On the variation of the root number in families of elliptic curves

Desjardins, Julie

doi:10.1112/jlms.12168

Cited by 16 publications

(20 citation statements)

References 23 publications

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“…In the ph.D thesis of the author [Des16a] and in [Des16b], we prove the following theorem. This work is based on a preprint of Helfgott [Hel03], revisited and completed with a different approach.…”

Section: Known Resultsmentioning

confidence: 91%

“…We already have evidence (see [Man95,Hel03,Des16b]) that the rational points should be dense when the elliptic surface is non-isotrivial, because of the variation of the root number of the fibers. The articles mentioned above additionally use two conjectures of analytic number theory, the squarefree conjecture and Chowla's conjecture, which are known only for polynomials of low degree.…”

Section: Introductionmentioning

confidence: 99%

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On the density of rational points on rational elliptic surfaces

Desjardins

2019

Acta Arith.

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Let E → P 1 Q be a non-trivial rational elliptic surface over Q with base P 1 Q (with a section). We conjecture that any non-trivial elliptic surface has a Zariski-dense set of Q-rational points. In this paper we work on solving the conjecture in case E is rational by means of geometric and analytic methods. First, we show that for E rational, the set E (Q) is Zariski-dense when E is isotrivial with non-zero j-invariant and when E is non-isotrivial with a fiber of type II * , III * , IV * or I * m (m ≥ 0). We also use the parity conjecture to prove analytically the density on a certain family of isotrivial rational elliptic surfaces with j = 0, and specify cases for which neither of our methods leads to the proof of our conjecture.

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Section: Known Resultsmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

On the density of rational points on rational elliptic surfaces

Desjardins

2019

Acta Arith.

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“…Thus, the study of the Zariski density of E(Q) in E cannot, in general, be addressed by rank considerations only. However, in general the root number should be, under the Chowla and squarefree conjectures, equidistributed and hence there should exist infinitely many specializations with rank ≥ 1 over Q, implying (conditionally) the Zariski density (see [Hel09,Des19b]). This strategy does not apply whenever the root number is not equidistributed: this happens to be the case for potentially parity biased families as studied in [BDD18], where it is proved that there are essentially 6 different classes of such non-isotrivial families of the form (1.1) with deg(α 2 ), deg(α 4 ), deg(α 6 ) ≤ 2.…”

Section: Introductionmentioning

confidence: 99%

On the typical rank of elliptic curves over ${\mathbb Q}(T)$

Battistoni¹,

Bettin²,

Delaunay³

2021

Preprint

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We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture [Cow20] in the case m, n ≤ 2. 2020 Mathematics Subject Classification. 11G05, 14G05. Key words and phrases. Rational elliptic surface, rank, elliptic curves, rational points. 1 Cowan stated the conjecture for the Mahler measure µ, mentioning that one could also choose the height instead. By the inequalities n [n/2] −1 P ≤ µ(P ) ≤ √ n + 1 P for P of degree n (see [BG06, Lemma 1.6.7]) the two choices are equivalent.

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“…The case where F is potentially parity-biased was also considered by Rizzo [Riz03] in two examples which already contain several of the important ideas for the general result. We also mention the recent work of Desjardins [Des16b] who revisited Helfgott's result, and relaxed some of the assumptions.…”

Section: Introductionmentioning

confidence: 99%

Non-isotrivial elliptic surfaces with non-zero average root number

Bettin

David

Delaunay

2018

Journal of Number Theory

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We consider the problem of finding non-isotrivial 1-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in Z. We classify all such families when the degree of the coefficients (in the parameter t) is less than or equal to 2 and we compute the rank over Q(t) of all these families. Also, we compute explicitly the average of the root numbers for some of these families highlighting some special cases. Finally, we prove some results on the possible values average root numbers can take, showing for example that all rational number in [−1, 1] are average root numbers for some non-isotrivial 1-parameter family.if the limit exists (and where we define ε F (t) = 0 if F (t) is not an elliptic curve). 1 The work of Helfgott ([Hel03, Hel09]) implies conjecturally (and unconditionally in some cases) that Av Z (ε F ) = 0 as soon as there exists a place, other than − deg, of multiplicative reduction of F over Q(t). Indeed, assuming the square-free sieve conjecture, one sees that in this case ε F (t) behaves roughly like λ(M (t)) where λ is the Liouville's function and M (t) is a certain non-constant square-free polynomial, so that Chowla's conjecture implies that Av Z (ε F ) = 0. This is the typical case, which occurs for "most" families F .2010 Mathematics Subject Classification. 11G05, 11G40. Key words and phrases. Rational elliptic surface, rank, root number, average root number. 1 Alternatively one could define Av Z (ε F ) with the symmetric average 1 2T |t|≤T replaced by 1 T 0≤t≤T . All the same considerations we make in the paper works in this case as well mutatis mutandis.. So, the p-divisibility of | Sel p (F (t))| 2 Isotrivial means that the j-invariant of F is constant. For isotrivial families, one can take, for instance, the quadratic twist of a fixed elliptic curve E/Q by a polynomial d(t) ∈ Z[t], E d(t) : d(t)y 2 = y 2 = x 3 + a 2 x 2 + a 4 x + a 6 , where a i ∈ Z for i = 2, 5, 6. In this case, it is easier to deal with the root number, for example if d(t) is coprime with the conductor of E, then the root number is simply given by some congruence relations.3 See the end of the introduction for the precise definition of the average root number over Q.2. The classification of potentially parity-biased families of low degree 2.1. The work of Helfgott and its consequences. We start with a more detailed discussion of the work of Helfgott ([Hel09, Hel03]) which gives (conditionally) a necessary condition for a family to be potentially parity-biased. First, we state the following conjectures.Conjecture 1 (Chowla's conjecture). Let P (x) ∈ Z[x] be square-free. Then, n≤N λ(P (n)) = o(N ) as N → ∞, where λ(n) is the Liouville function λ(n) := p|n (−1) vp(n) .Moreover, by strong Chowla's conjecture for a polynomial P we mean the assumption that Chowla's conjecture holds for P (ax + b) for all a, b ∈ Z, a = 0.Conjecture 2 (Square-free sieve conjecture). Let P (x) be a square-free polynomial in Z[x]. Then, the set of integers n such that P (n) is divisible by the square of a...

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On the variation of the root number in families of elliptic curves

Cited by 16 publications

References 23 publications

On the density of rational points on rational elliptic surfaces

On the density of rational points on rational elliptic surfaces

On the typical rank of elliptic curves over ${\mathbb Q}(T)$

Non-isotrivial elliptic surfaces with non-zero average root number

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