Galois Theory and Diophantine geometry
نویسنده
چکیده
To a large extent, the investigations to be brought up today arise from a curious inadequacy having to do with the arrow on the left. On the one hand, it is widely acknowledged that the theory of motives finds a strong source of inspiration in Diophantine geometry, inasmuch so many of the structures, conjectures, and results therein have as model the conjecture of Birch and Swinnerton-Dyer, where the concern is with rational points on elliptic curves that can be as simple as
منابع مشابه
A History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids
This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...
متن کاملTernary Diophantine Equations via Galois Representations and Modular Forms
In this paper, we develop techniques for solving ternary Diophantine equations of the shape Axn + Byn = Cz2 , based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C . We conclude with an application of our results to certain classical polynomial-exponential equat...
متن کاملRational Points on Atkin-lehner Twists of Modular Curves
These are the (more detailed) notes accompanying a talk that I am to give at the University of Pennsylvania on July 21, 2006. The topic is rational points on Atkin-Lehner twists of the modular curves X0(N). Apart from being an interesting Diophantine problem in its own right, there is an ulterior motive: Q-rational points correspond to “elliptic Q-curves” and thus to projective Galois represent...
متن کاملDensities of Quartic Fields with Even Galois Groups
Let N(d,G,X) be the number of degree d number fields K with Galois group G and whose discriminant DK satisfies |DK | ≤ X. Under standard conjectures in diophantine geometry, we show that N(4, A4, X) X2/3+ , and that there are N3+ monic, quartic polynomials with integral coefficients of height ≤ N whose Galois groups are smaller than S4, confirming a question of Gallagher. Unconditionally we hav...
متن کاملRECIPES FOR TERNARY DIOPHANTINE EQUATIONS OF SIGNATURE (p, p, k)
In this paper, we survey recent work on ternary Diophantine equations of the shape Axn + Byn = Czm for m ∈ {2, 3, n} where n ≥ 5 is prime. Our goal is to provide a simple procedure which, given A, B, C and m, enables us to decide whether techniques based on the theory of Galois representations and modular forms suffice to ensure that corresponding ternary equations lack nontrivial solutions in ...
متن کامل