# On a probabilistic local-global principle for torsion on elliptic curves

@article{Cullinan2020OnAP, title={On a probabilistic local-global principle for torsion on elliptic curves}, author={John Cullinan and Meagan Kenney and John Voight}, journal={Journal de th{\'e}orie des nombres de Bordeaux}, year={2020} }

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid\#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability that $m$ divides the the order of the torsion subgroup of $E(\mathbb{Q})$: we find it is nonzero for all $m \in \{ 1, 2, \dots, 10, 12, 16\}$ and we compute it exactly when $m \in \{ 1,2,3,4,5,7 \}$. As a supplement, we give an asymptotic count of elliptic…

## 12 Citations

### Probabilistic behaviors of elliptic curves with torsion points

- Mathematics
- 2020

We study probabilistic behaviors of elliptic curves with torsion points. First, we compute the probability for elliptic curves over the rationals with a non-trivial torsion subgroup $G$ whose size…

### Counting elliptic curves with prescribed level structures over number fields

- MathematicsJournal of the London Mathematical Society
- 2022

Harron and Snowden (J. reine angew. Math. 729 (2017), 151–170) counted the number of elliptic curves over Q$\mathbb {Q}$ up to height X$X$ with torsion group G$G$ for each possible torsion group G$G$…

### Elliptic curves with Galois-stable cyclic subgroups of order 4

- Mathematics
- 2020

Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the…

### Counting elliptic curves with a rational $N$-isogeny for small $N$

- Mathematics
- 2020

We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method…

### Correction: Examples of abelian surfaces failing the local-global principle for isogenies

- MathematicsResearch in Number Theory
- 2022

We provide examples of abelian surfaces over number fields $K$ whose reductions at almost all good primes possess an isogeny of prime degree $\ell$ rational over the residue field, but which…

### N T ] 1 2 A ug 2 02 0 COUNTING ELLIPTIC CURVES WITH PRESCRIBED LEVEL STRUCTURES OVER NUMBER FIELDS

- Mathematics
- 2020

Harron and Snowden [8] counted the number of elliptic curves over Q up to height X with torsion group G for each possible torsion group G over Q. In this paper we generalize their result to all…

### Counting elliptic curves over the rationals with a 7-isogeny

- Mathematics, Computer Science
- 2022

. We count by height the number of elliptic curves over the rationals, both up to isomorphism over the rationals and over an algebraic closure thereof, that admit a cyclic isogeny of degree 7.

### RATIONAL POINTS OF BOUNDED HEIGHT ON GENUS ZERO MODULAR CURVES AND AVERAGE ANALYTIC RANKS OF ELLIPTIC CURVES OVER NUMBER FIELDS

- Mathematics
- 2022

We give asymptotics for the number of isomorphism classes of elliptic curves over arbitrary number fields with certain prescribed level structures and prescribed local conditions. In particular, we…

### Rational Points of Bounded Height on Some Genus Zero Modular Curves over Number Fields

- Mathematics
- 2022

. We give asymptotics for the number of isomorphism classes of elliptic curves over arbitrary number ﬁelds with certain prescribed level structures. In particular, we count the number of points of…

### The average analytic rank of elliptic curves with prescribed torsion

- MathematicsJournal of the London Mathematical Society
- 2022

We show that average analytic rank of elliptic curves with prescribed torsion G is bounded for every torsion group G under GRH for elliptic curve L-functions.

## References

SHOWING 1-10 OF 36 REFERENCES

### Elliptic curves with Galois-stable cyclic subgroups of order 4

- Mathematics
- 2020

Infinitely many elliptic curves over ${\bf Q}$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect in their subgroups of order 2. Let $N_i(X)$ denote the…

### A classification of isogeny-torsion graphs of elliptic curves over $\mathbb{Q}$

- Mathematics, Computer Science
- 2020

The main result of the article is a classification of all the possible isogeny-torsion graphs that occur for elliptic curves defined over the rationals.

### Counting elliptic curves with prescribed torsion

- Mathematics
- 2013

Mazur's theorem states that there are exactly 15 possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs.…

### On Lifting Kleinian Groups to SL (2, ℂ)

- Mathematics
- 1985

The exact sequence of groups and group homomorphisms
$$1 \to \{ \pm I\} \to SL(2,\mathbb{C})\xrightarrow{\mathcal{P}}PSL(2,\mathbb{C}) \to 1$$
(0.1)
does not split. If in this sequence we…

### Possible indices for the Galois image of elliptic curves over Q

- Mathematics
- 2015

For a non-CM elliptic curve E/Q, the Galois action on its torsion points can be expressed in terms of a Galois representation ρE : GalQ := Gal(Q/Q) → GL2(Ẑ). A well-known theorem of Serre says that…

### Computing Classical Modular Forms for Arbitrary Congruence Subgroups.

- Mathematics
- 2020

In this paper, we prove the existence of an efficient algorithm for the computation of $q$-expansions of modular forms of weight $k$ and level $\Gamma$, where $\Gamma \subseteq SL_{2}({\mathbb{Z}})$…

### A CLASSIFICATION OF ISOGENY-TORSION GRAPHS OF ELLIPTIC CURVES OVER Q

- Mathematics, Computer Science
- 2020

The main result of the article is a classification of all the possible isogeny-torsion graphs that occur for elliptic curves defined over the rationals.

### Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

- Mathematics
- 2010

### Abelian L-adic representation and elliptic curves

- MathematicsAdvanced book classics
- 1989

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent…

### The canonical ring of a stacky curve

- MathematicsMemoirs of the American Mathematical Society
- 2022

Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an explicit Gröbner basis. We work in a general…