developed the analytic theory of modular forms and their associated L-functions. Hecke noted striking formal similarities between the L-functions of cusp forms of weight 2 and the L-functions arising from elliptic curves, which are Riemann surfaces of genus one. These L-functions, constructed via Hecke operators, exhibited properties analogous to those expected from the zeta functions of genus one curves, suggesting a deeper connection between analytic objects like modular forms and algebraic varieties such as elliptic curves.^[21] In the 1950s, Yutaka Taniyama advanced these ideas during the International Symposium on Algebraic Number Theory in Tokyo-Nikko in 1955, where he proposed a series of problems linking elliptic curves to modular forms. Specifically, Taniyama conjectured that the zeta function of an elliptic curve should coincide with the L-function of a modular form of weight 2, parametrizing analytic families of abelian varieties through modular forms. This formulation posited that abelian varieties could be constructed uniformly from modular data, extending Hecke's observations to a broader arithmetic framework.^[21] Goro Shimura refined Taniyama's conjecture in the 1960s, focusing on elliptic curves defined over the rational numbers \mathbb{Q}. In his work on complex multiplication, Shimura proposed that every elliptic curve over \mathbb{Q} arises as a quotient of the Jacobian of a modular curve, establishing a precise correspondence between such curves and weight-2 newforms. This refinement emphasized the role of modular curves as moduli spaces and provided a geometric interpretation, making the conjecture more amenable to algebraic verification.^[21]^[22] André Weil formalized these developments in the 1960s, dubbing the statement the Taniyama-Shimura-Weil conjecture. Weil's contribution clarified the conditions under which the L-function of an elliptic curve over \mathbb{Q} matches that of a corresponding modular form, including analytic continuation and functional equations. He highlighted the conjecture's implications for reciprocity laws and class field theory, while noting remaining challenges in its general validity.^[21] In the 1970s, Jean-Pierre Serre shifted attention to the residual representations associated with elliptic curves, posing questions about their modularity modulo primes. In 1975, Serre conjectured that every odd, irreducible, two-dimensional residual Galois representation of \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) arises from a modular form, with the level, weight, and nebentypus determined by the representation's local behavior. This modularity conjecture for residual representations provided a foundational tool for studying the original conjecture through reductions modulo primes, influencing subsequent arithmetic investigations.^[23]

Key Milestones Leading to Proof

In the 1980s, Barry Mazur advanced the understanding of Galois representations attached to elliptic curves through his development of deformation theory, which systematically studied liftings of residual representations to characteristic zero. This framework was pivotal in joint work with Andrew Wiles, where they analyzed p-adic analytic families of Galois representations arising from elliptic curves, establishing connections between universal deformation rings and Hecke algebras that foreshadowed modularity lifting techniques.^[24] A landmark contribution came in 1983 from the Langlands–Tunnell theorem, which proved that every irreducible odd two-dimensional Galois representation over \mathbb{Q} with dihedral (solvable) image corresponds to a weight 1 modular form. This resolved Artin's conjecture in this case and provided early evidence for the modularity of elliptic curves whose residual representations have solvable image, supporting the Taniyama–Shimura conjecture for a specific class of curves.^[25] The year 1986 marked a turning point with Gerhard Frey's introduction of "Frey curves," a geometric construction associating a semistable elliptic curve to any hypothetical solution of Fermat's equation x^n + y^n = z^n for integers x, y, z > 0 and n \geq 3. Frey showed that such a curve would have conductor 2 and minimal discriminant -2^{4}(xyz)^{2n}, and he conjectured that no such curve could be modular, thereby forging a direct link between the non-existence of Fermat solutions and the Taniyama-Shimura conjecture. This idea built on earlier work tying Diophantine equations to elliptic curves but highlighted the potential of modularity to resolve Fermat's Last Theorem. That same year, Kenneth Ribet proved Serre's epsilon conjecture, establishing a level-lowering result for modular Galois representations. Ribet's theorem demonstrated that if an elliptic curve over the rationals admits a non-modular residual representation at a prime p, then any associated modular form must have higher level unless the representation satisfies specific irreducibility conditions; crucially, for Frey curves, this implied the attached Galois representation could not arise from a modular form, reducing Fermat's Last Theorem to the modularity of semistable elliptic curves. The proof, leveraging Mazur's deformation theory and properties of Hecke algebras, appeared in print in 1990 but was announced in 1986, galvanizing efforts toward a full modularity proof.^[26] These milestones in the 1980s and early 1990s, rooted in the Taniyama-Shimura conjecture first articulated in the 1950s, provided the theoretical and motivational foundation for Andrew Wiles' subsequent strategy, transforming the abstract modularity question into a concrete pathway for proving Fermat's Last Theorem.

Proof Strategy

Ribet's Level-Lowering

Ribet's level-lowering theorem provides a crucial technique for reducing the level of modular residual Galois representations, serving as a foundational step in establishing connections between elliptic curves and modular forms in the proof of the modularity theorem. Specifically, the theorem asserts that if \overline{\rho}: \Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \GL_2(\mathbb{F}_\ell) is an irreducible representation arising from a modular form of level Mp, where p \nmid M, \ell is an odd prime (the characteristic of \mathbb{F}_\ell), and \overline{\rho} is finite at p (meaning it arises from a finite flat group scheme over \mathbb{Z}_p), then \overline{\rho} is modular of level M provided either \ell \nmid M or p \not\equiv 1 \pmod{\ell}. This reduction removes the prime p from the level while preserving the associated Galois representation up to congruence modulo \ell. The base case for modularity of such irreducible representations, particularly when the level is minimal or 1, relies on the Langlands–Tunnell theorem, which establishes that every continuous, odd, irreducible 2-dimensional Galois representation over \mathbb{Q} with coefficients in a finite field of characteristic an odd prime is modular. This result, building on the Artin conjecture for \GL_2, ensures that the lowered representation corresponds to a newform of the reduced level, allowing iterative application of level-lowering to reach the minimal conductor. In the application to Frey curves, Ribet's technique demonstrates the non-modularity of the elliptic curve E: y^2 = x(x - a^p)(x + b^p) attached to a hypothetical primitive solution (a, b, c) to Fermat's equation a^p + b^p = c^p with p \geq 5 an odd prime. The associated residual Galois representation \overline{\rho}_{E,\ell} (for suitable \ell) is irreducible and odd, with conductor dividing $2p. Assuming modularity at level $2p, level-lowering at p forces modularity at level 2; however, no cuspidal newform of weight 2 and level 2 exists, yielding a contradiction. This level-lowering approach also resolves the epsilon conjecture, which posits that Frey curves arising from non-trivial Fermat solutions are non-modular. By showing that modularity would imply the existence of a non-existent weight-2 newform at the lowered level, Ribet proves the conjecture, thereby linking the modularity theorem directly to the negation of Fermat's Last Theorem. Furthermore, it implies that non-semistable elliptic curves (like Frey curves, which have non-minimal reduction at p) cannot be modular without violating the level-minimality conditions, reinforcing the assumption of semistability in modularity proofs.

Wiles' Approach and Fixes

Wiles' strategy for proving the modularity theorem centered on the comparison of two rings: the universal deformation ring R parametrizing lifts of a residual Galois representation \bar{\rho}: \Gal(\bar{\Q}/\Q) \to \GL_2(\F_p) attached to an elliptic curve E over \Q, and the Hecke algebra T, which acts on the space of modular forms of level dividing the conductor of E and weight 2. Under suitable conditions, including minimality of the adjoint Selmer group H^1_{\ad}(G_{\Q}, \ad \bar{\rho}), Wiles established that R and T are isomorphic as complete local \Z_p-algebras at the maximal ideal corresponding to \bar{\rho}. This R = T theorem implies that the p-adic Galois representation of E arises from a modular form, as the Hecke action on modular forms deforms compatibly with the Galois action on the deformation space. A core innovation was the Taylor-Wiles method, which addresses the potential non-finiteness or lack of flatness in the map T \to R by introducing auxiliary primes Q of good reduction where \det \bar{\rho}(\Frob_q) = 1 for q \in Q. For such sets Q of cardinality growing with the patching level, Taylor and Wiles constructed patched Hecke algebras T_Q and deformation rings R_Q that are finite flat over a fixed Iwasawa algebra \Lambda, ensuring T_Q is Cohen-Macaulay and T_Q \otimes_{\Lambda} R \cong R_Q after base change. This patching argument, combined with a numerical criterion comparing the minimal number of generators of R and T via the Wiles defect formula, proves the desired isomorphism R \cong T when the adjoint Euler characteristic vanishes. The method guarantees finite generation and flatness by minimizing the dimension of certain Selmer groups through the choice of auxiliary primes.^[27] However, the initial R = T argument relied on the Hecke algebra being Gorenstein, but there was a gap when this might fail due to a non-trivial Eisenstein quotient. Wiles resolved this by employing the "3-5 switch," which transfers modularity from a congruent modular form at level Np (where N is the conductor) to one at level N \cdot 3 or N \cdot 5, avoiding problematic cases while preserving the residual representation via Mazur's deformation theory of modular curves. Specifically, if \bar{\rho} is modular at level N \cdot 3, the switch uses the action of Atkin-Lehner operators and Hecke correspondences to lift to a form at level Np congruent modulo the Eisenstein ideal.^[27] The proof initially covered semistable elliptic curves but left open cases with wild ramification at 2 and 3. In 2001, Breuil, Conrad, Diamond, and Taylor completed the modularity theorem for all elliptic curves over \Q by developing integral methods using Breuil modules, which classify potentially crystalline lifts of residual representations at these primes.^[28] Their approach extends the Taylor-Wiles patching to the 2-adic setting with ordinary conditions and handles the wild 3-adic case via explicit computations of local deformation rings, ensuring the global R = T holds without additional assumptions.^[28] Central to the ring comparison is the congruence between the characteristic polynomials: for a prime \ell \nmid Np, the characteristic polynomial of Frobenius \Frob_\ell acting on the p-adic Tate module of the universal deformation equals the reverse characteristic polynomial of the Hecke operator T_\ell on the space of modular forms, reflecting the Langlands reciprocity encoded in the isomorphism R \cong T. This equality, verified through the determinant of the action on cohomology, confirms that the Galois representation deforms as a modular representation.

Examples and Illustrations

A Specific Elliptic Curve

A concrete example illustrating the modularity theorem is the elliptic curve E defined by the minimal Weierstrass equation

y^2 + y = x^3 - x^2 - 10x - 20.

This curve has conductor N = 11 and j-invariant j(E) = -2^{15}/11 = -32768/11.^[29] The prime dividing the conductor is p = 11, where E has split multiplicative reduction.^[30] The modularity theorem associates E to the unique newform f of weight 2, level 11, and trivial character in S_2(\Gamma_0(11)), with q-expansion

f(q) = q - 2q^2 - q^3 + 2q^4 + 2q^5 + 2q^6 - 2q^7 - 2q^8 - q^9 - 2q^{10} + \cdots.

^[31] The Hecke eigenvalues a_p(f) of this form coincide with the traces of Frobenius a_p(E) = p + 1 - \#E(\mathbb{F}_p) for all primes p of good reduction (i.e., p \neq 11). For instance, a_2(f) = -2, corresponding to \#E(\mathbb{F}_2) = 2 + 1 - (-2) = 5; a_3(f) = -1, corresponding to \#E(\mathbb{F}_3) = 3 + 1 - (-1) = 5; and a_5(f) = 2, corresponding to \#E(\mathbb{F}_5) = 5 + 1 - 2 = 4. These matches for small primes exemplify the isogeny correspondence between E and the modular Jacobian.^[29]^[31] The L-functions satisfy L(E, s) = L(f, s), as guaranteed by the modularity theorem; this equality is verified computationally via tables of special values, such as the analytic rank 0 and L(E, 1) = L(f, 1) \approx 0.25384186 > 0.^[29]^[31]

Connection to Fermat's Last Theorem

The modularity theorem plays a pivotal role in the proof of Fermat's Last Theorem through the celebrated Frey-Ribet strategy, which links hypothetical solutions to the Fermat equation with properties of elliptic curves and modular forms. Suppose there exists a primitive integer solution a, b, c to the equation a^n + b^n = c^n, where n > 2 is an odd prime, with \gcd(a, b, c) = 1, a odd, and b, c even. Gerhard Frey proposed associating to this solution the elliptic curve

E: y^2 = x(x - a^n)(x + b^n),

known as the Frey curve, whose conductor divides $2(abc)^2.^[32] Ken Ribet established that the mod-n Galois representation attached to this Frey curve is irreducible and, under the assumptions of semistability, can be shown to arise from a modular form of weight 2 and level 2 via level-lowering techniques. However, the space of cusp forms of weight 2 and level \Gamma_0(2) is empty, leading to a contradiction if the Frey curve were modular.^[33] This implies that no such elliptic curve can exist under the modularity theorem, and thus no primitive solutions to the Fermat equation exist for odd primes n > 2.^[32] For the specific case n = 3, the Frey curve attached to a hypothetical primitive solution a^3 + b^3 = c^3 would similarly possess an irreducible mod-3 Galois representation that level-lowers to a nonexistent modular form of level 2, directly contradicting modularity and ruling out such solutions.^[33] Combined with the known solution for n = 2 and infinite descent arguments for composite exponents, this establishes Fermat's Last Theorem in full.^[32]

Generalizations and Extensions

Beyond Rational Coefficients

The modularity theorem, originally established for elliptic curves over the rational numbers \mathbb{Q}, has inspired efforts to extend the correspondence to elliptic curves over more general number fields, though complete results remain elusive beyond \mathbb{Q}. Over imaginary quadratic fields, modularity is known for elliptic curves with complex multiplication (CM). These curves are associated to grossencharacters (Hecke characters of infinite type) on the CM field, yielding L-functions that match those of CM modular forms. The groundbreaking work of Gross and Zagier establishes a precise formula relating the Néron-Tate height of Heegner points on the modular curve to the central derivative of the L-function of the CM elliptic curve, enabling proofs of the Birch and Swinnerton-Dyer conjecture in the rank-one case for such curves. In contrast, full modularity has been proved for all elliptic curves over real quadratic fields by Freitas, Le Hung, and Siksek in 2016.^[34] For elliptic curves over arbitrary number fields K, the full modularity theorem is open, with only partial progress. Naive attempts to generalize the level-lowering arguments from the \mathbb{Q}-case fail in general, as counterexamples demonstrate that certain residual Galois representations over K do not lift to modular forms in the expected way. Nonetheless, conditional modularity results follow from the Fontaine-Mazur conjecture, which posits that a continuous, irreducible, p-adic Galois representation of \mathrm{Gal}(\overline{K}/K) is automorphic (hence modular for dimension 2) if and only if it is de Rham at all primes above p with non-critical Hodge-Tate weights. This conjecture implies modularity for elliptic curves over K whose associated Galois representations satisfy the local conditions. Significant advances have been made in the integral setting, particularly for ordinary primes. The works of Bellaïche on the structure of Hecke algebras acting on modular symbols modulo p provide essential tools for analyzing congruences between modular forms and their Galois representations at ordinary primes. Complementing this, Diamond's contributions to modularity lifting theorems for ordinary residual representations enable the passage from mod p modularity to characteristic zero under ordinary conditions, extending the Taylor-Wiles method to this context. Modularity lifting theorems for potentially Barsotti-Tate Galois representations over finite extensions of \mathbb{Q} form a cornerstone of these extensions. A representation is potentially Barsotti-Tate if it becomes Barsotti-Tate (arising from the Tate module of an abelian variety) after restriction to a finite extension. Conrad, Diamond, and Taylor proved that certain 2-dimensional, odd, irreducible p-adic Galois representations over \mathbb{Q} that are potentially Barsotti-Tate at p and finite at all other primes are modular, resolving cases of the Fontaine-Mazur conjecture. Kisin extended this to 2-adic representations, establishing lifting under crystalline conditions at p. Thorne further generalized these results to representations over totally real fields, proving modularity for potentially Barsotti-Tate cases with minimal ramification. These theorems underpin modularity over extensions of \mathbb{Q} by allowing lifts from known modular residual representations. Serre's modularity conjecture, formulated in 1978, posits that every irreducible two-dimensional odd Galois representation of the absolute Galois group of the rationals with coefficients in a finite field of characteristic p > 2 is modular, meaning it arises from a modular form of level dividing the conductor of the representation and weight at most p.^[35] This conjecture extends the Taniyama-Shimura-Weil conjecture (now theorem) to modular representations, bridging Galois theory and modular forms for a broader class of residual characteristics. The conjecture was fully proved by Chandrashekhar Khare and Jean-Pierre Wintenberger in 2009, employing the Taylor-Wiles method of constructing modular deformation rings that match the Hecke algebra, adapted to handle the potentially non-minimal deformations at primes of bad reduction.^[36] Their proof first establishes modularity for representations with odd conductor and p \neq 2, then extends to the remaining cases using linked auxiliary primes and ordinary lifting techniques.^[37] The Fontaine-Mazur conjecture, proposed in 1995, addresses the modularity of crystalline p-adic Galois representations of dimension two over the rationals that are unramified outside a finite set of primes (including p) and de Rham at p with distinct Hodge-Tate weights.^[38] It predicts that such representations, which satisfy necessary local conditions for arising from geometry, are precisely those attached to cuspidal eigenforms, providing a p-adic analogue to the modularity theorem and constraining the possible global behaviors of these representations.^[39] Partial progress includes proofs for representations with small residual image or under Serre weight assumptions, often relying on potential automorphy and base change to CM fields.^[40] The conjecture remains open in general but has been verified for p=3 in the regular case as of 2024.^[41] The modularity theorem serves as a key instance of the Artin conjecture within the Langlands program for \mathrm{GL}_2(\mathbb{Q}), where it establishes the holomorphicity and functional equation of Artin L-functions for irreducible odd two-dimensional representations via their identification with L-functions of modular forms, incorporating reciprocity laws that equate Galois parameters with automorphic data.^[42] This correspondence realizes the global Langlands duality for \mathrm{GL}_2, transforming non-abelian Artin representations into automorphic forms and enabling functoriality transfers, such as symmetric powers, that underpin broader reciprocity in the program.^[19] The work of Christopher Skinner and Eric Urban on the Iwasawa main conjecture for \mathrm{GL}_2 connects modularity to the arithmetic of p-adic L-values, proving that the Selmer group of a modular representation over a p-adic Lie extension matches the characteristic ideal generated by a two-variable p-adic L-function under ordinary assumptions at p.^[43] Their 2014 proof uses Euler systems from Beilinson-Kato classes and control theorems for Hecke algebras to establish both divisibility and injectivity, linking the conjecture's \Lambda-adic formulation to the modularity lifting theorems of Kisin and others.^[44] This resolves the conjecture for a wide class of elliptic modular forms, with implications for non-vanishing of p-adic L-values and Birch-Swinnerton-Dyer ranks in Iwasawa theory.^[45]

References

[1]
[PDF] on the modularity of elliptic curves over q: wild 3-adic exercises.
ON THE MODULARITY OF ELLIPTIC CURVES OVER Q: WILD 3-ADIC EXERCISES. CHRISTOPHE BREUIL, BRIAN CONRAD, FRED DIAMOND, AND RICHARD TAYLOR. Introduction. In this ...
[2]
Modular elliptic curves and Fermat's Last Theorem
Modular elliptic curves and Fermat's Last Theorem from Volume 141 (1995), Issue 3 by Andrew Wiles. No abstract available for this article.
[3]
[PDF] Elliptic Curves and Modularity - DPMMS
The modularity conjecture for elliptic curves over Q was stated with increasing degrees of precision by Taniyama, Shimura, and Weil in the 1950's and 60's.
[4]
[PDF] The Modularity Theorem Jerry Shurman y + xy + y = x
This talk discusses a result called the Modu- larity Theorem: All rational elliptic curves arise from modular forms. Taniyama first suggested in the 1950's ...
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Elliptic Curve -- from Wolfram MathWorld
An elliptic curve is a type of cubic curve whose solutions are confined to a region of space that is topologically equivalent to a torus.
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Elliptic Discriminant -- from Wolfram MathWorld
Delta=-16(4A^3+27B^2). (9). DiscriminantEllipticCurve. Algebraically, the discriminant is nonzero when the right-hand side has three distinct roots.
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[PDF] elliptic curves - UMSL
Group Law: Adding points on an Elliptic Curve. Let P1 = (x1,y1) and P2 = (x2,y2) be points on an elliptic curve E given by y2 = x3 + Ax + B. Define P3 = (x3 ...
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[PDF] Elliptic Curves and the Mordell-Weil Theorem - UChicago Math
Sep 26, 2013 · Main Theorem (Mordell-Weil). The group of Q-points on an elliptic curve defined over Q is finitely generated. This generalizes to the K-points ...
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https://mathworld.wolfram.com/j-Invariant.html
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[PDF] LECTURE 24: FROM ZETA TO L-FUNCTIONS - Vanderbilt University
Conjecture (Hasse-Weil). The function L(E,s) has an analytic continuation to the complex plane and satisfies a functional equation. where f is a theta function ...
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L-series of an elliptic curve - PlanetMath.org
Mar 22, 2013 · L-series of an elliptic curve. Let E E be an elliptic curve over Q ℚ with Weierstrass equation: y2+ ...Missing: rationals | Show results with:rationals<|separator|>
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[PDF] 25 Modular forms and L-functions - MIT Mathematics
May 15, 2017 · level N, meaning that it contains Γ(N), then Γ contains the matrix ... must be a modular form of weight 2; but there are additional constraints.
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[PDF] Mod p Galois representations attached to modular forms
Apr 7, 2006 · The group G` is reducible when ` = 2,3,5,7,619 and irreducible but of order prime to ` when ` = 23. Note that ∆ is of weight k = 12. Looking at.
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[PDF] 25 Modular forms and L-series - MIT Mathematics
May 12, 2015 · This is main motivation for introducing the notion of weight, it allows us to generalize modular functions in an interesting way, by ...
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[PDF] An Explicit Dimension Formula and Classification of Trivial Newspaces
Let N ≥ 1, k ≥ 2, and χ be a Dirichlet character modulo N such that χ(−1) = (−1)k. The space of cuspforms of level N, weight k, and nebentypus χ is denoted by ...
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[PDF] Lecture 19 : Eichler-Shimura Theory (Cntd.)
... modular forms of weight k = 2 arising from compatible systems of λ-adic representations {Vf,λ}. We remark that for the case when k > 2, one can realize the ...
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https://link.springer.com/book/10.1007/978-0-387-21695-8
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[PDF] The Shimura-Taniyama conjecture (d'apr`es Wiles)
Sep 9, 2007 · This paper is entirely expository. It owes everything to the ideas of Wiles, and to a course he and his students gave at Princeton ...Missing: original | Show results with:original
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[PDF] modularity of galois representations and langlands functoriality
Introduction. In this article, we survey some recent work on Langlands reciprocity and func- toriality in which Galois representations play a central role.
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Modularity of Galois Representations and Langlands Functoriality
Jul 25, 2022 · In this article, we survey some recent work on Langlands reciprocity and functoriality in which Galois representations play a central role.
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[PDF] 18.783 Elliptic Curves Lecture 25
Dec 12, 2023 · In 1984 Frey suggested that any such Ea,b,c could not be modular. Serre gave a more precise formulation of Frey's suggestion known as the ...
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Some History of the Shimura-Taniyama Conjecture
I shall deal specifically with the history of the conjecture which asserts that every elliptic curve over Q (the field of rational numbers) is modular.
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Abelian Varieties with Complex Multiplication and Modular Functions
### Summary of Shimura's Refinement of the Conjecture for Elliptic Curves over Q in the 1960s
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[PDF] SUR LES REPRÉSENTATIONS MODULAIRES DE DEGRÉ 2 DE ...
(i) Si E a bonne réduction, on a k = 2 . (ii) Si E a mauvaise réduction de type multiplicatif, on a. SUR LES REPRÉSENTATIONS MODULAIRES DE DEGRÉ 2 DE ...
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[PDF] On p-adic analytic families of Galois representations - Numdam
MAZUR. A. WILES. On p-adic analytic families of Galois representations. Compositio Mathematica, tome 59, no 2 (1986), p. 231-264. <http://www.numdam.org/item?id ...
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A classical Diophantine problem and modular forms of weight 3/2
A classical Diophantine problem and modular forms of weight 3/2. Published: June 1983. Volume 72, pages 323–334, (1983); Cite this ...Missing: Jerrold | Show results with:Jerrold
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On modular representations of Gal (.../Q) arising from ... - EUDML
Ribet, K.A.. "On modular representations of Gal (.../Q) arising from modular forms.." Inventiones mathematicae 100.2 (1990): 431-476.Missing: epsilon | Show results with:epsilon
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Ring-theoretic properties of certain Hecke algebras
Ring-theoretic properties of certain Hecke algebras. Pages 553-572 from Volume 141 (1995), Issue 3 by Richard Taylor, Andrew Wiles.
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[PDF] Reciprocity in the Langlands program since Fermat's Last Theorem.
however, the classicality theorems ...
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AMS :: Journal of the American Mathematical Society
Brian Conrad, Fred Diamond, and Richard Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 2 ...
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Elliptic curve with LMFDB label 11.a1 (Cremona label 11a2)
This is the curve of minimal conductor and minimal absolute discriminant to have trivial Mordell-Weil group. This project is supported by grants from the US ...
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[PDF] ELLIPTIC CURVE ALGORITHMS 3.1 Terminology and notation For ...
3.1 Terminology and notation. For reference in the following sections, we collect here the notation, terminology and formulae concerning elliptic curves ...
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Newform orbit 11.2.a.a - LMFDB
This is the first weight 2 newform when ordered by level. Newspace parameters.Missing: coefficients | Show results with:coefficients
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[PDF] Modular elliptic curves and Fermat's Last Theorem
An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form X0(N).
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[PDF] On modular representations of Gal (.../Q) arising from modular forms.
On modular representations of Gal (.../Q) arising from modular forms. by Ribet, K.A. in Inventiones mathematicae volume 100; pp. 431 - 476. Page 2. Page 3 ...
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[PDF] serre's modularity conjecture (ii) - UCLA Department of Mathematics
of the residual representation, which by a misunderstanding we thought to be unnecessary. When this hypothesis will be removed, we will get a strictly.
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Serre's modularity conjecture (I) | Inventiones mathematicae
Jul 4, 2009 · This paper is the first part of a work which proves Serre's modularity conjecture. We first prove the cases and odd conductor, and p=2 and weight 2.
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Serre's modularity conjecture (II) | Inventiones mathematicae
Jul 4, 2009 · We provide proofs of Theorems 4.1 and 5.1 of Khare and Wintenberger (Invent. Math., doi:10.1007/s00222-009-0205-7, 2009).Missing: paper | Show results with:paper
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the fontaine-mazur conjecture for gl2 - American Mathematical Society
Jan 21, 2009 · the modularity of certain p-adic Galois representations, assuming the mod p re- duction was modular. Subsequently a number of authors ...
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[PDF] Remarks on a conjecture of Fontaine and Mazur
May 23, 2000 · Hilbert-Blumental modular varieties over totally real fields in which p and ... number field M over a perfect field K. Suppose also that X is a ...
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[PDF] THE FONTAINE-MAZUR CONJECTURE FOR GL2 Mark Kisin ...
They showed how one could deduce the modularity of certain p-adic Galois representations, assuming the mod p re- duction was modular. Subsequently a number of ...
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[2412.06812] On the Fontaine-Mazur conjecture for $p=3$ - arXiv
Nov 30, 2024 · In this article, we prove the remaining open cases of the Fontaine-Mazur conjecture on two-dimensional regular Galois representations over \Gal ...
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[PDF] Galois representations associated to modular forms | MIT
• holomorphic cusp form of weight > 2 −→ ℓ-adic Galois representation: Deligne. • holomorphic cusp form of weight 1 −→ Galois representation: Deligne–Serre.<|control11|><|separator|>
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[PDF] The Iwasawa main conjecture for GL2 - Columbia Math Department
In this paper we prove the Iwasawa-Greenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The Iwasawa-Greenberg Main Conjecture.
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Main Conjectures and Modular Forms - Project Euclid
This paper reports on joint work with E. Urban that completes 1 a proof of the Main Conjecture of Iwasawa Theory for a broad class of elliptic curves and ...Missing: modularity | Show results with:modularity
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The Iwasawa Main Conjectures for ${\rm GL}_2$ and derivatives of ...
Jan 12, 2020 · Abstract:We prove under mild hypotheses the three-variable Iwasawa main conjecture for $p$-ordinary modular forms in the indefinite setting.Missing: modularity | Show results with:modularity