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Modular Elliptic Curves and Fermat's Last Theorem

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... In his work on modularity of elliptic curves and Fermat's last theorem, Wiles [17] discovered a numerical criterion for certain noetherian local rings A to be complete intersections. Diamond [8] generalized Wiles' result by establishing a criterion for modules M over A to be free and for A to be a complete intersection; see the discussion below for the precise statements of their results. ...
... In Section 5, we give a more streamlined proof of a formula for δ A (A), formulated (albeit in the setting of certain derived rings) by Venkatesh [16], and proved in [10], in terms of certain André-Quillen cohomology modules. We end by explaining how this formula gives another proof of the isomorphism criterion for maps between complete intersection rings due to Wiles [17] and Lenstra [13]. ...
... The question of comparing congruence modules for A and M has been studied extensively, in the context of the theory of congruences between modular forms. This theory plays a key role in the breakthrough work of Wiles [17]. As recalled above, one studies a Hecke algebra T acting on H 1 (X 0 (N), O) m that is isomorphic to M⊕M where M is T-module that is finite f lat over O, and hence of positive depth, and of generic rank one as a T-module. ...
... The patching method introduced in the work of Wiles [49] and Taylor-Wiles [46] on modularity lifting theorems, and instrumental in Wiles' proof of Fermat's Theorem, has undergone intense development. In the original approach of Wiles, patching was combined with level raising results, and a numerical criterion [49,Appendix,Proposition], extended by Lenstra [34], for surjective maps between rings R → T, with T finite flat over O, to be isomorphisms of complete intersections. ...
... The patching method introduced in the work of Wiles [49] and Taylor-Wiles [46] on modularity lifting theorems, and instrumental in Wiles' proof of Fermat's Theorem, has undergone intense development. In the original approach of Wiles, patching was combined with level raising results, and a numerical criterion [49,Appendix,Proposition], extended by Lenstra [34], for surjective maps between rings R → T, with T finite flat over O, to be isomorphisms of complete intersections. In applications in [49], the ring R is a deformation ring for Galois representations and T a Hecke algebra. ...
... In the original approach of Wiles, patching was combined with level raising results, and a numerical criterion [49,Appendix,Proposition], extended by Lenstra [34], for surjective maps between rings R → T, with T finite flat over O, to be isomorphisms of complete intersections. In applications in [49], the ring R is a deformation ring for Galois representations and T a Hecke algebra. Diamond [20,Theorem 2.4] developed the results of Wiles and Lenstra by proving a numerical criterion for freeness of R-modules M finite flat over O, in terms of an augmentation λ : R → O supported on M , that is to say, M p = 0 for p the kernel of λ. ...
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We define a congruence module $\Psi_A(M)$ associated to a surjective $\mathcal{O}$-algebra morphism $\lambda\colon A \to \mathcal{O}$, with $\mathcal{O}$ a discrete valuation ring, $A$ a complete noetherian local $\mathcal{O}$-algebra regular at $\mathfrak{p}$, the kernel of $\lambda$, and $M$ a finitely generated $A$-module. We establish a numerical criterion for $M$ to have a free direct summand over $A$ of positive rank. It is in terms of the lengths of $\Psi_A(M)$ and the torsion part of $\mathfrak{p}/\mathfrak{p}^2$. It generalizes results of Wiles, Lenstra, and Diamond, that deal with the case when the codimension of $\mathfrak{p}$ is zero. Number theoretic applications include integral (non-minimal) $R=\mathbb T$ theorems in situations of positive defect conditional on certain standard conjectures. Here $R$ is a deformation ring parametrizing certain Galois representations and $\mathbb T$ is a Hecke algebra. An example of a positive defect situation is that of proving modularity lifting for 2-dimensional $\ell$-adic Galois representations over an imaginary quadratic field. The proofs combine our commutative algebra results with a generalization due to Calegari and Geraghty of the patching method of Wiles and Taylor--Wiles and level raising arguments that go back to Ribet. The results provide new evidence in favor of the intriguing, and as yet fledgling, torsion analog of the classical Langlands correspondence.
... Wiles [5], and Taylor and Wiles [4], used Galois representations, Frey's elliptic curves and the modular forms associated with them, and Wiles' demonstration of the validity of the Shimura-Taniyama Conjecture, to prove the theorem [ The proof by contradiction I offer below employs algebra that would have been available to Fermat in the 17th century. Using an alternative to the standard binomial expansion, (a + b) n = a n + b n i=1 a n−i (a + b) i−1 , a and b nonzero integers, n a positive integer, I ...
... (5) In order for equation (5) to sum to zero, however, the negative term must be equal in value to the sum of all of the other terms. Thus the last line of equation (5), with its terms regrouped, must sum to less than zero, ...
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Fermat's Last Theorem: A Proof by Contradiction Benson Schae�er* Portland, OR, USA Abstract In this paper I o�er an algebraic proof by contradiction of Fermat's Last Theorem. Using an alternative to the standard binomial expansion, (a+b)n = an + b Pn i=1 an􀀀i(a + b)i􀀀1, a and b nonzero integers, n a positive integer, I show that a simple rewrite of the equation stating the theorem, Ap + (A + b)p = (2A + b 􀀀 c)p; A; b and c positive integers, entails the contradiction of two positive integers that sum to less than zero, (2f + g)(f + g)(f + g + b) Xp􀀀2 i=1 (2f + g)p􀀀2􀀀i(3f + 2g + b)i􀀀1 + (f + b)(f + g)(3f + 2g + b)p􀀀2 + fb(3f + 2g + b)p􀀀2 < 0; f and g positive integers. This contradiction shows that the rewrite has no non-trivial positive integer solutions and proves Fermat's Last Theorem. AMS 2020 subject classi�cation: Diophantine equations, Fermat's equation
... Number theory has been developed leaps and bounds by the immense contributions by a lot of mathematicians. Finally, after 350 years, the theorem was completely proved by Prof. Andrew Wiles, using highly complicated mathematical tools and advanced number theory [2], [3]. ...
... )( )( ) 3 ...
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Fermat's Last Theorem states that it is impossible to find positive integer A, B and C satisfying the equation A n + B n = C n where n is any integer > 2. Taking the proofs of Fermat for the index n = 4, and Euler for n = 3, it is sufficient to prove the theorem for n = p, any prime > 3 [1]. We hypothesize that all r, s and t are non-zero integers in the equation r p + s p = t p and establish contradiction, and prove that st = 0. Just for supporting the proof in the above equation, we have another equation x 3 + y 3 = z 3 Without loss of generality, we assert that both x and y as non-zero integers; z 3 a non-zero integer; z and z 2 irrational. By trial and error method, we create the transformed equations to the above two equations, solving by which we prove the theorem.
... The most famous example of a Diophantine equation appears in Fermat's Last Theorem. This is the statement, asserted by Fermat in 1637 without proof, that the Diophantine equation a n + b n = c n has no solutions in whole numbers when n is at least 3, other than the trivial solutions which arise when abc = 0. Andrew Wiles famously proved the Fermat's Last Theorem in 1995 in his paper "Modular elliptic curves and Fermat's Last Theorem" [34]. The proof is by contradiction employing techniques from algebraic geometry and number theory to prove a special case of the modularity theorem for elliptic curves, which together with Ribet's level lowering theorem gives the long-waited result. ...
... In each of the cases, we can associate a so-called Frey elliptic curve E a,b,c /Q and let ρ E,p be its mod p Galois representation, where E = E a,b,c . Then ρ E,p is irreducible by Mazur [18] and modular by Wiles and Taylor [34] and [29]. Applying Ribet's level lowering theorem [21] one gets that that ρ E,p arises from a weight 2 newform of level 32 for (i) and level 27 for (ii). ...
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Let $K$ be a totally real number field and consider a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature $(p,p,2)$ and $(p,p,3)$ using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of I\c{s}ik, Kara and Ozman. For example, consider $K$ a totally real field of degree $n$ with $2 \nmid h_K^+$ and $2$ inert. Moreover, suppose there is a prime $q\geq 5$ which totally ramifies in $K$ and satisfies $\gcd(n,q-1)=1$, then we know that the equation $a^p+b^p=c^2$ has no primitive, non-trivial solutions $(a,b,c) \in \mathcal{O}_K^3$ with $2 | b$ for $p$ sufficiently large.
... When the residual representationρ f is absolutely irreducible and p-distinguished, it follows from the results of [Wil95] and [Hid88] that C(f ) is generated by a nonzero element η f ∈ I. ...
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Let $E_{/\mathbf{Q}}$ be a CM elliptic curve and $p>3$ a prime of good ordinary reduction for $E$. Assume $L(E,1)=0$ with sign $+1$ (so ${\rm ord}_{s=1}L(E,s)\geq 2$) and $E$ has a rational point of infinite order. We prove that if ${\rm Sel}(\mathbf{Q},V_pE)$ is $2$-dimensional, then a certain generalised Kato class $\kappa_p\in{\rm Sel}(\mathbf{Q},V_pE)$ is nonzero; and conversely, the nonvanishing of $\kappa_p$ implies that ${\rm Sel}(\mathbf{Q},V_pE)$ is $2$-dimensional. For non-CM elliptic curves, a similar result was proved in a joint work with M.-L. Hsieh. The proof in this paper is completely different, and should extend to other settings. Moreover, combined with work of Rubin, we can exhibit explicit bases for $2$-dimensional ${\rm Sel}(\mathbf{Q},V_pE)$.
... The modular degree m E of E is the minimum degree of all modular parametrizations φ : X 0 (N ) → E over Q. The modularity Theorem [24,20,2] implies that it is well-defined. In 2002 Watkins [23] conjectured that for every elliptic curve E over Q we have r ≤ ν 2 (m E ), where ν 2 denotes the 2-adic valuation and r := rank Z (E(Q)). ...
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In 2002 Watkins conjectured that given an elliptic curve defined over $\mathbb{Q}$, its Mordell-Weil rank is at most the $2$-adic valuation of its modular degree. We consider the analogous problem over function fields of positive characteristic, and we prove it in several cases. More precisely, every modular semi-stable elliptic curve over $\mathbb{F}_q(T)$ after extending constant scalars, and every quadratic twist of a modular elliptic curve over $\mathbb{F}_q(T)$ by a polynomial with sufficiently many prime factors satisfy the analogue of Watkins' conjecture. Furthermore, for a well-known family of elliptic curves with unbounded rank due to Ulmer, we prove the analogue of Watkins' conjecture.
... Now let E 0 /Q be an elliptic curve of conductor N, E 0 : y 2 + e 1 xy + e 3 y = x 3 + e 2 x 2 + e 4 x + e 6 , e i ∈ Z, which is modular by [8], [26], [30]. Assume further that the modular parametrization µ : X 0 (N) → E 0 is optimal, i.e., E 0 is a strong Weil curve. ...
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We analyze log-algebraic power series identities for formal groups of elliptic curves over $\mathbb{Q}$ which arise from modular parametrizations. We further investigate applications to special values of elliptic curve $L$-functions.
... This gives an algorithm to list all the solutions of an arbitrary genus 1 equation. In particular, by the genus-degree formula (35), this result covers all 2-variable cubic equations. While Baker's bounds are enormous and the corresponding algorithm is impractical, a practical method for finding all integer solutions to genus 1 equations was later developed by Stroeker and Tzanakis [31]. ...
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This paper reports on the current status of the project in which we order all polynomial Diophantine equations by an appropriate version of "size", and then solve the equations in that order. We list the "smallest" equations that are currently open, both unrestricted and in various families, like the smallest open symmetric, 2-variable or 3-monomial equations. All the equations we discuss are amazingly simple to write down but some of them seem to be very difficult to solve.
... The most important progress in the field of the Diophantine equations has been with Wiles' proof of Fermat's Last Theorem [18,20]. His proof is based on deep results about Galois representations associated to elliptic curves and modular forms. ...
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Let k be a positive integer. In this paper, using the modular approach, we prove that if k≡0(mod4), 30<k<724 and 2k−1 is an odd prime power, then under the GRH, the equation x2+(2k−1)y=kz has only one positive integer solution (x,y,z)=(k−1,1,2). The above results solve some difficult cases of Terai’s conecture concerning this equation.
... Hence NAFL requires that FLT, if true, must necessarily have an elementary proof in NPA. Wiles's proof of FLT [9,10] assumes the consistency of classical theories stronger than PA and hence cannot be formalized in NPA. ...
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Non-Aristotelian finitary logic (NAFL) is a finitistic paraconsistent logic that redefines finitism. It is argued that the existence of nonstandard models of arithmetic is an artifact of infinitary classical semantics, which must be rejected by the finitist, for whom the meaning of ``finite'' is not negotiable. The main postulate of NAFL semantics defines formal truth as time-dependent axiomatic declarations of the human mind, an immediate consequence of which is the following metatheorem. If the axioms of an NAFL theory T are pairwise consistent, then T is consistent. This metatheorem, which is the more restrictive counterpart of the compactness theorem of classical first-order logic, leads to the diametrically opposite conclusion that T supports only constructive existence, and consequently, nonstandard models of T do not exist, which in turn implies that infinite sets cannot exist in consistent NAFL theories. It is shown that arithmetization of syntax, Godel's incompleteness theorems and Turing's argument for the undecidability of the halting problem, which lead classically to nonstandard models, cannot be formalized in NAFL theories. The NAFL theories of arithmetic and real numbers are defined. Several paradoxical phenomena in quantum mechanics, such as, quantum superposition, entanglement, the quantum Zeno effect and wave-particle duality, are shown to be justifiable in NAFL, which provides a logical basis for the incompatibility of quantum mechanics and infinitary (by the NAFL yardstick) relativity theory. Finally, Zeno's dichotomy paradox and its many variants, which pose a problem for classical infinitary reasoning, are shown to be resolvable in NAFL.
... This fact follows from results of Deligne [13] and Deligne-Serre [14]. In particular, landmark work on modularity in [7,48,51] shows that if E/Q is an elliptic curve, then L(s, E) is an analytic L-function. This property was leveraged by Fiorilli [16] to obtain so-called highly biased prime number races in the context of elliptic curves under certain hypotheses. ...
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The Riemann Hypothesis, one of the Millennium Prize Problems, was formulated by Bernhard Riemann in 1859 as part of his attempts to understand how prime numbers are distributed along the number line. In this expository article, we delve into the Deep Riemann Hypothesis for the general linear group $\mathrm{GL}_{n}$, which was posed by the third author and evinces the convergence of partial Euler products of $L$-functions on the critical line. As applications of the Deep Riemann Hypothesis, one can improve on the error term in the Prime Number Theorem obtained from the Riemann Hypothesis and verify a similar phenomenon to Chebyshev's bias for Satake parameters. The aim of introducing the notion of the Deep Riemann Hypothesis is to rid ourselves of a quagmire of the Riemann Hypothesis.
... Which is in contradiction with Fermat's Last Theorem [7]. Therefore, for all p ≥ 3 there is no operation ν on N * , such that, H θp (ν) = +. ...
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The aim of this article is to revolutionize completely the way of understanding the different links between the structures of two given non-empty sets $G$ and $G^{'}$. Indeed, for any operation $\tau$ defined on a non empty set $G$, and for all $\theta$ an injective mapping from $G$ into $G^{'}$, there exists a \textit{unique operation} $\mathcal{H}_{\theta}(\tau)$ on $\theta(G)$ that makes $\theta$ a \textit{homomorphism} from $(G,\tau)$ to $(\theta(G),\mathcal{H}_{\theta}(\tau))$. Hence, if $\theta$ is bijective, therefore, $G$ is isomorphic to $G^{'}$. We establish that the symmetric group of $G$ \textit{acts} on the set of all operations on $G$, then, we can define the \textit{geometry of operations} for any nonempty set $G$. If, there is an operation $\tau$ on $G$, such that, for all $\theta$ in $\mathcal{S}_{G}, \,\mathcal{H}_{\theta}(\tau)=\tau $, then, $\mathcal{S}_{G}$ becomes the group of \textit{automorphisms} of $G$.
... Thus, we may assume that x ≥ 3. By Wiles' solution of Fermat's last theorem [23] it follows that x = y. ...
... Let E be an elliptic curve defined over Q. The modularity theorem [26,24,4] ensures the existence of a non-constant morphism φ : X 0 (N ) → E defined over Q. Denote by φ E the morphism, up to sign, which has minimal degree and which sends the cusp i∞ to the neutral point of E. The modular degree m E of E is the degree of φ E . There are many relevant conjectures in Number Theory about this invariant. ...
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Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with rational $2$-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form $y^2=x^3-dx$, with $d$ a biquadratefree integer.
... The celebrated Fermat's Last Theorem was proven in [22] [19] by Wiles and Taylor-Wiles. Ever since then, it has been natural to attempt to use the same methods to tackle more general forms of the Fermat equation, particularly the still unresolved Beal conjecture [16]. ...
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We prove Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$ and ${\mathbb Q}(\sqrt{17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer-Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of ${\mathbb Q}(\sqrt{5})$ is a generalization to a real quadratic base field of the one used by Chen-Siksek. For the case of ${\mathbb Q}(\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.
... This formula follows Poitou-Tate duality and the global Euler-Poincaré characteristic formula of Tate. This particular statement is a special case of [23, 8.7.9], which is itself a generalization of a formula of Wiles [41]. We note that, in the enumeration of places, each pair of complex embeddings of F corresponds to one place in V 0 . ...
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We study the distribution of fixed point Selmer groups in the twist family of a given Galois module over a number field. Building off the work on higher Selmer groups in the author's preceding paper, we find conditions under which we can compute the distribution of the $\ell^\infty$-Selmer groups for a given degree $\ell$ twist family. Along the way, we show that the average rank in the quadratic twist family of any given abelian variety over a number field is bounded.
... As everyone knows, the study of the existence of solutions for Fermat type equations has always been an important and interesting problem. The famous Fermat's Last Theorem has attracted the attention of many mathematical scholars [1,2]. About 60 years ago or even earlier, Montel [3] and Gross [4] had considered the equation f m + g m = 1 and obtained that the entire solutions of f 2 + g 2 = 1 are f = cos ζðzÞ, g = sin ζðzÞ for the case m = 2, where ζðzÞ is an entire function, and this equation does not admit any nonconstant entire solution for any positive integer m > 2. ...
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Our main aim is to describe the entire solutions of several systems of α 1 f 1 z 2 + α 2 f 2 z + c 2 = 1 , β 1 f 2 z 2 + β 2 f 1 z + c 2 = 1 , α 1 ∂ f 1 / ∂ z 1 n 1 + α 2 f 2 z + c m 1 = 1 , β 1 ∂ f 2 / ∂ z 1 n 2 + β 2 f 1 z + c m 2 = 1 , and α 1 ∂ f 1 / ∂ z 1 2 + α 2 f 2 z + c 2 = 1 , β 1 ∂ f 2 / ∂ z 1 2 + β 2 f 1 z + c 2 = 1 , α 1 ∂ f 1 / ∂ z 1 2 + α 2 f 2 z + c + α 3 f 1 z 2 = 1 , β 1 ∂ f 2 / ∂ z 1 2 + β 2 f 1 z + c + β 3 f 2 z 2 = 1 , where α j , β j j = 1 , 2 , 3 are nonzero constants in ℂ and m j , n j j = 1 , 2 are positive integers. We obtain several theorems on the existence and the forms of solutions for these systems, which are some improvements and supplements of the previous theorems given by Xu and Cao, Gao, and Liu and Yang. Moreover, we give some examples to explain the existence of solutions for such systems.
... Notably, important results in number theory have recently become so loaded with complicated techniques that mathematicians have begun to question whether the proofs extrapolated Peano's axioms. This is the case of Fermat's last theorem and the weak Goldbach conjecture, proved respectively by Andrew Wiles [21,22] and by Harald Helfgott [23]. This type of question is akin to the program of reverse mathematics and has drawn the attention of mathematicians like Harvey Friedman. ...
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We consider the foundational relation between arithmetic and set theory. Our goal is to criticize the construction of standard arithmetic models as providing grounds for arithmetic truth. Our method is to emphasize the incomplete picture of both theories and to treat models as their syntactical counterparts. Insisting on the incomplete picture will allow us to argue in favor of the revisability of the standard-model interpretation. We start briefly characterizing the expansion of arithmetic ‘truth’ provided by the interpretation in a set theory. Interpreted versions of an arithmetic theory into set theories generally have more theorems than the original. This theorem expansion is not complete however. Using this, the set theoretic multiversalist concludes that there are multiple legitimate standard models of arithmetic. We suggest a different multiversalist conclusion: while there is a single arithmetic structure, its interpretation in each universe may vary or even not be possible. We continue by defining the coordination problem. We consider two independent communities of mathematicians responsible for deciding over new axioms for ZF and PA. How likely are they to be coordinated regarding PA’s interpretation in ZF? We prove that it is possible to have extensions of PA not interpretable in a given set theory ST. We further show that the number of extensions of arithmetic is uncountable, while interpretable extensions in ST are countable. We finally argue that this fact suggests that coordination can only work if it is assumed from the start.
... This result does imply that the elliptic curve given above is modular. Therefore, proving Fermat's Last Theorem [20,21]. Many mathematicians are still heavenly involved on studying Fermat's Last Theorem [22,23,24]. ...
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We construct sequences of triples of circulant matrices with positive integers as entries which are solutions of the equation. We introduce Mouanda's choice function for matrices which allows us to construct galaxies of sequences of triples of circulant matrices with positive integers as entries. We give many examples of galaxies of circulant matrices. The characterization of the matrix solutions of the equation allows us to show that the equation 2) has no circulant matrix with positive integers as entries solutions. This allows us to prove that, in general, the equation 3) has no circulant matrix with positive integers as entries solutions. We prove Fermat's Last Theorem for eigenvalues of circulant matrices. Also, we prove Fermat's Last Theorem for complex polynomials over D associated to circulant matrices.
... We also note that the theta blocks associated to elliptic curves given in Theorem 4.16 have first Taylor coefficients whose square when evaluated at certain CM points are essentially the central L-value of the L-function of the elliptic curve as shown in [24]. When we say a weight 2 newform corresponds to an elliptic curve it is meant via the modularity theorem of Taylor and Wiles [29,31]. ...
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We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi form which specializes to $\eta(\tau)^{26}$ which we present to highlight an open question of Dyson and Serre. We give other examples and applications of Jacobi forms with complex multiplication including constructing theta blocks associated to elliptic curves with complex multiplication and new families of congruences and cranks for certain partition functions.
... The celebrated Fermat's last theorem [17] elaborates that it do not exist nonzero rational numbers x, y and an integer n ≥ 3 such that x n + y n = 1. The equation x 2 + y 2 = 1 does admit nontrivial rational solutions. ...
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In this paper, we investigate some systems of the Fermat type differential-difference equations with polynomial coefficients and obtain the condition for the existence of finite order transcendental entire solutions and the expression for the entire solutions. We also give some corresponding examples.
... The well-known "Fermat's Last Theorem", which was proved by Wiles [1] and Taylor and Wiles [2], states that there do not exist non-zero rational numbers x and y and an integer m > 2, such that x m + y m = 1. Analogous to this result of number theory, there have been similar function theory investigations. ...
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We describe the entire solutions for Fermat type functional equations with functional coefficients in Cn , i.e., h f p +k g q = 1, where p, q 2 are two integers. We then apply the result to obtain that entire function solutions f , g of f 2 + g 2 = 1 in n are constant if D f −1 (0) ⊆ D g −1 (0) with ignoring multiplicities, where D := n j=1 z j ∂ ∂ z j is the Euler operator. Meromorphic function solutions of f 3 + g 3 = 1 in n and applications to nonlinear (ordinary and partial) differential equations are also discussed.
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We consider the Kisin variety associated to an n-dimensional absolutely irreducible mod p Galois representation ρ ¯ of a p-adic field K together with a cocharacter μ. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin’s conjecture holds if K is totally ramified with n = 3 or μ is of a very particular form. As an application, we get a connectedness result for the deformation ring associated to ρ ¯ of given Hodge–Tate weights. We also give counterexamples to show Kisin’s conjecture does not hold in general.
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Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows exponentially as the argument expands. When a mathematician encounters a proof, how does she come to believe it? Here we show that, under a cognitively-plausible belief formation mechanism combining deductive and abductive reasoning, belief in mathematical arguments can undergo what we call an epistemic phase transition: a dramatic and rapidly-propagating jump from uncertainty to near-complete confidence at reasonable levels of claim-to-claim error rates. To show this, we analyze an unusual dataset of forty-eight machine-aided proofs from the formalized reasoning system Coq, including major theorems ranging from ancient to 21st Century mathematics, along with five hand-constructed cases including Euclid, Apollonius, Hernstein's Topics in Algebra, and Andrew Wiles's proof of Fermat's Last Theorem. Our results bear both on recent work in the history and philosophy of mathematics on how we understand proofs, and on a question, basic to cognitive science, of how we justify complex beliefs.
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We prove under mild hypotheses the three-variable Iwasawa Main Conjecture for p-ordinary modular forms base changed to an imaginary quadratic field K in which p splits in the indefinite setting (in the definite setting this is a result due to Skinner–Urban). Being in a setting encompassing Heegner points and their variation in p-adic families, our main result has new applications to Greenberg's nonvanishing conjecture for central derivatives of p-adic L-functions of Hida families with root number −1.
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Two well-studied Diophantine equations are those of Pythagorean triples and elliptic curves, for the first we have a parametrization through rational points on the unit circle, and for the second we have a structure theorem for the group of rational solutions. Recently, Yekutieli discussed a connection between these two problems, and described the group structure of Pythagorean triples and the number of triples for a given hypotenuse. In arXiv:2112.03663 we generalized these methods and results to Pell's equation. We find a similar group structure and count on the number of solutions for a given $z$ to $x^2 + Dy^2 = z^2$ when $D$ is 1 or 2 modulo 4 and the class group of $\mathbb{Q}[\sqrt{-D}]$ is a free $\mathbb{Z}_2$ module, which always happens if the class number is at most 2. In this paper, we discuss the main results of arXiv:2112.03663 using some concrete examples in the case of $D=105$.
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We study ring structure of the big ordinary Hecke algebra T with the modular deformation ρT:Gal(Q¯/Q)→GL2(T) of an induced Artin representation IndFQφ from a real quadratic field F with a fundamental unit ε, varying a prime p≥3 split in F. Under mild assumptions (H0)–(H3) given in the Introduction (on the prime p), we prove that T is an integral domain free of even rank e>0 over Λ for the weight Iwasawa algebra Λ étale outside Spec(Λ/p(⟨ε⟩-1)) for ⟨ε⟩:=(1+T)logp(ε)/logp(1+p)∈Zp[[T]]⊂Λ. If p∤e, T is shown to be a normal noetherian domain of dimension 2 with ramification locus exactly given by (⟨ε⟩-1). Moreover, only under p-distinguishedness (H0), we prove that any modular specialization of weight ≥2 of ρT is indecomposable over the inertia group at p (solving a conjecture of Greenberg without exception).
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We present a new Java package, named binMeta, for the development and the study of meta-heuristic searches for global optimization. The solution space for our optimization problems is based on a discrete representation, but it does not restrict to combinatorial problems, for every representation on computer machines finally reduces to a sequence of bits. We focus on general purpose meta-heuristics, which are not tailored to any specific subclass of problems. Although we are aware that this is not the first attempt to develop one unique tool implementing more than one meta-heuristic search, we are motivated by the following three main research lines on meta-heuristics. First, we plan to collect several implementations of meta-heuristic searches, developed by several programmers under the common interface of the package, where a particular attention is given to the common components of the various meta-heuristics. Second, the discrete representation for the solutions that we employ allows the user to perform a preliminary study on the degrees of freedom that is likely to give a positive impact on the performance of the meta-heuristic searches. Third, the choice of Java as a programming language is motivated by its flexibility and the use of a high-level objective-oriented paradigm. Finally, an important point in the development of binMeta is that a meta-heuristic search implemented in the package can also be seen as an optimization problem, where its parameters play the role of decision variables.
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En el mundo globalizado se requiere contar con ingenieros líderes de proyectos, que puedan realizar labores de evaluadores de proyectos con éxito. El sistema propuesto permite hacer un análisis comparativo de los diferentes proyectos participantes en eventos de invención, innovación y creatividad,basados en sus características de calidad en uso, funcionalidad y usabilidad, mediante un plan de métricas externas y de calidad en uso. El modelo está basado en normas internacionales (ISO/IEC 9126, 14598, IEEE 1061) y modelos mexicanos (MECHDAV), y software propuesto, es desarrollado en un ambiente visual WEB, para dispositivos móviles (tabletas), permiten evaluar genéricamente la calidad de los proyectos-productos-servicios que participan en los concursos mencionados; este sistema proporciona unsoporte a las personas evaluadoras (jurados) para emitir dictámenes imparciales con mayor precisión cuantitativa. Este sistema está dirigido a organizaciones, empresas y usuarios finales que necesiten seleccionar, fácilmente, los proyectos desarrollados con más calidad, para ser los ganadores en estos concursos. Se proporciona una guía para la instrumentación concreta de la evaluación, así como sus rangos, la presentación, procedimientos y documentación.
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«Ratio formalis» seems to be a useful classical notion for contemporary metaphysics that currently debates whether the foundation of reality is formaliter the «substance» or the «structure». For this reason, it is necessary to make the notion explicit. First, I will summarize the positions of the debate; then, I will present the general sense of our notion; I will show its precisions according to three authors (Thomas Aquinas, Cajetan and John of Saint Thomas); once I present their subtleties, I will continue by summarizing what has been obtained; almost to finish, I will use mathematics as an example to show the use of the regained notion; I will conclude by showing how our notion is pertinent for the mentioned debate.
Preprint
In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of $\GL_2$-type over totally real fields. The original program was based on hard open conjectures, which has made it difficult to apply in practice. In this paper, building on the progress surrounding the modular method from the last two decades, we analyze and expand the current limits of this program by developing all the necessary ingredients to use Frey abelian varieties for two Diophantine applications. In the first application, we use a multi-Frey approach combining two Frey elliptic curves over totally real fields, a Frey hyperelliptic over~$\Q$ due to Kraus, and ideas from the Darmon program to give a complete resolution of the generalized Fermat equation $$x^7 + y^7 = 3 z^n$$ for all integers $n \ge 2$. The use of higher dimensional Frey abelian varieties allows a more efficient proof of the above result due to additional structures that they afford. In the second application, we use some of the additional structures that Frey abelian varieties possess to give an asymptotic resolution of the generalized Fermat equation $$x^{11} + y^{11} = z^p$$ for solutions locally away from $xy=0$ and where $p$ is a prime exponent. In this application, the use of higher dimensional Frey abelian varieties helps to overcome the computational difficulties arising when working in large spaces of Hilbert modular forms.
Article
This paper reports on the current status of the project in which we order all polynomial Diophantine equations by an appropriate version of "size", and then solve the equations in that order. We list the "smallest" equations that are currently open, both unrestricted and in various families, like the smallest open symmetric, 2-variable or 3-monomial equations. All the equations we discuss are amazingly simple to write down but some of them seem to be very difficult to solve.
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In [13], the authors showed the importance of studying simplicial generalizations of Galois deformation functors. They established a precise link between the simplicial universal deformation ring R pro-representing such a deformation problem (with local conditions) and a derived Hecke algebra. Here we focus on the algebraic part of their study which we complete in two directions. First, we introduce the notion of simplicial pseudo-characters and prove relations between the (derived) deformation functors of simplicial pseudo-characters and that of simplicial Galois representations. Secondly, we define the relative cotangent complex of a simplicial deformation functor and, in the ordinary case, we relate it to the relative complex of ordinary Galois cochains. Finally, we recall how the latter can be used to relate the fundamental group of R to the ordinary dual adjoint Selmer group, by a homomorphism already introduced in [13] and studied in greater generality in [26].
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PROOFS OF BEALS CONJECTURE , FERMAT'S LAST THEOREM, COLLATZ CONJECTURE , GOLDBACHS CONJECTURE AND TWIN PRIME CONJECTURE.
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The Eichler-Shimura isomophism gives a relation between holomorphic cusp forms and period polynomials. In [21], Paşol and Popa introduced vector-valued period polynomials and studied various properties of such polynomials. In this paper, we find a basis for the space of vector-valued period polynomials consisting of Hecke eigenpolynomials, and relate the basis to a Miller-like basis of the space of weakly holomorphic cusp forms by calculation of the matrix representation of the Hecke operator T˜l. Furthermore, we find the relation between our Hecke eigenpolynomials and the even and odd parts of the period polynomials induced from holomorphic Hecke eigenforms, and examine algebracity of the Hecke eigenpolynomials.
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The Riemann-Roch theorem is of utmost importance and a vital tool to the fields of complex analysis and algebraic geometry, specifically in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this paper is to give two proofs of this important theorem and explore some of its numerous consequences. As an application, we compute the genus of some interesting algebraic curves or Riemann surfaces.
Article
For any positive integer $N$, let $J_0(N)$ be the Jacobian variety of the modular curve $X_0(N)$ over ${\mathbb {Q}}$ and ${\mathcal {C}}_N$ its cuspidal subgroup. Let $F_N$ denote the splitting field of ${\mathcal {C}}_N$, which is the smallest number field whose absolute Galois group acts trivially on ${\mathcal {C}}_N$. Let ${\mathcal {J}}_N=J_0(N)(F_{N})_{\textrm {tor}}$ be the torsion subgroup of the group of $F_N$-rational points on $J_0(N)$. We prove that ${\mathcal {J}}_N$ coincides with ${\mathcal {C}}_N$ outside $6N[F_N:{\mathbb {Q}}]$.
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This survey reports on some of the recent developments in the area of Galois representations and automorphic forms, with a particular focus on the author and Thorne’s work on symmetric power functoriality for modular forms.
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Why is it that both complex and simple solutions that have proved to be effective have low rates of adoption? The literature on innovation (i.e., a specific category of solutions) management has provided some clues, identifying barriers of several types: organizational, technological, economic, human behavior and the nature of the innovation. We suggest that one reason is the misalignment between the degrees of complexity i.e., the degree of knowledge embedded, of the problem and its solution. A solution perceived to be too simple for a complex problem falls into the category of what might be called “Columbus' egg”. At the basis of this effect there is the tendency to minimize expected frustration as the difference between the effort made in looking for a solution and the obtained reward. When the solution is too complex for a simple problem, this is the case of the “Engineer's effect”. This effect has its cognitive underpinnings in the tendency to minimize decision-making costs. We discuss and illustrate these phenomena and propose some guidelines for technology developers and product innovation managers, as well as for forecasting solutions adoption.
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A geometric approach to the proof of Fermat’s last theorem is proposed. Instead of integers a, b, c, Fermat’s last theorem considers a triangle with side lengths a, b, c. It is proved that in the case of right-angled and obtuse-angled triangles Fermat's equation has no solutions. When considering the case when a, b, c are sides of an acute triangle, it is proved that Fermat's equation has no entire solutions for p>2. The numbers a=k, b=k+m, c=k+n, where k, m, n are natural numbers satisfying the inequalities n>m, n<k+m, exhaust all possible variants of natural numbers a, b, c which are the sides of the triangle. The proof in this case is carried out by introducing a new auxiliary function f(k,p)=kp+(k+m)p–(k+n)p of two variables, which is a polynomial of degree in the variable . The study of this function necessary for the proof of the theorem is carried out. A special case of Fermat’s last theorem is proved, when the variables a, b, c take consecutive integer values. The proof of Fermat’s last theorem was carried out in two stages. Namely, all possible values of natural numbers k, m, n, p were considered, satisfying the following conditions: firstly, the number (np–mp) is odd, and secondly, this number is even, where the number (np–mp) is a free member of the function f(k, p). Another proof of Fermat’s last theorem is proposed, in which all possible relationships between the supposed integer solution of the equation f(k, p)=0 and the number corresponding to this supposed solution are considered. The proof is carried out using the mathematical apparatus of the theory of integers, elements of higher algebra and the foundations of mathematical analysis. These studies are a continuation of the author's works, in which some special cases of Fermat’s last theorem are proved
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In this paper, we détermine thé structure of a certain /?-di visible group of thé jacobian of X^ (N ff} as a module ovcr thé Hecke algebra, under rathcr weak hypothèses. This will allows us to generalize in a forthcoming paper some rcsults of Hida (7) on />-adic interpolation of spécial values of L séries.
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dThe action of the derivation e = q ~ on the q-expansions of modular forms in characteristic p is one of the fundamental tools in the Serre/Swinnerton-Dyer theory of mod p modular forms. In this note, we extend the basic results about this action, already known for P > 5 and level one, to arbitrary p and arbitrary prime-to-p level. !. Review of modular forms in characteristic p We fix an algebraically closed field K of characteristic p > 0, an integer N > 3 prime to p, and a primitive N'th root of unity ~ ~ K. The moduli problem "elliptic curves E over N structure ~ of determinant {" is K-algebras with level represented by
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Let K be an imaginary quadratic eld and let O be the ring of integers of K. Let E be an elliptic curve deened over Q with complex multiplication by O. Let be the Grr ossencharacter attached to the curve E over K by the theory of complex multiplication, and let L(k ; s) be the complex Hecke L-function attached to the powers of , k = 1; 2; ; here we have xed an embedding of K in C. In a previous paper 7] we studied the general Selmer groups for an ordinary l-adic representation over K or over a Z p-extension K 1 of K. Here we will give some applications of these results to the arithmetic of the Hecke L-functions L(k ; s). Let p be a prime of Q such that p 6 = 2; 3 and E has good, ordinary reduction at p. Fix a Weierstrass model for E y 2 = 4x 3 ? g 2 x ? g 3 (1) such that g 2 and g 3 belong to Zand the discriminant of (1) is prime to p. Let L be the period lattice of the Weierstrass }-function associated with our model, and choose an element 1 2 L such that L = 1 O. Then, if ?d K denotes the discriminant of K, Damerell's theorem shows that the numbers (2 p d K) j ?(j+k) 1 L(j+k ; k) are algebraic and in fact belong to K for integers j and k satisfying 0 j < k. Since E is deened over Q, it follows that these numbers are actually in Q. We would like to address the following two questions: 1. How one can give an algebraic interpretation of the K-values when they are not zero? 2. If these values are zero, how one can give an algebraic interpretation of the order of vanishing? In the special case when j = 0, k = 1, the answer to these questions is suggested by the conjecture of Birch and Swinnerton-Dyer. In general, these questions should t into the framework of the Bloch-Kato Conjecture and the Beilinson-Bloch Conjecture.
Article
The aim of the paper is to prove an elliptic analogue of a deep theorem of Iwasawa on cyclotomic fields. Subject classification ( Amer. Math. Soc. ( MOS ) 1970 ): primary 12 A 35, 12 A 65.
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Soit X le demi-plan de Poincaré $$ X = \{ z \in \mathbb{C}|Im(z) > 0\} . $$ Le groupe SL(2, ℝ) agit sur X par transformations homographiques $$ z \mapsto \frac{{az + b}} {{cz + d}}. $$ Si Γ est un sous-groupe de SL(2, ℤ,) défini par des conditions de congruence, la surface de Riemann X/Γ est le complément d’un ensemble fini de points («à l’infini») dans une surface de Riemann compacte. C’est donc une courbe algébrique. Ses points sont en correspondance bijective avec les classes d’isomorphic de courbes elliptiques munies d’une «structure de niveau» d’espèce convenable. On sait qu’il résulte de cette interprétation qu’elle admet pour corps de définition un sous-corps d’un corps cyclotomique. Dans cet article, nous étudions la structure à l’infini et la réduction modulo p de X/Γ.
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Thesis (Ph. D.)--Princeton University, 1992. Includes bibliographical references (leaves 35-36).
Functions and Tamagawa Numbers of Motives, The Grothendieck Festschrift
  • S Bloch
  • K Kato
S. BLOCH and K. KATO, L-Functions and Tamagawa Numbers of Motives, The Grothendieck Festschrift, Vol. 1 (P. Cartier et al. eds.), Birkhduser, 1990. [BLR]
Linear Groups with an Exposition of the Galois Field Theory, Teub-ner, Leipzig, 1901. This content downloaded from 130.127.238.233 on Wed
  • L E Dickson
L. E. DICKSON, Linear Groups with an Exposition of the Galois Field Theory, Teub-ner, Leipzig, 1901. This content downloaded from 130.127.238.233 on Wed, 25 Sep 2013 07:52:30 AM All use subject to JSTOR Terms and Conditions MODULAR ELLIPTIC CURVES AND FERMAT'S LAST THEOREM 549 [Dr]
Theory of p-adic Hecke algebras and Galois representations
  • Hi
Hi2], Theory of p-adic Hecke algebras and Galois representations, Sugaku Ex-positions 2-3 (1989), 75-102.