L-functions of twisted Legendre curves

Let C/$Fq$ a proper smooth geometrically connected curve, K_C = $Fq$(C) and E/K_C be an elliptic curve with non-constant j-invariant. The L-function L(T, E/K_C) is a polynomial in 1 + T · $Z$[T]. For any N > 1 invertible in $Fq$ and finite subgroup $T$ ⊂ E(K_C) of order N, we compute the mod-N reduction of L( T, E/K_C) and determine an upper-bound for the analytic rank of E/K_C. We construct infinite families of curves of rank zero by constructing infinitely many twin-prime pairs (Λ, Λ − 1) ∈ $Fq$[λ] × $Fq$ [λ] with deg(Λ) odd, provided q is an odd prime power such that q ≡ 1 mod l for some odd prime l, and pulling back the Legendre curve $ELeg/Fq$(λ) along the finite flat map Λ: $P1$ →$P1$. We also construct infinitely many quadratic twists with minimal analytic rank, half of which have rank zero and half have (analytic) rank one. In both cases we bound the analytic rank by letting $T$ ≅ $Z$/2 ⊕ $Z$/2 and studying the mod-4 reduction of L( T, E/K_C).