Jacobians of Genus One Curves

By S. An, S. Kim, David C. Marshall, S. Marshall, William G. McCallum, Alexander R. Perlis. 2001

107 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e5b5513aca3722a7e780ac92f709b8333c2261c5

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Manjul Bhargava
2. A. Shankar
3. Michael Stoll
4. Noam D. Elkies
5. John Cremona
6. Abhinav Kumar
7. Hernan Piragua
8. Tom A. Fisher

In Geometry and Topology

1. Manjul Bhargava
2. A. Shankar
3. Michael Stoll
4. Noam D. Elkies
5. John Cremona
6. Abhinav Kumar
7. Tom A. Fisher
8. Hernan Piragua

Citations in the corpus

• The invariants of a genus one curve (2006, background): “These formulae have already been surveyed in [1], but are included here to demonstrate our preferred choice of scalings.”
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1 (2013, background): “, [1] for an excellent treatment over general fields).”
• Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves (2010, methodology): “iminant is an acceptable function. We then have the following version of Theorem 2.12, in which we allow weights to be defined by certain infinite sets of congruence conditions: Theorem 2.21 Let φ: VZ →[0,1] be an acceptable function that is defined by congruence conditions via the local functions φp: ”
• F-theory on all toric hypersurface fibrations and its Higgs branches (2014, methodology): “The algorithm for computing J(XF1) is well known in the mathematics literature, see for example [76], from where we calculate f and g, given explicitly in (B.”
• MINIMISATION AND REDUCTION OF 2-, 3- AND 4-COVERINGS OF ELLIPTIC CURVES (2009, methodology): “Φ)x−54c 6(Φ). Proof: The invariants c 4, c 6 and ∆ were known to the nineteenth century invariant theorists. The observation that they give a formula for the Jacobian is due to Weil [We1], [We2]. See [AKM3P] for a brief survey, or [Fi4] for a proof of the theorem exactly as it is stated here. ✷ As w”
• K3 surfaces and equations for Hilbert modular surfaces (2012, methodology): “We may use standard formulas to write down the Weierstrass equation of the Jacobian (see [An et al. 2001] and the references cited therein), and this is the desired Weierstrass equation for the elliptic fibration with fiber F .”