Elliptic curves over 
                
                  
                
                
                  
                    ℚ
                  
                
                
					$\mathbb {Q}$
				
               and 2-adic images of Galois

Rouse, Jeremy; Zureick-Brown, David

doi:10.1007/s40993-015-0013-7

Cited by 55 publications

(87 citation statements)

References 35 publications

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“…This, of course, is impossible for an elliptic curve of type [8X4,13B] since any such curve already has a 13-isogeny and there are no elliptic curves over Q with a 26-isogeny. Lastly, using the database in [36] we see that in order for an elliptic curve to have surjective image mod 2, be of type [8X4,13B], and to not have surjective image mod 4 is for the mod 4 image to be contained in the group associated to 4X7. This would mean that E also has type [4X7,13B], but our computations show that the modular curve corresponding to [4X7,13B] is genus 3 and so there can be at most finitely many such Q-isomorphism classes.…”

Section: Proof Of the Theorem 17 Theorem 19 Corollary 18 And Coromentioning

confidence: 99%

Serre’s Constant of Elliptic Curves Over the Rationals

Daniels

González–Jiménez

2019

Experimental Mathematics

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Let E be an elliptic curve without complex multiplication defined over the rationals. The purpose of this article is to define a positive integer A(E), that we call the Serre's constant associated to E, that gives necessary conditions to conclude that ρE,m, the mod m Galois representation associated to E, is non-surjective. In particular, if there exists a prime factor p of m satisfying valp(m) ≥ valp(A(E)) > 0 then ρE,m is non-surjective. Conditionally under Serre's Uniformity Conjecture, we determine all the Serre's constants of elliptic curves without complex multiplication over the rationals that occur infinitely often. Moreover, we give all the possible combination of mod p Galois representations that occur for infinitely many non-isomorphic classes of non-CM elliptic curves over Q, and the known cases that appear only finitely. We obtain similar results for the possible combination of maximal non-surjective subgroups of GL2(Zp). Finally, we conjecture all the possibilities of these combinations and in particular all the possibilities of these Serre's constants.Serre's Uniformity Question. If E/Q is a non-CM elliptic curve, then must it be that ρ E,p is surjective for any prime p ≥ 41?Nowadays, an affirmative answer to the above question has received the name of Serre's Uniformity Conjecture (or sometimes just Uniformity Conjecture) despite the fact that Serre himself never conjectured it to be true.

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Section: Proof Of the Theorem 17 Theorem 19 Corollary 18 And Coromentioning

confidence: 99%

Serre’s Constant of Elliptic Curves Over the Rationals

Daniels

González–Jiménez

2019

Experimental Mathematics

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“…As in the previous remark, the corresponding modular curve has genus 0 but no noncuspidal rational points. More generally, the ten pointless conics noted in [52] that are models of modular curves associated to subgroups of GL 2 (2 n ) lack rational points for this reason.…”

Section: Cyclic Casesmentioning

confidence: 99%

“…REMARK 5.28. Lemma 5.27 does not apply to composite integers m. Indeed, for m = 8 there may be as many as 20 nonconjugate G E F (m) that arise as F ranges over quadratic extensions of K ; see [52] for examples.…”

Section: Quadratic Twistsmentioning

confidence: 99%

Computing Images of Galois Representations Attached to Elliptic Curves

Sutherland

2016

Forum of Mathematics, Sigma

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Let E be an elliptic curve without complex multiplication (CM) over a number field K , and let G E ( ) be the image of the Galois representation induced by the action of the absolute Galois group of K on the -torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G E ( ) up to local conjugacy for all primes by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G E ( ) up to one of at most two isomorphic conjugacy classes of subgroups of GL 2 (Z/ Z) that have the same semisimplification, each of which occurs for an elliptic curve isogenous to E. Under the GRH, their running times are polynomial in the bit-size n of an integral Weierstrass equation for E, and for our Monte Carlo algorithm, quasilinear in n. We have applied our algorithms to the non-CM elliptic curves in Cremona's tables and the Stein-Watkins database, some 140 million curves of conductor up to 10 10 , thereby obtaining a conjecturally complete list of 63 exceptional Galois images G E ( ) that arise for E/Q without CM. Under this conjecture, we determine a complete list of 160 exceptional Galois images G E ( ) that arise for non-CM elliptic curves over quadratic fields with rational j-invariants. We also give examples of exceptional Galois images that arise for non-CM elliptic curves over quadratic fields only when the j-invariant is irrational.

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“…The number of fields grows over time as new data are contributed. For example, in 2015 Jeremy Rouse offered to provide information concerning the 2-adic Galois Representation attached to every elliptic curve over Q, after developing and implementing an algorithm to determine this jointly with David Zureick-Brown (see [9]). He provided us with a Magma script of their implementation, we ran it and uploaded the data, and added a corresponding section on the home page of every curve showing these additional data.…”

Section: Sample Database Entrymentioning

confidence: 99%

The L-Functions and Modular Forms Database Project

Cremona

2016

Found Comput Math

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The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-function is the Riemann zeta function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive, and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website Communicated by

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Elliptic curves over ℚ $\mathbb {Q}$ and 2-adic images of Galois

Cited by 55 publications

References 35 publications

Serre’s Constant of Elliptic Curves Over the Rationals

Serre’s Constant of Elliptic Curves Over the Rationals

Computing Images of Galois Representations Attached to Elliptic Curves

The L-Functions and Modular Forms Database Project

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