|
|
| Line 1: |
Line 1: |
| In [[mathematics]], '''elliptic units''' are certain units of [[abelian extension]]s of [[imaginary quadratic field]]s constructed using singular values of [[modular function]]s, or division values of [[elliptic function]]s. They were introduced by Gilles Robert in 1973, and were used by [[John Coates (mathematician)|John Coates]] and [[Andrew Wiles]] in their work on the [[Birch–Swinnerton-Dyer conjecture]]. Elliptic units are an analogue for imaginary quadratic fields of [[cyclotomic unit]]s. They form an example of an [[Euler system]].
| | Nice to satisfy you, my name is Ling and I totally dig that name. To play badminton is some thing he truly enjoys performing. Some time ago he chose to reside in Idaho. Bookkeeping has been his day occupation for a whilst.<br><br>Feel free to surf to my web page - auto warranty [[http://Www.nanumgdc.com/forest/board_xUbo69/841259 Recommended Web page]] |
| | |
| A system of elliptic units may be constructed for an [[elliptic curve]] ''E'' with [[complex multiplication]] by the ring of integers ''R'' of an [[imaginary quadratic field]] ''F''. For simplicity we assume that ''F'' has [[Class number (number theory)|class number]] one. Let '''a''' be an ideal of ''R'' with generator α. For a Weierstrass model of ''E'', define
| |
| | |
| :<math>\Theta_{\mathbf{a}} = \alpha^{-12} \Delta_E^{N\mathbf{a} - 1} \prod_{\mathbf{a}P=0, P\ne0} (x-x(P))^{-6} \ . </math>
| |
| | |
| where Δ is the discriminant and ''x'' is the X-coordinate on the Weierstrass model. The function Θ is independent of the choice of model, and is defined over the field of definition of ''E''.
| |
| | |
| Let '''b''' be an ideal of ''R'' coprime to ''a'' and ''Q'' an ''R''-generator of the '''b'''-torsion. Then Θ<sub>'''a'''</sub>(''Q'') is defined over the [[ray class field]] ''K''('''b'''), and if '''b''' is not a prime power then Θ<sub>'''a'''</sub>(''Q'') is a global unit: if '''b''' is a power of a prime '''p''' then Θ<sub>'''a'''</sub>(''Q'') is a unit away from '''p'''.
| |
| | |
| The function Θ<sub>'''a'''</sub> satisfies a ''distribution relation'' for '''b''' = (β) coprime to '''a''':
| |
| | |
| :<math> \prod_{\mathbf{b}Q=0} \Theta_{\mathbf{a}}(P+R) = \Theta_{\mathbf{a}}(\beta P) \ . </math>
| |
| | |
| ==See also==
| |
| *[[Modular unit]]
| |
| | |
| ==References==
| |
| * {{cite book | first1=J.H. | last1=Coates | authorlink1=John Coates (mathematician) | first2=R. | last2=Greenberg | first3=K.A. | last3=Ribet | authorlink3=Kenneth Alan Ribet | first4=K. | last4=Rubin | authorlink4=Karl Rubin | title=Arithmetic Theory of Elliptic Curves | series=Lecture Notes in Mathematics | volume=1716 | publisher=[[Springer-Verlag]] | year=1999 | isbn=3-540-66546-3 }}
| |
| * {{cite journal | first1=John | last1=Coates | authorlink1=John H. Coates | first2=Andrew | last2=Wiles | authorlink2=Andrew Wiles | title=On the conjecture of Birch and Swinnerton-Dyer | journal=[[Inventiones Mathematicae]] | volume=39 | year=1977 | number=3 | pages=223–251 | zbl=0359.14009 |doi = 10.1007/BF01402975}}
| |
| *{{Cite book | last1=Kubert | first1=Daniel S. | last2=Lang | first2=Serge | author2-link=Serge Lang | title=Modular units | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften | isbn=978-0-387-90517-4 | mr=648603 | year=1981 | volume=244 | zbl=0492.12002 }}
| |
| *Robert, Gilles ''Unités elliptiques.'' (Elliptic units) Bull. Soc. Math. France, Supp. Mém. No. 36. Bull. Soc. Math. France, Tome 101. Société Mathématique de France, Paris, 1973. 77 pp.
| |
| | |
| [[Category:Algebraic number theory]]
| |
| [[Category:Modular forms]]
| |
| | |
| | |
| {{numtheory-stub}}
| |
Nice to satisfy you, my name is Ling and I totally dig that name. To play badminton is some thing he truly enjoys performing. Some time ago he chose to reside in Idaho. Bookkeeping has been his day occupation for a whilst.
Feel free to surf to my web page - auto warranty [Recommended Web page]