Difference between revisions of "Zombies Ate My Neighbors"

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{{SNES| title = Zombies Ate My Neighbors
1Q: What is the current status of Fermat's last theorem?
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|image = [[Image:ZAMN Title.PNG|center]]
 +
|name = ZOMBIES ATE MY NEIGHB
 +
|company = Konami
 +
|header = None
 +
|bank = LoROM
 +
|interleaved = No
 +
|sram = 0 Kb
 +
|type = Normal
 +
|rom = 8 Mb
 +
|country = USA
 +
|video = NTSC
 +
|romspeed = 200ns (SlowROM)
 +
|revision = 1.0
 +
|checksum = Good 0xF50C
 +
|crc32 = 7CFC0C7C
 +
|game = Zombies Ate My Neighbors
 +
}}
  
 +
==Utilities==
 +
* [http://www.romhacking.net/utils/363/ ZAMN-Edit]
 +
* [https://code.google.com/archive/p/zamn-editor/ ZAMN Level Editor] ([http://www.romhacking.net/utilities/1178/ RHDN mirror])
  
and
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==Hacks==
 +
* [http://acmlm.no-ip.org/archive2/thread.php?id=9051 Ultimate Zombies Ate My Neighbors] - All 55 levels changed. ([http://www.romhacking.net/hacks/59/ RHDN mirror])
 +
* [http://www.romhacking.net/hacks/623/ Oh No! More Zombies Ate My Neighbors!]
 +
* [http://www.romhacking.net/hacks/2745/ Sky's Zombies Ate My Neighbors Hack]
  
    Did Fermat prove this theorem?
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==Translations==
     
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* None known.
 
 
    Fermat's Last Theorem:
 
  
    There are no positive integers x,y,z, and n > 2 such that x^n + y^n = z^n.
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==Miscellaneous==
 +
* No miscellaneous yet.
  
    I heard that <insert name here> claimed to have proved it but later
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==Known Dumps==
    on the proof was found to be wrong. ...
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* Zombies (Beta)
 +
* Zombies (E)
 +
* Zombies Ate My Neighbors (U) [!]
  
A:  The status of FLT has remained remarkably constant.  Every few
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==External Links==
    years, someone claims to have a proof ... but oh, wait, not quite.
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* No external links yet.
 
 
    UPDATE... UPDATE... UPDATE
 
 
 
    Andrew Wiles, a researcher at Princeton, Cambridge claims to have
 
    found a proof.
 
 
 
    SECOND UPDATE...
 
 
 
    A mistake has been found. Wiles is working on it. People remain
 
    mildly optimistic about his chances of fixing the error.
 
 
 
    The proposed proof goes like this:
 
 
 
    The proof was presented in Cambridge, UK during a three day seminar
 
    to an audience including some of the leading experts in the field.
 
 
 
    The manuscript has been submitted to INVENTIONES MATHEMATICAE, and
 
    is currently under review.  Preprints are not available until the
 
    proof checks out.  Wiles is giving a full seminar on the proof this
 
    spring.
 
 
 
    The proof is long and cumbersome, but here are some of the first
 
    few details:
 
 
 
    *From Ken Ribet:
 
 
 
    Here is a brief summary of what Wiles said in his three lectures.
 
 
 
    The method of Wiles borrows results and techniques from lots and lots
 
    of people.  To mention a few: Mazur, Hida, Flach, Kolyvagin, yours
 
    truly, Wiles himself (older papers by Wiles), Rubin...  The way he does
 
    it is roughly as follows.  Start with a mod p representation of the
 
    Galois group of Q which is known to be modular.  You want to prove that
 
    all its lifts with a certain property are modular.  This means that the
 
    canonical map from Mazur's universal deformation ring to its "maximal
 
    Hecke algebra" quotient is an isomorphism.  To prove a map like this is
 
    an isomorphism, you can give some sufficient conditions based on
 
    commutative algebra.  Most notably, you have to bound the order of a
 
    cohomology group which looks like a Selmer group for Sym^2 of the
 
    representation attached to a modular form.  The techniques for doing
 
    this come from Flach; you also have to use Euler systems a la
 
    Kolyvagin, except in some new geometric guise.
 
 
 
    CLARIFICATION: This step in Wiles' manuscript, the Selmer group
 
    bound, is currently considered to be incomplete by the reviewers.
 
    Yet the reviewers (or at least those who have gone public) have
 
    confidence that Wiles will fill it in.  (Note that such gaps are
 
    quite common in long proofs.  In this particular case, just such
 
    a bound was expected to be provable using Kolyvagin's techniques,
 
    independently of anyone thinking of modularity.  In the worst of
 
    cases, and the gap is for real, what remains has to be recast, but
 
    it is still extremely important number theory breakthrough work.)
 
 
 
 
 
   
 
    If you take an elliptic curve over Q, you can look at the
 
    representation of Gal on the 3-division points of the curve.  If you're
 
    lucky, this will be known to be modular, because of results of Jerry
 
    Tunnell (on base change).  Thus, if you're lucky, the problem I
 
    described above can be solved (there are most definitely some
 
    hypotheses to check), and then the curve is modular.  Basically, being
 
    lucky means that the image of the representation of Galois on
 
    3-division points is GL(2,Z/3Z).
 
   
 
    Suppose that you are unlucky, i.e., that your curve E has a rational
 
    subgroup of order 3.  Basically by inspection, you can prove that if it
 
    has a rational subgroup of order 5 as well, then it can't be
 
    semistable.  (You look at the four non-cuspidal rational points of
 
    X_0(15).)  So you can assume that E[5] is "nice." Then the idea is to
 
    find an E' with the same 5-division structure, for which E'[3] is
 
    modular.  (Then E' is modular, so E'[5] = E[5] is modular.)  You
 
    consider the modular curve X which parameterizes elliptic curves whose
 
    5-division points look like E[5].  This is a "twist" of X(5).  It's
 
    therefore of genus 0, and it has a rational point (namely, E), so it's
 
    a projective line.  Over that you look at the irreducible covering
 
    which corresponds to some desired 3-division structure.  You use
 
    Hilbert irreducibility and the Cebotarev density theorem (in some way
 
    that hasn't yet sunk in) to produce a non-cuspidal rational point of X
 
    over which the covering remains irreducible.  You take E' to be the
 
    curve corresponding to this chosen rational point of X.
 
   
 
   
 
    *From the previous version of the FAQ:
 
   
 
    (b) conjectures arising from the study of elliptic curves and
 
    modular forms. -- The Taniyama-Weil-Shmimura conjecture.
 
   
 
    There is a very important and well known conjecture known as the
 
    Taniyama-Weil-Shimura conjecture that concerns elliptic curves.
 
    This conjecture has been shown by the work of Frey, Serre, Ribet,
 
    et. al. to imply FLT uniformly, not just asymptotically as with the
 
    ABC conj.
 
   
 
    The conjecture basically states that all elliptic curves can be
 
    parameterized in terms of modular forms.
 
 
 
    There is new work on the arithmetic of elliptic curves. Sha, the
 
    Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the way
 
    an interesting aspect of this work is that there is a close
 
    connection between Sha, and some of the classical work on FLT. For
 
    example, there is a classical proof that uses infinite descent to
 
    prove FLT for n = 4. It can be shown that there is an elliptic curve
 
    associated with FLT and that for n=4, Sha is trivial. It can also be
 
    shown that in the cases where Sha is non-trivial, that
 
    infinite-descent arguments do not work; that in some sense 'Sha
 
    blocks the descent'. Somewhat more technically, Sha is an
 
    obstruction to the local-global principle [e.g. the Hasse-Minkowski
 
    theorem].
 
 
 
    *From Karl Rubin:
 
 
 
    Theorem.  If E is a semistable elliptic curve defined over Q,
 
      then E is modular.
 
 
 
    It has been known for some time, by work of Frey and Ribet, that
 
    Fermat follows from this.  If u^q + v^q + w^q = 0, then Frey had
 
    the idea of looking at the (semistable) elliptic curve
 
    y^2 = x(x-a^q)(x+b^q).  If this elliptic curve comes from a modular
 
    form, then the work of Ribet on Serre's conjecture shows that there
 
    would have to exist a modular form of weight 2 on Gamma_0(2).  But
 
    there are no such forms.
 
   
 
    To prove the Theorem, start with an elliptic curve E, a prime p and let
 
   
 
        rho_p : Gal(Q^bar/Q) -> GL_2(Z/pZ)
 
   
 
    be the representation giving the action of Galois on the p-torsion
 
    E[p].  We wish to show that a _certain_ lift of this representation
 
    to GL_2(Z_p) (namely, the p-adic representation on the Tate module
 
    T_p(E)) is attached to a modular form.  We will do this by using
 
    Mazur's theory of deformations, to show that _every_ lifting which
 
    'looks modular' in a certain precise sense is attached to a modular form.
 
   
 
    Fix certain 'lifting data', such as the allowed ramification,
 
    specified local behavior at p, etc. for the lift. This defines a
 
    lifting problem, and Mazur proves that there is a universal
 
    lift, i.e. a local ring R and a representation into GL_2(R) such
 
    that every lift of the appropriate type factors through this one.
 
   
 
    Now suppose that rho_p is modular, i.e. there is _some_ lift
 
    of rho_p which is attached to a modular form.  Then there is
 
    also a hecke ring T, which is the maximal quotient of R with the
 
    property that all _modular_ lifts factor through T.  It is a
 
    conjecture of Mazur that R = T, and it would follow from this
 
    that _every_ lift of rho_p which 'looks modular' (in particular the
 
    one we are interested in) is attached to a modular form.
 
   
 
    Thus we need to know 2 things:
 
 
 
      (a)  rho_p is modular
 
      (b)  R = T.
 
   
 
    It was proved by Tunnell that rho_3 is modular for every elliptic
 
    curve.  This is because PGL_2(Z/3Z) = S_4.  So (a) will be satisfied
 
    if we take p=3.  This is crucial.
 
   
 
    Wiles uses (a) to prove (b) under some restrictions on rho_p.  Using
 
    (a) and some commutative algebra (using the fact that T is Gorenstein,
 
    'basically due to Mazur')  Wiles reduces the statement T = R to
 
    checking an inequality between the sizes of 2 groups.  One of these
 
    is related to the Selmer group of the symmetric square of the given
 
    modular lifting of rho_p, and the other is related (by work of Hida)
 
    to an L-value.  The required inequality, which everyone presumes is
 
    an instance of the Bloch-Kato conjecture, is what Wiles needs to verify.
 
   
 
    He does this using a Kolyvagin-type Euler system argument.  This is
 
    the most technically difficult part of the proof, and is responsible
 
    for most of the length of the manuscript.  He uses modular
 
    units to construct what he calls a 'geometric Euler system' of
 
    cohomology classes.  The inspiration for his construction comes
 
    from work of Flach, who came up with what is essentially the
 
    'bottom level' of this Euler system.  But Wiles needed to go much
 
    farther than Flach did.  In the end, _under_certain_hypotheses_ on rho_p
 
    he gets a workable Euler system and proves the desired inequality.
 
    Among other things, it is necessary that rho_p is irreducible.
 
   
 
    Suppose now that E is semistable.
 
   
 
    Case 1.  rho_3 is irreducible.
 
    Take p=3.  By Tunnell's theorem (a) above is true.  Under these
 
    hypotheses the argument above works for rho_3, so we conclude
 
    that E is modular.
 
   
 
    Case 2.  rho_3 is reducible.
 
    Take p=5.  In this case rho_5 must be irreducible, or else E
 
    would correspond to a rational point on X_0(15).  But X_0(15)
 
    has only 4 noncuspidal rational points, and these correspond to
 
    non-semistable curves.  _If_ we knew that rho_5 were modular,
 
    then the computation above would apply and E would be modular.
 
   
 
    We will find a new semistable elliptic curve E' such that
 
    rho_{E,5} = rho_{E',5} and rho_{E',3} is irreducible.  Then
 
    by Case I, E' is modular.  Therefore rho_{E,5} = rho_{E',5}
 
    does have a modular lifting and we will be done.
 
   
 
    We need to construct such an E'.  Let X denote the modular
 
    curve whose points correspond to pairs (A, C) where A is an
 
    elliptic curve and C is a subgroup of A isomorphic to the group
 
    scheme E[5].  (All such curves will have mod-5 representation
 
    equal to rho_E.)  This X is genus 0, and has one rational point
 
    corresponding to E, so it has infinitely many.  Now Wiles uses a
 
    Hilbert Irreducibility argument to show that not all rational
 
    points can be images of rational points on modular curves
 
    covering X, corresponding to degenerate level 3 structure
 
    (i.e. im(rho_3) not GL_2(Z/3)).  In other words, an E' of the
 
    type we need exists.  (To make sure E' is semistable, choose
 
    it 5-adically close to E.  Then it is semistable at 5, and at
 
    other primes because rho_{E',5} = rho_{E,5}.)
 
 
 
 
 
    Referencesm:
 
 
 
 
 
    American Mathematical Monthly
 
    January 1994.
 
 
 
    Notices of the AMS, Februrary 1994.   
 
 
 
   
 
###########################################################################
 

Latest revision as of 23:48, 22 February 2016


Zombies Ate My Neighbors
ZAMN Title.PNG
Name ZOMBIES ATE MY NEIGHB
Company Konami
Header None
Bank LoROM
Interleaved No
SRAM 0 Kb
Type Normal
ROM 8 Mb
Country USA
Video NTSC
ROM Speed 200ns (SlowROM)
Revision 1.0
Checksum Good 0xF50C
CRC32 7CFC0C7C
ROM map | RAM map | Text table | Notes | Tutorials

Utilities

Hacks

Translations

  • None known.

Miscellaneous

  • No miscellaneous yet.

Known Dumps

  • Zombies (Beta)
  • Zombies (E)
  • Zombies Ate My Neighbors (U) [!]

External Links

  • No external links yet.