Samar Jaafar and Kamal Khuri - Makdisi

Computational Methods

Jaafar and Kamal Khuri - Makdisi

We study pullbacks of modular forms of weight 1 from the modular curve X(4) to the modular curve X(4p), where p is an odd prime. We find the extent to which such modular forms separate points on X(4p). Our main result is that these modular forms give rise to a morphism F from the quotient of X(4p) by a certain involution ι to projective space, such that F is a projective embedding of X(4p)/ι aw...

Fast Jacobian group operations for C_{3,4} curves over a large finite field

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Mathematics of Computation). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g) field operations using fast linear algebr...

An exact sequence in the representation theory of SL(2)

Let V be the standard two-dimensional representation of the algebraic group G = SL(2,C), and write Vn = SymnV for the irreducible (n+1)-dimensional representation of G on the nth symmetric tensor power of V . Also write Wn = V ⊗n for the nth tensor power of V (which has dimension 2n). Then, as virtual representations, it is known that each Vn can be written in terms of W0, . . . , Wn as Vn = Wn...

Asymptotically fast group operations on Jacobians of general curves

Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of dimension at most O(g log g), and so take O(g3+ ) field operations in k, using Gaussian elimination. Using ...