The Kernel of the Eisenstein Ideal

✍ Scribed by János A. Csirik

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 204 KB
• Volume: 92
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2001.2709

✦ Synopsis

Let N be a prime number, and let J 0 (N) be the Jacobian of the modular curve X 0 (N). Let T denote the endomorphism ring of J 0 (N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I ı T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J 0 (N) [I] of this ideal (the set of points in J 0 (N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q ¯/Q) on J 0 (N) [I]. Our results were previously known in the special case where N -1 is not divisible by 16.