Conic bundles and Mordell--Weil ranks of elliptic surfaces

Abstract

Let $k$ be a number field and $\mathcal{E}$ an elliptic curve defined over the function field $k(T)$ given by an equation of the form $y^2 = a_3x^3 + a_2x^2 + a_1x + a_0$, where $a_i \in k[T]$ and $deg(a_i) \leq 2$. We explore the conic bundle structure over the $x$-line to obtain lower and upper bounds for the Mordell--Weil rank of $\mathcal{E}(k(T))$.

Figures (2)

• Figure 1: Blowing-down a fiber of type $A_n$ to a fiber of type $A_2$
• Figure 2: Blowing-down a fiber of type $D_n$ to a fiber of type $A_2$

Theorems & Definitions (91)

• Theorem \ref{theorem_rank_k}
• Conjecture \ref{theorem_rank_k}
• Theorem \ref{theorem_rank_k}: Rosen, Silverman
• Theorem \ref{theorem_rank_k}
• proof
• Theorem \ref{theorem_rank_k}: Battistoni, Bettin, Delaunay
• proof
• Theorem \ref{theorem_rank_k}: Battistoni, Bettin, Delaunay
• proof
• Theorem \ref{theorem_rank_k}: Battistoni, Bettin, Delaunay