Explicit L-functions and a Brauer-Siegel theorem for Hessian elliptic curves

For a finite field F-q of characteristic p >= 5 and K = F-q(t), we consider the family of elliptic curves E-d over K given by y(2) + xy - t(d)y = x(3) for all integers d coprime to q. We provide an explicit expression for the L-functions of these curves. Moreover, we deduce from this calculation that the curves E-d satisfy an analogue of the Brauer-Siegel theorem. Precisely, we show that, for d -> infinity ranging over the integers coprime with q, one has log (vertical bar sic(E-d/K)vertical bar . Reg(E-d/K)) similar to log H(E-d/K), where H(E-d/K) denotes the exponential differential height of E-d, sic(E-d/K) its Tate-Shafarevich group and Reg(E-d/K) its Neron-Tate regulator.