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Register a new userThe conjectures of Artin-Tate and Birch-Swinnerton-Dyer
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We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of the generic fibre. As a byproduct, we obtain a new proof of a theorem of Geisser relating the orders of the Brauer group and the Tate-Shafarevich group.
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Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here we extend the original empirical approach, to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine-learning and topological data analysis. Key quantities such as rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point-cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstrass coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient boosted trees, machine learning is applied in finding inter-correlation amongst the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics.
We take an approach toward counting the number of n for which the curves E_n: y^2=x^3-n^2x have 2-Selmer groups of a given size. This question was also discussed in a pair of Invent. Math. papers by Roger Heath-Brown. We discuss the connection between computing the size of these Selmer groups and verifying cases of the Birch and Swinnerton-Dyer Conjecture. The key ingredient for the asymptotic formulae is the ``independence of the Legendre symbol evaluated at the prime divisors of an integer with exactly k prime factors.
This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem.
We explain how to reconstruct the category of Artin-Tate $mathbb{R}$-motivic spectra as a deformation of the purely topological $C_2$-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate category of $C_2$-equivariant sheaves on the moduli stack of formal groups. As such, our results directly generalize the cofiber of $tau$ philosophy that has revolutionized classical stable homotopy theory. A key observation is that the Artin-Tate subcategory of $mathbb{R}$-motivic spectra is easier to understand than the previously studied cellular subcategory. In particular, the Artin-Tate category contains a variant of the $tau$ map, which is a feature conspicuously absent from the cellular category.
Let k be a number field, and let S be a finite set of k-rational points of P^1. We relate the Deligne-Goncharov contruction of the motivic fundamental group of X:=P^1_k- S to the Tannaka group scheme of the category of mixed Tate motives over X.
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