Galois criterion for torsion points of Drinfeld modules

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

Hessian matrices, automorphisms of p-groups, and torsion points of elliptic curves

We describe the automorphism groups of finite p-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of j-invariant 1728 given in Weierstrass form. We interpret these orders in terms of the numbers of 3-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arising in these examples are polynomial on Frobenius sets and vary with the primes in a nonquasipolynomial manner.

Mazur’s rational torsion result for pointless genus one curves: examples

This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\mathbb {Q}$$ Q corresponding to the torsion orders 7, 8, 9, 10, 12 (and hence, all possibilities) occurring in Mazur’s theorem. In fact three distinct methods are proposed for constructing such examples, each involving different in our opinion quite nice ideas from the arithmetic of elliptic curves or from algebraic geometry.

p-adic Integration on Bad Reduction Hyperelliptic Curves

In this paper, we introduce an algorithm for computing $p$-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of $p$-adic integration: Berkovich–Coleman integrals, which can be performed locally, and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve with basic wide-open sets, we reduce the computation of Berkovich–Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya and to Balakrishnan and Besser for regular and meromorphic $1$-forms, respectively. We then employ tropical geometric techniques due to the 1st-named author with Rabinoff and Zureick-Brown to convert the Berkovich–Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish.

On the moments of torsion points modulo primes and their applications

Let [Formula: see text] be the group of [Formula: see text]-torsion points of a commutative algebraic group [Formula: see text] defined over a number field [Formula: see text]. For a prime [Formula: see text] of [Formula: see text], we let [Formula: see text] be the number of [Formula: see text]-solutions of the system of polynomial equations defining [Formula: see text] when reduced modulo [Formula: see text]. Here, [Formula: see text] is the residue field at [Formula: see text]. Let [Formula: see text] denote the number of primes [Formula: see text] of [Formula: see text] such that [Formula: see text]. We then, for algebraic groups of dimension one, compute the [Formula: see text]th moment limit [Formula: see text] by appealing to the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of [Formula: see text] on [Formula: see text] copies of [Formula: see text] by an application of Burnside’s Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on [Formula: see text] copies of a set.