Number fields and function fields: coalescences, contrasts and emerging applications
نویسندگان
چکیده
منابع مشابه
Statistics of Number Fields and Function Fields
We discuss some problems of arithmetic distribution, including conjectures of Cohen-Lenstra, Malle, and Bhargava; we explain how such conjectures can be heuristically understood for function fields over finite fields, and discuss a general approach to their proof in the function field context based on the topology of Hurwitz spaces. This approach also suggests that the Schur multiplier plays a ...
متن کاملElliptic Curves and Analogies Between Number Fields and Function Fields
Well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we discuss traffic of this sort, in both directions, in the theory of elliptic curves. In the first part of the paper, we consider various works on Heegner points and Gross–Zagier formulas in the function field context; these works lead to a ...
متن کاملBounded Gaps between Primes in Number Fields and Function Fields
The Hardy–Littlewood prime k-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field Fq(t).
متن کاملHILBERT’S TENTH PROBLEM FOR FUNCTION FIELDS OF VARIETIES OVER NUMBER FIELDS AND p-ADIC FIELDS
Let k be a subfield of a p-adic field of odd residue characteristic, and let L be the function field of a variety of dimension n ≥ 1 over k. Then Hilbert’s Tenth Problem for L is undecidable. In particular, Hilbert’s Tenth Problem for function fields of varieties over number fields of dimension ≥ 1 is undecidable.
متن کاملNumber of Points of Function Fields over Finite Fields
Definition 1. The category Mot∼ is the Karoubian envelope (or idempotent completion) of the quotient of Mot ∼ by the ideal consisting of morphisms factoring through an object of the form M ⊗L, where L is the Lefschetz motive. This is a tensor additive category. If M ∈ Mot ∼ , we denote by M̄ its image in Mot∼. Lemma 1 ([6, Lemmas 5.3 and 5.4]). Let X, Y be two smooth projective irreducible k-var...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
سال: 2015
ISSN: 1364-503X,1471-2962
DOI: 10.1098/rsta.2014.0315