We prove the exceptional zero conjecture is true for semistable elliptic curves E/Q over number fields of the form F ( e2πi/q n ,∆ 1/q 1 , . . . ,∆ 1/q d ) where F is a totally real field, and the split multiplicative prime p 6= 2 is inert in F (e2πi/q n ) ∩ R. In 1986 Mazur, Tate and Teitelbaum [9] attached a p-adic L-function to an elliptic curve E/Q with split multiplicative reduction at p. ...

In 1989, Koblitz proposed using the Jacobian of a hyperelliptic curve defined over a finite field to implement discrete logarithm cryptographic protocols. The discrete logarithm problem of the Jacobian is called hyperelliptic curve discrete logarithm problem (HCDLP). For a hyperelliptic curve of genus g over the finite field Fq, the group order of the Jacobian is ( ) g O q which is larger than ...

We recall here some of the definitions given in [7]. Definition 1. The ’discrete plane’ Q′ with respect to some fixed point z′ = (x′, y′) in the first quadrant, is defined by the set of lattice points, Q′ = {(pmx′, qny′) : m,n ∈ Z the set of integers}. Definition 2. Two lattice points zi, zi+1 ∈ Q′ are said to be ’adjacent’ if zi+1 is one of (pxi, yi), (p−1xi, yi), (xi, qyi) or (xi, q−1yi). Def...

Three lectures on elliptic surfaces and curves of high rank Noam D. Elkies Over the past two years we have improved several of the (Mordell–Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P i ∈ E(Q) (the previous record was 24, by R. Martin and W. McMillen 2000), and the fi...

Our concern is a nonsingular plane curve defined over a finite field Fq which includes all the Fq-rational points of the projective plane. The possible degree of such a curve is at least q + 2. We prove that nonsingular plane curves of degree q + 2 having the property actually exist. More precisely, we write down explicitly all of those curves. Actually, Giuseppe Tallini studied such curves in ...

We construct a nite-dimensional vector space of functions of two complex variables attached to a smooth algebraic curve C over a nite eld Fq , q odd, and a level. These functions collect the analytic information about the cohomology of the curve and its quadratic twists that is encoded in the corresponding L-functions; they are double Dirichlet series in two independent complex variables s;w. W...

If E is a minimal elliptic curve defined over Z, we obtain a bound C, depending only on the global Tamagawa number of E, such that for any point P ∈ E(Q), nP is integral for at most one value of n > C. As a corollary, we show that if E/Q is a fixed elliptic curve, then for all twists E′ of E of sufficient height, and all torsion-free, rank-one subgroups Γ ⊆ E′(Q), Γ contains at most 6 integral ...

Let C be a smooth absolutely irreducible curve of genus g ≥ 1 defined over Fq, the finite field of q elements, and let #C(Fqn) be the number of Fqn-rational points on C. Under a certain condition, which, for example, satisfied by all ordinary elliptic curves, we obtain an asymptotic formula for the number of ratios (#C(Fqn)−q−1)/2gq, n = 1, . . . , N , inside of a given interval I ⊆ [−1, 1] . T...

Let C be a smooth absolutely irreducible curve of genus g ≥ 1 defined over Fq, the finite field of q elements, and let #C(Fqn) be the number of Fqn-rational points on C. Under a certain condition, which for example, satisfied by all ordinary elliptic curves, we obtain an asymptotic formula for the number of ratios (#C(Fqn)−q−1)/2gq, n = 1, . . . , N , inside of a given interval I ⊆ [−1, 1] . Th...