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Register a new userLarge Galois images for Jacobian varieties of genus 3 curves
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Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation rho_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l).
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We discuss the $ell$-adic case of Mazurs Program B over $mathbb{Q}$, the problem of classifying the possible images of $ell$-adic Galois representations attached to elliptic curves $E$ over $mathbb{Q}$, equivalently, classifying the rational points on the corresponding modular curves. The primes $ell=2$ and $ellge 13$ are addressed by prior work, so we focus on the remaining primes $ell = 3, 5, 7, 11$. For each of these $ell$, we compute the directed graph of arithmetically maximal $ell$-power level modular curves, compute explicit equations for most of them, and classify the rational points on all of them except $X_{{rm ns}}^{+}(N)$, for $N = 27, 25, 49, 121$, and two level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$. Aside from the $ell$-adic images that are known to arise for infinitely many $overline{mathbb{Q}}$-isomorphism classes of elliptic curves $E/mathbb{Q}$, we find only 22 exceptional subgroups that arise for any prime $ell$ and any $E/mathbb{Q}$ without complex multiplication; these exceptional subgroups are realized by 20 non-CM rational $j$-invariants. We conjecture that this list of 22 exceptional subgroups is complete and show that any counterexamples must arise from unexpected rational points on $X_{rm ns}^+(ell)$ with $ellge 17$, or one of the six modular curves noted above. This gives us an efficient algorithm to compute the $ell$-adic images of Galois for any non-CM elliptic curve over $mathbb{Q}$. In an appendix with John Voight we generalize Ribets observation that simple abelian varieties attached to newforms on $Gamma_1(N)$ are of ${rm GL}_2$-type; this extends Kolyvagins theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.
Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell 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, quasi-linear in n. We have applied our algorithms to the non-CM elliptic curves in Cremonas 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(ell) that arise for E/Q without CM. Under this conjecture we determine a complete list of 160 exceptional Galois images G_E(ell) the 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.
Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $mathfrak{p}$ of $M$ such that the stable reduction of $C$ at $mathfrak{p}$ contains three irreducible components of genus $1$.
We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus two. In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields.
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