On Mordell's Inverse Problem in Dimension Three

✍ Scribed by G. Ramharter

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 747 KB
• Volume: 58
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0083

✦ Synopsis

Given a lattice L=AZ 3 with determinant d(L)= |det(A)| >0, let }(L)= sup[vol(P)Â(8d(L)] where the supremum is taken over all o-symmetric parallelepipeds P with faces parallel to the coordinate axes such that P & L= [o]. We prove the following conjecture by Gruber: The absolute minimum of the function }(L) has value 8Â7 cos 2 (?Â7) cos(2?Â7)=0.578416...=}(L*) and is uniquely attained at the critical lattice L* of the star body |x 1 x 2 x 3 | 1. Moreover, this minimum is isolated: There exists a positive ' such that }(L)>}(L*)+' holds for any lattice which is not equivalent to L*. We also state a conjecture concerning higher minima of }(L).