On the Eisenstein ideal over function fields

Abstract

We study the Eisenstein ideal of Drinfeld modular curves of small levels, and the relation of the Eisenstein ideal to the cuspidal divisor group and the component groups of Jacobians of Drinfeld modular curves. We prove that the characteristic of the function field is an Eisenstein prime number when the level is an arbitrary non square-free ideal of F q [T ] not equal to a square of a prime.

Explicit computation of the modular parametrization of elliptic curves over function fields by Drinfeld modular curves

“…Using Theorem 26, we compute card(E(F p )) for all primes p = (P ) with deg(P ) ≤ 15 and obtain that the image of c is a torsion point in E of order dividing 16. Note that in this example, the bound given by Papikan-Wei in [19] is 4, which is better than ours.…”

“…As before, we compute card(E(F p )) for prime ideals p = (P ) such that deg(P ) ≤ 10 and P ≡ 1 mod n. By Theorem 26, we obtain that the order of image of a cusp divides 8. This bound is sharper than the one of Papikian-Wei [19] which is 24 in this example.…”

“…Remark. In specific cases, explicit bounds for the size of the cuspidal subgroup of the Jacobian are known, for example when n is irreducible by Pál ([18]) or deg(n) = 3 by Papikian and Wei ( [19]). These bounds, which in contrast with Theorem 26 depend only on n, also are upper bounds for the order of torsion points on the elliptic curve coming from evaluating Φ at cusps.…”

The Eisenstein ideal and Jacquet-Langlands isogeny over function fields

“…Remark 2.22. In [44], we have extended the statement of Proposition 2.21 to arbitrary n ✁ A of degree 3. More precisely, we proved that the pairing (2.5) is perfect if deg(n) = 3.…”

On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves $X_0(\mathfrak{p}^r)$

“…In this article, we are interested in the structure of C(n) for prime power n = p r ∈ A. In [8], we have the following: Theorem 2.5 (Papikian and Wei).…”