Anni, Samuele; Dokchitser, Vladimir; (2020) Constructing hyperelliptic curves with surjective Galois representations. Transactions of the American Mathematical Society , 373 (2) pp. 1477-1500. 10.1090/tran/7995.

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Abstract

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes ` the image of the mod ` Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the `-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 ` 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 is expected to hold for all g 6= 0, 1, 2, 3, 4, 5, 7 and 13). Then there is an explicit N ∈ Z and an explicit monic polynomial f0(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[`])/Q) ∼= GSp2g (F`) for all odd primes ` and Gal(Q(J[2])/Q) ∼= S2g+2, whenever f(x) ∈ Z[x] is monic with f(x) ≡ f0(x) mod N and with no roots of multiplicity greater than 2 in Fp for any p - N.

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