Title: | Mathematics at DEC |
Moderator: | RUSURE::EDP |
Created: | Mon Feb 03 1986 |
Last Modified: | Fri Jun 06 1997 |
Last Successful Update: | Fri Jun 06 1997 |
Number of topics: | 2083 |
Total number of notes: | 14613 |
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
T.R | Title | User | Personal Name | Date | Lines |
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975.1 | don't ask me, I just post them | AITG::DERAMO | Daniel V. {AITG,ZFC}:: D'Eramo | Mon Nov 14 1988 21:19 | 4 |
Might even be interesting, if I knew what it was talking about. Dan | |||||
975.2 | Glurp | HERON::BUCHANAN | fragmentary blue | Mon Nov 14 1988 21:25 | 5 |
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 | |||||
975.3 | AITG::DERAMO | Daniel V. {AITG,ZFC}:: D'Eramo | Mon Nov 14 1988 21:45 | 7 | |
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 | |||||
975.4 | SSDEVO::LARY | One more thin gypsy thief | Tue Nov 15 1988 05:51 | 10 | |
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)? | |||||
975.5 | Pure bullshit. Must be midterm time. | POOL::HALLYB | The smart money was on Goliath | Tue Nov 15 1988 11:51 | 0 |
975.6 | A clue? | ATLAST::FRAZER | Je suis prest! | Tue Nov 15 1988 14:10 | 7 |
>< 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 ^^ ===================================================+ | |||||
975.7 | AITG::DERAMO | Daniel V. {AITG,ZFC}:: D'Eramo | Tue Nov 15 1988 14:31 | 9 | |
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 | |||||
975.8 | That's easy for them to say... | CTCADM::ROTH | If you plant ice you'll harvest wind | Thu Nov 17 1988 15:01 | 4 |
It must be true since no hoaxer would have the imagination to come up with such jargon on their own. - Jim | |||||
975.9 | Who is Shafarevich ? | SNOFS1::MOHAN | Tue Oct 17 1989 06:29 | 5 | |
Yes, I have heard of Shafarevich. He is the leading algebraic geometrician from Russia. - Raj |