On universal elliptic curves over Igusa curves
type: Item
instance of: scholarly article
title: On universal elliptic curves over Igusa curves
zbMATH Open document ID: 0705.14024
DOI: 10.1007/BF01234424
published in: Inventiones Mathematicae
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143763
review text: Let E be the universal elliptic curve over an Igusa curve. Arithmetic properties of E are studied aiming toward the Birch and Swinnerton-Dyer conjecture. Let K denote the function field of an Igusa curve, and let E be the universal elliptic curve over an Igusa curve. Weierstrass models for E/K, the discriminant and the Hasse invariant of E over \({\mathbb{F}}_ p(j)\) are explicitly determined. Then the Hasse-Weil L-function of E over \({\mathbb{F}}_ p(j)\) is computed in terms of modular forms: Proposition. \(L(E/{\mathbb{F}}_ p(j),s)=H_ q(S_ 3(\Gamma_ 0(p),(\frac{.}{p})),q^{-s})\) where q is a power of p, \((\frac{.}{p})\) the Legendre symbol, and \(H_ q\) is the Hecke polynomial. Further the invariants of E over \({\mathbb{F}}_ p(j)\) such as torsion points, local invariants and Tamagawa numbers are explicitly computed. The functional equation for E is proved: Proposition. Let \(\Lambda(s)=N_ E^{s/2}D^{2s}_{{\mathbb{F}}_ q(j)}L(E/{\mathbb{F}}_ q(j),s)\). Then \(\Lambda(s)= \pm \Lambda(2-s).\) The Tate-Shafarevich group \(\text{Russian {Sh}}({\mathbb{F}}_ p(j),E)\) is computed for some primes p, and the Birch and Swinnerton-Dyer conjecture is verified numerically for these examples.
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 4156632
zbMATH Keywords: universal elliptic curve over an Igusa curve
zbMATH Keywords: Birch and Swinnerton-Dyer conjecture
author: Douglas Ulmer
reviewed by: Noriko Yui
MaRDI profile type: Publication
cites work: L-functions of p-adic characters and geometric Iwasawa theory
cites work: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper
cites work: Heegner points and the modular curve of prime level
cites work: Q4758296
cites work: On the algebraic theory of elliptic modular functions
cites work: Hecke polynomials as congruence \(\zeta\) functions in elliptic modular case
cites work: Arithmetic Moduli of Elliptic Curves. (AM-108)
cites work: The Hasse-Witt matrix of an algebraic curve
cites work: On a conjecture of Artin and Tate
cites work: Q5628266
cites work: On the eigenvalues of Hecke operators
cites work: On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields
cites work: Q5625265
cites work: Algorithm for determining the type of a singular fiber in an elliptic pencil
cites work: Group schemes of prime order
cites work: L-Functions of Universal Elliptic Curves Over Igusa Curves
cites work: Adèles and algebraic groups. (Appendix 1: The case of the group \(G_2\), by M. Demazure. Appendix 2: A short survey of subsequent research on Tamagawa numbers, by T. Ono)
description: scientific article; zbMATH DE number 4156632