[Paper Review] Mordell-Weil groups and Selmer groups of two types of elliptic curves

Derong Qiu, Xianke Zhang|ArXiv.org|Mar 15, 2001

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper explicitly computes the 2-Selmer, φ-Selmer, and φ̂-Selmer groups for a family of elliptic curves $ E_\sigma: y^2 = x(x + \sigma p)(x + \sigma q) $, where $ p, q $ are twin primes with $ q = p+2 $. It determines the Mordell-Weil rank and the 2-primary part of the Shafarevich-Tate group based on $ p \mod 8 $, showing that the rank plus the dimension of the 2-torsion in the Shafarevich-Tate group is 0, 1, or 2 depending on the residue of $ p \mod 8 $, and proves modularity with conductor $ 2^5 p q $.

ABSTRACT

Consider elliptic curves $ E=E_σ: y^2 = x (x+σp) (x+σq), $ where$ σ=\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(ϕ)}(E/{\mathbf{Q})}$, and $\ S^{(\hatϕ)}(E/{\mathbf{Q})} $ are explicitly determined, e.g., $\ S^{(2)}(E_{+1}/{\mathbf{Q}})= $ $({\mathbf{Z}}/2{\mathbf{Z}})^2; $ $ ({\mathbf{Z}}/2{\mathbf{Z}})^3; $ or $ ({\mathbf{Z}}/2{\mathbf{Z}})^4 $ when $p\equiv 5; 1 $ or $3; $ or $ 7 ({\mathrm{mod}} 8)$ respectively. (2) When $p\equiv 5 (3, 5$ for $σ=-1) ({\mathrm{mod}} 8), $ it is proved that the Mordell-Weil group $ E({\mathbf{Q})} \cong $ $ {\mathbf{Z}}/2{\mathbf{Z}} \oplus{\mathbf{Z}}/2{\mathbf{Z}} $ having rank $0, $ and Shafarevich-Tate group {\CC ':} $(E/{\mathbf{Q}})[2]=0. $ (3) In any case, the sum of rank$E({\mathbf{Q})}$ and dimension of {\CC ':} $(E/{\mathbf{Q}})[2] $ is given, e.g., $0; 1; 2 $ when $p\equiv 5; 1 $ or $3; 7 ({\mathrm{mod}} 8)$ for $σ=1$. (4) The Kodaira symbol, the torsion subgroup $E(K)_{tors}$ for any number field $K$, etc. are also obtained. This paper is a revised version of ANT-0229.

Motivation & Objective

• Understand the structure of Selmer and Mordell-Weil groups for a specific family of elliptic curves defined by twin primes.
• Characterize the Mordell-Weil rank and the 2-primary part of the Shafarevich-Tate group for these curves.
• Determine the torsion subgroup over arbitrary number fields and the Kodaira type of the Néron model.
• Establish modularity and functional equations for the L-function of these curves.
• Provide explicit classification of the 2-Selmer group and related Selmer groups based on $ p \mod 8 $.

Proposed method

• The 2-Selmer group $ S^{(2)}(E/\mathbb{Q}) $ is computed via explicit descent using rational points and 2-isogenies.
• Selmer groups $ S^{(\varphi)}(E/\mathbb{Q}) $ and $ S^{(\hat{\varphi})}(E/\mathbb{Q}) $ are determined using isogeny descent techniques and the 2-torsion structure.
• Kodaira symbols and Tamagawa numbers are computed using the Néron model and reduction types at primes $ \ell = 2, p, q $.
• The conductor $ N_E = 2^5 p q $ is derived from local exponents $ f_\ell $, and modularity is established via the functional equation of the L-function.
• Exact sequences involving Selmer and Shafarevich-Tate groups are used to relate $ \cyrshape(E/\mathbb{Q})[2] $ to the rank and Selmer group dimensions.
• Rational points on auxiliary curves $ C' $ are used to link solutions of Diophantine equations to points on $ E $, enabling descent arguments.

Experimental results

Research questions

• RQ1 What is the structure of the 2-Selmer group $ S^{(2)}(E/\mathbb{Q}) $ for the elliptic curve $ E_\sigma: y^2 = x(x + \sigma p)(x + \sigma q) $ with twin primes $ p, q $?
• RQ2 How does the Mordell-Weil rank of $ E(\mathbb{Q}) $ depend on the residue of $ p \mod 8 $?
• RQ3 What is the dimension of the 2-primary part of the Shafarevich-Tate group $ \cyrshape(E/\mathbb{Q})[2] $, and how does it relate to the rank?
• RQ4 What are the Kodaira symbols and Tamagawa numbers at primes $ \ell = 2, p, q $?
• RQ5 How does the modularity of these curves manifest in their L-function and functional equation?

Key findings

• The 2-Selmer group $ S^{(2)}(E_{+1}/\mathbb{Q}) $ is isomorphic to $ (\mathbb{Z}/2\mathbb{Z})^2 $, $ (\mathbb{Z}/2\mathbb{Z})^3 $, or $ (\mathbb{Z}/2\mathbb{Z})^4 $, depending on whether $ p \equiv 5, 1, 3, $ or $ 7 \mod 8 $.
• When $ p \equiv 5 \mod 8 $, the Mordell-Weil group $ E(\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} $, so the rank is 0 and $ \cyrshape(E/\mathbb{Q})[2] = 0 $.
• For $ \sigma = 1 $, the sum of the rank of $ E(\mathbb{Q}) $ and the dimension of $ \cyrshape(E/\mathbb{Q})[2] $ is 0, 1, or 2 when $ p \equiv 5, 1 \text{ or } 3, \text{ or } 7 \mod 8 $, respectively.
• The Kodaira symbol at $ \ell = 2 $ is $ III $, and at $ \ell = p, q $ it is $ I_2 $, with Tamagawa number 2 at each.
• The conductor of $ E $ is $ N_E = 2^5 p q $, and the L-function satisfies the functional equation $ \xi(E, 2-s) = \pm \xi(E, s) $, confirming modularity.
• The torsion subgroup $ E(K)_{\text{tors}} $ over any number field $ K $ is either $ \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} $ or $ \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/6\mathbb{Z} $, depending on ramification at 3.