Varieties via their L-functions

Farmer, David W.; Koutsoliotas, S.; Lemurell, Stefan

doi:10.48550/arxiv.1502.00850

Cited by 2 publications

(4 citation statements)

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“…The isogeny classes 561.a and 737.a likely correspond to the Prym varieties listed in [2, Table2], while the isogeny classes 550.a, 702.a, 732.a are likely to be three of the eight "unknown" isogeny classes corresponding to paramodular cuspidal newforms of weight 2 and level N ≤ 1000 listed in the tables of Poor and Yuen[29]. We have verified that the Euler factors of isogeny class 550.a match those listed in[12, Table…”

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confidence: 52%

A database of nonhyperelliptic genus-3 curves over ℚ

Sutherland¹

2019

Open Book Series

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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

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confidence: 52%

A database of nonhyperelliptic genus-3 curves over ℚ

Sutherland¹

2019

Open Book Series

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“…One can treat all of the a n as unknowns (not only the ones for even n), subject only to the assumptions that they are integer valued and satisfy the Ramanujan bound. This was carried out by Farmer-Koutsoliotas-Lemurell [17], who found all solutions to (5.2.1) with N ≤ 500. Assuming the Hasse-Weil conjecture, this includes a complete classification, up to isogeny, of the abelian surfaces with conductor ≤ 500.…”

Section: Resultsmentioning

confidence: 99%

“…These all appear to lie in the 33 isogeny classes identified by Brumer-Kramer using Jacobians of curves with absolute discriminants below 10 6 . -In a similar vein, Farmer-Koutsoliotas-Lemurell [17] used analytic methods to determine a complete list of the integers N ≤ 500 that can arise as the conductor of degree-4 Lfunctions under a standard set of hypotheses. Excluding L-functions arising from products of elliptic curves or classical modular forms with real quadratic character, they list 12 values of N ≤ 500 that may arise for the L-function of a genus 2 curve [17, Theorem 2.1] (the 9 odd values match the Brumer-Kramer list up to 500).…”

Section: Completenessmentioning

confidence: 99%

“…So it remains to compute ord 2 (N ), and we do so analytically, using methods that go back at least to Dokchitser [15] and the first author [4]; see also recent work of Farmer-Koutsoliotas-Lemurell [17]. The result of our computation will be that if the genus 2 curve has an L-function with analytic continuation and functional equation, then there is provably only one possibility for the conductor and bad L-factor.…”

Section: Analytically Computing the Conductormentioning

confidence: 99%

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