Mazur’s Eisenstein Descent
نویسنده
چکیده
The purpose of these notes is to illustrate the descent technique used in [Maz72] and [Maz77] to bound the rank of abelian varieties A/Q; this method is used in [Maz77] to prove that the Eisenstein quotient of J0(N)/Q (where N is a prime number) has rank 0. We begin with a brief review of the standard method of descent. We then explain how fppf cohomology can be used following this model to prove our main result (Theorem 3.1); an easy corollary is that an elliptic curve E/Q has rank 0, if it has a non-zero torsion point of order prime to N , good reduction outside a prime N , and multiplicative reduction at N . Finally, the last two sections of the notes provide some examples of abelian varieties to which the theorem applies, constructed using Tate normal form and modular curves.
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