Computing -series of geometrically hyperelliptic curves of genus three

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We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant. of this database: an enumeration of all smooth plane quartic curves with coefficients of absolute value at most B c := 9, with the aim of obtaining a set of unique Q-isomorphism class representatives for all such curves that have absolute discriminant at most B ∆ := 10 7 .Even after accounting for obvious symmetries, this involves more than 10 17.5 possible curve equations and requires a massively distributed computation to complete in a reasonable amount of time. Efficiently computing the discriminants of these equations is a non-trivial task, much more so than in the hyperelliptic case, and much of this article is devoted to an explanation of how this was done. Many of the techniques that we use can be generalized to other enumeration problems and may be of independent interest, both from an algorithmic perspective, and as an example of how cloud computing can be effectively applied to a research problem in number theory. A list of the curves that were found (more than 80 thousand) is available on the author's website [33].Remark 1.1. The informed reader will know that not every genus 3 curve over Q falls into the category of smooth plane quartics f (x, y, z) = 0 or curves with a hyperelliptic model y 2 + h(x) y = f (x). The other possibility is a degree-2 cover of a pointless conic; see [18] for a discussion of such curves and algorithms to efficiently compute their L-functions. We plan to conduct a separate search for curves of this form that will also become part of the genus 3 database in the LMFDB.1.1. Acknowledgments. The author is grateful to Nils Bruin, Armand Brumer, John Cremona, Tim Dokchitser, Jeroen Sijsling, Michael Stoll, and John Voight for their insight and helpful comments, and to the anonymous referees for their careful reading and suggestions for improvement. THE DISCRIMINANT OF A SMOOTH PLANE CURVELet C[x] d denote the space of ternary forms of degree d ≥ 1, as homogeneous polynomials in the variables x := (x 0 , x 1 , x 2 ). It is a C-vector space of dimension n d := d+2 2 equipped with a standard monomial basis

“…15] and the magma script in [33] for further details and more examples. 9 and det B ∈ Z[a] 18 . Computing the determinants of all the submatrices A and B takes only a few minutes.…”

Section: Sylvester's Resultant Formula For Ternary Formsmentioning

confidence: 99%

mentioning

confidence: 99%

A database of nonhyperelliptic genus-3 curves over ℚ

Sutherland¹

2019

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“…. , a d j −1 and use (15) to deduce the last row of W j 1 from the last row of B j . One might suppose that we could instead compute the last rows of the B j (a i ) instead of their first rows, but this is not enough to deduce B j .…”

Section: Translation Tricksmentioning

confidence: 99%

“…For p ∈ P let k( p) denote the index k of the m k for which m k = p. 14), use B j 1 to compute B j ∈ ‫ކ‬ d j ×d p via (15), and set the j, block of A p to B j as in (8). (5) Output A p ∈ ‫ކ‬ g×g p for all primes p ≤ N such that p m lc( f ) disc( f ).…”

Section: Algorithmsmentioning

confidence: 99%

Counting points on superelliptic curves in average polynomial time

Sutherland

2020

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We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over ‫ޑ‬ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curve y m = f (x) with m ≥ 2 and f ∈ ‫[ޚ‬x] any squarefree polynomial of degree d ≥ 3, along with a positive integer N . It can compute #X ‫ކ(‬ p ) for all p ≤ N not dividing mlc( f )disc( f ) in time O(md 3 N log 3 N log log N ). It achieves this by computing the trace of the Cartier-Manin matrix of reductions of X . We can also compute the Cartier-Manin matrix itself, which determines the p-rank of the Jacobian of X and the numerator of its zeta function modulo p.

“…This was not true of initial attempts that relied on a generic implementation of the balanced divisor approach included in Magma [2], which has not been optimized for hyperelliptic curves of genus 3. For the application in [17], the primary use of our addition formulas occurs as part of a baby-steps giant-steps search in which field inversions can easily be combined by taking steps in parallel [20, §4.1]. The incremental cost of a field inversion is then just three field multiplications, making affine coordinates preferable to projective coordinates (by a wide margin); we thus present our formulas in affine coordinates, although they can be readily converted to projective coordinates if desired.…”

Section: Introductionmentioning

confidence: 99%

Fast Jacobian arithmetic for hyperelliptic curves of genus 3

Sutherland

2019

We consider the problem of efficient computation in the Jacobian of a hyperelliptic curve of genus 3 defined over a field whose characteristic is not 2. For curves with a rational Weierstrass point, fast explicit formulas are well known and widely available. Here we address the general case, in which we do not assume the existence of a rational Weierstrass point, using a balanced divisor approach.