Locally imprimitive points on elliptic curves

Francesco Pappalardi; Nathan Jones; Peter Stevenhagen

arxiv: 2304.03964 · v1 · submitted 2023-04-08 · 🧮 math.NT · math.AG

Locally imprimitive points on elliptic curves

Nathan Jones , Francesco Pappalardi , Peter Stevenhagen This is my paper

Pith reviewed 2026-05-24 09:14 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords elliptic curvesGalois representationsprimitive pointscyclic reductionMordell-Weil grouplocal-global principlesnumber fields

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The pith

A Galois representation into GL_3 of the profinite integers detects when globally primitive points on elliptic curves fail to generate the local point groups at primes of cyclic reduction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines elliptic curves E over a number field K that possess infinitely many primes of cyclic reduction. It shows that a point P in E(K) which is globally primitive, meaning not a perfect power in the Mordell-Weil group, can still fail to generate the full group E(k_p) at many such primes p. The failure is captured by an associated three-dimensional Galois representation whose image controls the local generation behavior. The authors apply this description to produce explicit examples of points that are locally imprimitive at a positive density of primes. A reader cares because this supplies a concrete arithmetic setting where a local-global principle for group generation does not hold, paralleling the situation for primitive roots in number fields.

Core claim

Under the hypothesis that E/K has infinitely many primes of cyclic reduction, possibly conditional on GRH, there exist globally primitive points P in E(K) that do not generate E(k_p) for any prime p of cyclic reduction. The paper accounts for this by means of the Galois representation rho_{E/K,P} from the absolute Galois group of K to GL_3 of the profinite integers, whose properties determine the local imprimitivity. Concrete non-trivial examples of such points are constructed using this representation.

What carries the argument

The Galois representation rho_{E/K, P}: G_K to GL_3(Ẑ) associated to the elliptic curve and point, which encodes the Galois action on the point and the torsion and thereby detects the local generation failure.

If this is right

Non-trivial examples of globally primitive yet locally imprimitive points exist on elliptic curves satisfying the cyclic reduction hypothesis.
The set of primes at which such a point fails to generate has positive density.
The three-dimensional representation supplies a criterion for deciding when local imprimitivity occurs for a given point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same representation could be used to investigate local-global generation questions for points on abelian varieties of higher dimension.
For curves of small conductor, the image of the representation can be computed directly to locate additional examples.
The construction suggests that density statements for the exceptional primes may be provable unconditionally in some cases.

Load-bearing premise

The elliptic curve E over K has infinitely many primes of cyclic reduction, possibly under the generalized Riemann hypothesis.

What would settle it

An explicit elliptic curve E/K together with a globally primitive point P for which P generates E(k_p) at all but finitely many primes p of cyclic reduction.

read the original abstract

Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^*$) is a primitive root modulo a set of primes of $K$ of positive density. For elliptic curves $E/K$ that are known to have infinitely many primes $\mathfrak p$ of cyclic reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail to generate any of the point groups $E(k_{\mathfrak p})$. We describe this phenomenon in terms of an associated Galois representation $\rho_{E/K, P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper introduces a 3-dimensional Galois representation to construct explicit examples of globally primitive points that are locally imprimitive at cyclic reduction primes.

read the letter

The main takeaway is that Jones, Pappalardi and Stevenhagen introduce a three-dimensional Galois representation attached to an elliptic curve and a point to capture local imprimitivity, and they use it to produce explicit examples of points that are globally primitive but fail to generate the group at primes of cyclic reduction. This is new content. The representation ρ_{E/K, P} to GL_3 of the profinite integers seems to be the tool they develop for this, going beyond the background on primitive roots and cyclic reduction that they cite. The paper does a good job of making the failure of local generation concrete through this representation rather than leaving it as an abstract possibility. The soft spot is that the whole story is conditional on the curve having infinitely many primes of cyclic reduction, which holds unconditionally for some curves but requires GRH for others. The examples they construct are for curves where this is known or assumed, so the result is not fully unconditional. The approach avoids circularity by defining the representation and then finding points where it shows the imprimitivity. This paper is for number theorists working on elliptic curves over number fields and questions about generators of their reductions. A reader interested in analogies between multiplicative groups and elliptic curve groups will find the examples useful. I would recommend sending it to peer review. The construction is explicit and the representation provides a new way to study the problem.

Referee Report

2 major / 1 minor

Summary. The manuscript studies points P on elliptic curves E over number fields K that are globally primitive (not a multiple of another point in E(K)) but locally imprimitive, meaning P does not generate E(k_𝔭) for primes 𝔭 of cyclic reduction. It associates to such data a Galois representation ρ_{E/K,P}:G_K→GL_3(Ẑ) and, conditional on the existence of infinitely many primes of cyclic reduction (possibly under GRH), constructs explicit non-trivial examples of this phenomenon.

Significance. If the constructions are correct, the work supplies the first concrete examples of locally imprimitive points on elliptic curves together with a 3-dimensional Galois representation that encodes the failure of local generation. This supplies a precise arithmetic-geometric analogue of the failure of Artin’s primitive-root conjecture in the elliptic setting and furnishes falsifiable, explicitly constructed instances that can be checked at individual primes.

major comments (2)

[Abstract] Abstract and §1: the central examples rest on the existence of infinitely many primes of cyclic reduction for the chosen E/K. The manuscript must state explicitly, for each curve used in the constructions, whether this infinitude is known unconditionally or only under GRH; without this clarification the load-bearing hypothesis remains imprecise.
The definition of ρ_{E/K,P} (presumably in §2 or §3) must be given in sufficient detail to verify that the image of a Frobenius element at a prime of cyclic reduction lies outside the subgroup corresponding to the cyclic subgroup generated by P; the current outline does not yet permit this verification.

minor comments (1)

Notation: the hat on Ẑ in the target of ρ should be written consistently as Ẑ throughout; a single instance of Ẑ appears in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments on the manuscript. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

Referee: [Abstract] Abstract and §1: the central examples rest on the existence of infinitely many primes of cyclic reduction for the chosen E/K. The manuscript must state explicitly, for each curve used in the constructions, whether this infinitude is known unconditionally or only under GRH; without this clarification the load-bearing hypothesis remains imprecise.

Authors: We agree that the status of the infinitude hypothesis must be made explicit for each example. In the revised manuscript we will add, in §1 and in the section containing the explicit constructions, a clear statement for every curve E/K indicating whether the existence of infinitely many primes of cyclic reduction is known unconditionally or only under GRH. A short table summarizing this information for all examples will also be included. revision: yes
Referee: The definition of ρ_{E/K,P} (presumably in §2 or §3) must be given in sufficient detail to verify that the image of a Frobenius element at a prime of cyclic reduction lies outside the subgroup corresponding to the cyclic subgroup generated by P; the current outline does not yet permit this verification.

Authors: We will expand the definition of the representation ρ_{E/K,P} in §2. The revised text will include the explicit action on the three-dimensional module, the precise embedding of the cyclic subgroup generated by P, and the criterion showing that Frobenius elements at primes of cyclic reduction lie outside the corresponding subgroup. This will make the verification of local imprimitivity direct from the definition. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines an associated Galois representation ρ_{E/K,P} to describe the locally imprimitive phenomenon for globally primitive points on elliptic curves with infinitely many cyclic reduction primes, then uses this definition to construct explicit examples. This is a standard definitional framework for organizing and exhibiting mathematical objects rather than a derivation that reduces to its own inputs by construction. No fitted parameters, self-citation load-bearing steps, or renamings of known results appear in the provided material; the central claim rests on the existence assumption (possibly under GRH) which is stated as a hypothesis rather than derived internally.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so a full audit of free parameters, axioms, and invented entities cannot be performed. The GRH is invoked conditionally for the density statements, and the Galois representation is associated rather than newly postulated without evidence.

axioms (2)

domain assumption GRH for the density of primes where elements are primitive roots
Invoked in the first sentence of the abstract for the multiplicative group statement.
domain assumption Existence of infinitely many primes of cyclic reduction for E/K
Stated as a hypothesis for the elliptic curve case in the abstract.

pith-pipeline@v0.9.0 · 5659 in / 1274 out tokens · 22128 ms · 2026-05-24T09:14:36.723841+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

W einberger, On the construction of division chains in algebraic number rings, with applications to SL2, Comm

George Cooke and Peter J. W einberger, On the construction of division chains in algebraic number rings, with applications to SL2, Comm. Algebra 3 (1975), 481–524

work page 1975
[2]

Francesco Campagna and Peter Stevenhagen, Cyclic reduction densities for elliptic curves , ArXiv preprint 2001.00028, submitted

work page arXiv 2001
[3]

Ram Murty, Cyclicity and generation of points mod p on elliptic curves , Invent

Rajiv Gupta and M. Ram Murty, Cyclicity and generation of points mod p on elliptic curves , Invent. Math. 101 (1990), no. 1, 225–235. [4] , Primitive points on elliptic curves , Compositio Math. 58 (1986), no. 1, 13–44

work page 1990
[4]

Reine Angew

Christopher Hooley, On Artin ’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220

work page 1967
[5]

Dale Husem¨ oller, Elliptic curves, 2nd edition , Springer Verlag, New York, 2004

work page 2004
[6]

108 (2003), no

Leopoldo Kulesz, Families of elliptic curves of high rank with nontrivial tor sion group over Q, Acta Arith. 108 (2003), no. 4, 339–356

work page 2003
[7]

Serge Lang and Hale Trotter, Primitive points on elliptic curves , Bull. Amer. Math. Soc. 83 (1977), no. 2, 289–292

work page 1977
[8]

H. W. Lenstra Jr., On Artin ’s conjecture and Euclid’s algorithm in global ﬁeld s, Invent. Math. 42 (1977), 201–224

work page 1977
[9]

H. W. Lenstra Jr., P. Stevenhagen, and P. Moree, Character sums for primitive root densities , Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 489–511

work page 2014
[10]

Giulio Meleleo, Questions related to Primitive Points on Elliptic Curves an d Statistics for Biquadratic Curves over Finite Fields , Tesi di Dottorato, Universit` a Roma Tre (2015)

work page 2015
[11]

Lo ¨ ıc Merel,Bornes pour la torsion des courbes elliptiques sur les corps de nombres , Invent. Math. 124 (1996), no. 1-3, 437–449 (French)

work page 1996
[12]

Jean-Pierre Serre, Propri´ et´ es galoisiennes des points d’ordre ﬁni des courbes elliptiques , In- vent. Math. 15 (1972), no. 4, 259–331

work page 1972
[13]

, Abelian ℓ-adic representations and elliptic curves , Benjamin, New York, 1968

work page 1968
[14]

Reprint of the 2003 edition [of the 1986 or iginal MR0926691]

, R´ esum´ e des cours de 1977-1978, Springer Collected W orks in Mathematics, Springer, Heidelberg, 2013. Reprint of the 2003 edition [of the 1986 or iginal MR0926691]

work page 1977
[15]

2, 26–37

Peter Stevenhagen and Hendrik Lenstra, Chebotarev and his density theorem , The Mathe- matical Intelligencer 18 (1996), no. 2, 26–37

work page 1996
[16]

Peter Stevenhagen, The correction factor in Artin ’s primitive root conjecture, J. Th´ eor. Nom- bres Bordeaux 15 (2003), no. 1, 383–391 (English, with English and French sum maries). Les XXII` emes Journ´ ees Arithmetiques (Lille, 2001)

work page 2003
[17]

Jacques V´ elu,Isog´ enies entre courbes elliptiques, C. R. Acad. Sci. Paris S´ er. A-B 273 (1971), A238–A241 (French). Department of Mathematics, Statistics and Computer Scienc e, University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago, IL 60607, USA Dipartimento di Matematica, Universit `a Roma Tre, Largo S. L. Murialdo 1, I–00146 Rome, Ita...

work page 1971

[1] [1]

W einberger, On the construction of division chains in algebraic number rings, with applications to SL2, Comm

George Cooke and Peter J. W einberger, On the construction of division chains in algebraic number rings, with applications to SL2, Comm. Algebra 3 (1975), 481–524

work page 1975

[2] [2]

Francesco Campagna and Peter Stevenhagen, Cyclic reduction densities for elliptic curves , ArXiv preprint 2001.00028, submitted

work page arXiv 2001

[3] [3]

Ram Murty, Cyclicity and generation of points mod p on elliptic curves , Invent

Rajiv Gupta and M. Ram Murty, Cyclicity and generation of points mod p on elliptic curves , Invent. Math. 101 (1990), no. 1, 225–235. [4] , Primitive points on elliptic curves , Compositio Math. 58 (1986), no. 1, 13–44

work page 1990

[4] [4]

Reine Angew

Christopher Hooley, On Artin ’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220

work page 1967

[5] [5]

Dale Husem¨ oller, Elliptic curves, 2nd edition , Springer Verlag, New York, 2004

work page 2004

[6] [6]

108 (2003), no

Leopoldo Kulesz, Families of elliptic curves of high rank with nontrivial tor sion group over Q, Acta Arith. 108 (2003), no. 4, 339–356

work page 2003

[7] [7]

Serge Lang and Hale Trotter, Primitive points on elliptic curves , Bull. Amer. Math. Soc. 83 (1977), no. 2, 289–292

work page 1977

[8] [8]

H. W. Lenstra Jr., On Artin ’s conjecture and Euclid’s algorithm in global ﬁeld s, Invent. Math. 42 (1977), 201–224

work page 1977

[9] [9]

H. W. Lenstra Jr., P. Stevenhagen, and P. Moree, Character sums for primitive root densities , Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 489–511

work page 2014

[10] [10]

Giulio Meleleo, Questions related to Primitive Points on Elliptic Curves an d Statistics for Biquadratic Curves over Finite Fields , Tesi di Dottorato, Universit` a Roma Tre (2015)

work page 2015

[11] [11]

Lo ¨ ıc Merel,Bornes pour la torsion des courbes elliptiques sur les corps de nombres , Invent. Math. 124 (1996), no. 1-3, 437–449 (French)

work page 1996

[12] [12]

Jean-Pierre Serre, Propri´ et´ es galoisiennes des points d’ordre ﬁni des courbes elliptiques , In- vent. Math. 15 (1972), no. 4, 259–331

work page 1972

[13] [13]

, Abelian ℓ-adic representations and elliptic curves , Benjamin, New York, 1968

work page 1968

[14] [14]

Reprint of the 2003 edition [of the 1986 or iginal MR0926691]

, R´ esum´ e des cours de 1977-1978, Springer Collected W orks in Mathematics, Springer, Heidelberg, 2013. Reprint of the 2003 edition [of the 1986 or iginal MR0926691]

work page 1977

[15] [15]

2, 26–37

Peter Stevenhagen and Hendrik Lenstra, Chebotarev and his density theorem , The Mathe- matical Intelligencer 18 (1996), no. 2, 26–37

work page 1996

[16] [16]

Peter Stevenhagen, The correction factor in Artin ’s primitive root conjecture, J. Th´ eor. Nom- bres Bordeaux 15 (2003), no. 1, 383–391 (English, with English and French sum maries). Les XXII` emes Journ´ ees Arithmetiques (Lille, 2001)

work page 2003

[17] [17]

Jacques V´ elu,Isog´ enies entre courbes elliptiques, C. R. Acad. Sci. Paris S´ er. A-B 273 (1971), A238–A241 (French). Department of Mathematics, Statistics and Computer Scienc e, University of Illinois at Chicago, 851 S Morgan St, 322 SEO, Chicago, IL 60607, USA Dipartimento di Matematica, Universit `a Roma Tre, Largo S. L. Murialdo 1, I–00146 Rome, Ita...

work page 1971