Constructing hyperelliptic curves with surjective Galois representations

International audienceIn this paper we show how to explicitly write down equations of hyperelliptic curves over Q \mathbb {Q} such that for all odd primes ℓ \ell the image of the mod ℓ \ell Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the ℓ \ell -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 \ell Galois representations. The main result of the paper is the following. Suppose n = 2 g + 2 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 n (this hypothesis is expected to hold for all g ≠ 0 , 1 , 2 , 3 , 4 , 5 , 7 , g\neq 0,1,2,3,4,5,7, and 13 13 ). Then there is an explicit N ∈ Z N\in \mathbb {Z} and an explicit monic polynomial f 0 ( x ) ∈ Z [ x ] f_0(x)\in \mathbb {Z}[x] of degree n n , such that the Jacobian J J of every curve of the form y 2 = f ( x ) y^2=f(x) has Gal ⁡ ( Q ( J [ ℓ ] ) / Q ) ≅ GSp 2 g ⁡ ( F ℓ ) \operatorname {Gal}(\mathbb {Q}(J[\ell ])/\mathbb {Q})\cong \operatorname {GSp}_{2g}(\mathbb {F}_\ell ) for all odd primes ℓ \ell and Gal ⁡ ( Q ( J [ 2 ] ) / Q ) ≅ S 2 g + 2 \operatorname {Gal}(\mathbb {Q}(J[2])/\mathbb {Q})\cong S_{2g+2} , whenever f ( x ) ∈ Z [ x ] f(x)\in \mathbb {Z}[x] is monic with f ( x ) ≡ f 0 ( x ) mod N f(x)\equiv f_0(x) \bmod {N} and with no roots of multiplicity greater than 2 2 in F ¯ p \overline {\mathbb {F}}_p for any p ∤ N p\nmid N