We investigated maximal Prym varieties on finite fields by attaining their upper bounds on the number of rational points. This concept gave us a motivation for defining a generalized definition of maximal curves i.e. maximal morphisms. By MAGMA, we give some non-trivial examples of maximal morphisms that results in non-trivial examples of maximal Prym varieties.

In [14], D. Skabelund constructed a maximal curve over Fq4 as cyclic cover of the Suzuki curve. this paper we explicitly determine structure Weierstrass semigroup at any point P We show that its points are precisely Fq4-rational points. Also among points, two types occur: one for Fq-rational remaining For each these Apéry set is computed well generators.

Let C be a smooth projective curve and W symplectic bundle over . L Q e ( ) the Lagrangian Quot scheme parametrizing subsheaves E ⊂ of degree We give closed formula for intersection numbers on As special case, g ≥ 2 , we compute number subbundles maximal general stable bundle, when this is finite. This analogue Holla's enumeration in [14]

Previous results on genera g of Fq2 -maximal curves are improved: (1) Either g ≤ ⌊(q − q + 4)/6⌋ , or g = ⌊(q − 1)/4⌋ , or g = q(q − 1)/2 . (2) The hypothesis on the existence of a particular Weierstrass point in [2] is proved. (3) For q ≡ 1 (mod 3), q ≥ 13, no Fq2 -maximal curve of genus (q−1)(q−2)/3 exists. (4) For q ≡ 2 (mod 3), q ≥ 11, the non-singular Fq2 -model of the plane curve of equat...

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K− denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the gene...