LAUSR

Abstract We prove an analogue of the Brauer–Siegel theorem for the Legendre elliptic curves over K = F q ( t ) . Namely, denoting by E d the elliptic curve with model y 2 = x ( x + 1 ) ( x + t d ) over K, we show that, for d → ∞ ranging over the integers, one has log ⁡ ( | СХ ( E d / K ) | ⋅ Reg ( E d / K ) ) ∼ log ⁡ H ( E d / K ) ∼ log ⁡ q 2 ⋅ d . Here, H ( E d / K ) denotes the exponential differential height of E d , СХ ( E d / K ) its Tate–Shafarevich group (which is known to be finite), and Reg ( E d / K ) its Neron–Tate regulator.