Large Galois images for Jacobian varieties of genus 3 curves

Arias-de-Reyna, Sara; Armana, Cécile; Karemaker, Valentijn; Rebolledo, Marusia; Thomas, Lara; Vila, Núria Sala i

doi:10.4064/aa8250-4-2016

Cited by 3 publications

(3 citation statements)

References 23 publications

(32 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…this curve was produced in the recent paper of Arias-de-Reyna et al [Ari+16] as an example of a genus 3 curve whose Jacobian is proven to be generic by their Theorem 4.1. We ran a Sage program to collect the group orders #(J C ) p (F p ), with p ≤ 2 20 for the genus 2 curves, and p ≤ 6 • 10 4 for C 3 .…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

Almost prime values of the order of abelian varieties over finite fields

Bloom¹

2018

Preprint

View full text Add to dashboard Cite

Let E/Q be an elliptic curve, and denote by N (p) the number of Fp-points of the reduction modulo p of E. A conjecture of Koblitz, refined by Zywina, states that the number of primes p ≤ X at which N (p) is also prime is asymptotic to CE • X/ log(X) 2 , where CE is an arithmetically-defined non-negative constant. Following Miri-Murty ( 2001) and others, Y.R. Liu (2006) and David-Wu ( 2012) study the number of prime factors of N (p). We generalize their arguments to abelian varieties A/Q whose adelic Galois representation has open image in GSp 2g Z. Our main result, after David-Wu, finds a conditional lower bound on the number of primes at which #Ap(Fp) has few prime factors. We also present some experimental evidence in favor of a generalization of Koblitz's conjecture to this context.

show abstract

Section: Proof Of Main Resultsmentioning

confidence: 99%

Almost prime values of the order of abelian varieties over finite fields

Bloom¹

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…It is known that the Weil pairing induces a symplectic pairing in T ℓ (J(X)) ∼ = H 1 (X, Z ℓ ), [21, prop. 16.6], [1], [18] so that σa, σa ′ = χ ℓ (σ) a, a ′ .…”

Section: Arithmetic Intersectionmentioning

confidence: 99%

“…. , P s } ⊂ P 1 . The open curve Y 0 = Y − π −1 (Σ) is then a topological cover of X s = P 1 − Σ and can be seen as a quotient of the universal covering space Xs by the free subgroup R 0 = π 1 (Y 0 , y 0 ) of the free group π 1 (X s , x 0 ) = F s−1 (resp.…”

Section: On the Fundamental Group Of Cyclic Coversmentioning

confidence: 99%

Group Actions on cyclic covers of the projective line

Kontogeorgis

Paramantzoglou

2019

Geom Dedicata

View full text Add to dashboard Cite

show abstract

Constructing hyperelliptic curves with surjective Galois representations

Anni

Dokchitser

2019

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes ℓ the image of the mod ℓ Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the ℓ-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod ℓ Galois representations.The main result of the paper is the following. Suppose n = 2g +2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n (this hypothesis appears to hold for all g = 0, 1, 2, 3, 4, 5, 7 and 13). Then there is an explicit N ∈ Z and an explicit monic polynomial f 0 (x) ∈ Z[x] of degree n, such that the Jacobian J of every curve of the form y 2 = f (x) has Gal(Q(J[ℓ])/Q) ∼ = GSp 2g (F ℓ ) for all odd primes ℓ and Gal(Q(is monic with f (x) ≡ f 0 (x) mod N and with no roots of multiplicity greater than 2 in Fp for any p ∤ N .

show abstract

Large Galois images for Jacobian varieties of genus 3 curves

Cited by 3 publications

References 23 publications

Almost prime values of the order of abelian varieties over finite fields

Almost prime values of the order of abelian varieties over finite fields

Group Actions on cyclic covers of the projective line

Constructing hyperelliptic curves with surjective Galois representations

Contact Info

Product

Resources

About