Conference rusure::math

975.0. "Tate-Shafarevich Group" by AITG::DERAMO (Daniel V. {AITG,ZFC}:: D'Eramo) Mon Nov 14 1988 18:16

Newsgroups: sci.math
Path: decwrl!purdue!mailrus!uflorida!gatech!linus!bs
Subject: Tate-Shafarevich Group
Posted: 11 Nov 88 17:47:06 GMT
Organization: The MITRE Corporation, Bedford MA
 
 
A couple of weeks ago I posted a query regarding a rumor I heard that
someone had proved that SHA was finite. This is one of the more
important outstanding conjectures in mathematics today.
 
Well, the following was just posted to the Number Theory net:
 
=====================================================================
 
The purpose of this note is to announce the following
 
Theorem. Let E be a modular elliptic curve of conductor N defined over Q
without complex multiplication such that L(1,E) is not 0. Then exist
infinitely many imaginary quadratic fields K of discriminant D<0 such that
the twisted L-function L(s,E,chi_D) has a first order zero at s=1, where
chi_D is the quadratic Dirichlet character associated with K, and such that
every prime dividing N splits in K.
 
The significance of this is that, according to recent work of Kolyvagin,
the existence of such a twist implies the finiteness of E(Q), and of the
Tate-Shafarevich group of E over Q.
 
The method is the same as in our paper ``Eisenstein series on the
metaplectic group and nonvanishing theorems for automorphic  L-functions
and their derivatives.'' From the modular form associated with E we form
an Eisenstein series on the metaplectic group (the double cover of
GSp(4,R)).  We prove that the Whittaker models of this Eisenstein series
involve the values of L(s,E,chi_D).
 
To this Eisenstein series we apply an integral transform introduced by
Novodvorsky.  This yields a Dirichlet series with known analytic properties
involving an auxiliary variable u in which the coefficient of D-u is
essentially L(s,E,chi_D).  The theorem comes from analysis of the pole
at u=1 of the derivative at s=1 of this Dirichlet series.
 
--Daniel Bump, Solomon Friedberg and Jeffrey Hoffstein
 
 
 
========================================================================
Bob Silverman

     Might even be interesting, if I knew what it was talking
     about.
     
     Dan

	Sheesh, just shows how far from the frontiers we are!   I understand
almost none of terms used as nouns in the description, let alone what the
sentences are all about.

Andrew

     We should start a note where we each make up a result that
     sounds like real mathspeak but is actually nonsense.
     
     Or if the description fits, post a pointer to one of our
     previous entries. :-)
     
     Dan

Could this be a hoax? The only name I recognize in it is Dirichlet, and I
associate him with "classical" number theory - pre-algebraic-fields stuff.
You know, integers and all that.

The very density of indigestible terms also suggests the possibility of
a hoax, but then it could also suggest that I made the right choice in
becoming a Computer Bum in college instead of a mathematician...

Anyone recognize any of the other names? Or terms (other than the complex
variables stuff)?

>< Note 975.0 by AITG::DERAMO "Daniel V. {AITG,ZFC}:: D'Eramo" >
>                          -< Tate-Shafarevich Group >-
>
>Newsgroups: sci.math
>Path: decwrl!purdue!mailrus!uflorida!gatech!linus!bs
                                                   ^^
===================================================+

     re .6
     
     The "bs" presumably stands for Bob Silverman, the name at
     the bottom of .0.
     
     I've heard of "every prime dividing N splits in K" before in
     algebraic number theory.
     
     Dan

    It must be true since no hoaxer would have the imagination to come
    up with such jargon on their own.

    - Jim

    
    Yes, I have heard of Shafarevich. He is the leading algebraic
    geometrician from Russia. 
    
    - Raj