Topic
Global field
About: Global field is a research topic. Over the lifetime, 417 publications have been published within this topic receiving 4908 citations.
Papers published on a yearly basis
Papers
TL;DR: In this paper, it was shown that a positive proportion of hyperelliptic curves of even genus g. 2 over a global field k have a Jacobian with nonsquare #.. (if finite).
Abstract: Let (A, e) be a principally polarized abelian variety defined over a global field k, and let ..(A) be its Shafarevich-Tate group. Let ..(A)nd denote the quotient of ..(A) by its maximal divisible subgroup. Cassels and Tate constructed a nondegenerate pairing ..(A)nd . ..(A)nd > Q/Z. If A is an elliptic curve, then by a result of Cassels the pairing is alternating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on ..(A)nd. These criteria are expressed in terms of an element c . ..(A)nd that is canonically associated to the polarization e. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether #..(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g . 2 over Q have a Jacobian with nonsquare #.. (if finite). For example, it appears that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can be applied in other situations.
244 citations
TL;DR: In this article, it was shown that a hyperring of the form R / G (where R is a ring and G ⊂ R × is a subgroup of its multiplicative group) is a commutative hyperring extension of a global field K if and only if G ∪ { 0 }, where G ∈ R × denotes a subfield of R. This result applies to the adele class space which thus inherits the structure of the hyperring H K of K.
126 citations
TL;DR: In this article, the generalized Lefschetz trace formula in /-adic cohomology was used to give a global cohomological expression for the L-function of a smooth projective variety over a global field.
Abstract: For a smooth projective variety X over a global field k consider the completed L-function of H w(x). It is defined as the Euler product over all places v of k of the local L-factors L~(Hw(X), s) introduced in [Se]. In the function field case the generalized Lefschetz trace formula in /-adic cohomology gives a global cohomological expression for the L-function. This has been used on the following purposes:
126 citations
TL;DR: In this paper, the authors developed a theory of functorial $\F_1$-schemes which reconciles the previous attempts by C. Soul\'e and A. Deitmar.
Abstract: We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial $\F_1$-schemes which reconciles the previous attempts by C. Soul\'e and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of ad\`ele classes over an arbitrary global field, how to apply our functorial theory of $\Mo$-schemes to interpret conceptually the spectral realization of zeros of $L$-functions.
116 citations
TL;DR: In this article, the authors obtained counting and equidistribution results for the S-integral points of a symmetric variety defined over K and S a finite set of places of K. Their results are effective when K is a number field.
Abstract: Let K be a global field of characteristic not 2. Let Z = H\G be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the S-integral points of Z. Our results are effective when K is a number field.
90 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2024 | 1 |
| 2023 | 3 |
| 2022 | 8 |
| 2021 | 19 |
| 2020 | 22 |
| 2019 | 22 |