Numerical computation of endomorphism rings of Jacobians

Endomorphism algebras of Jacobians

where K is a subfield of even index at most 10 in a primitive cyclotomic field Q(ζp), or a subfield of index 2 in Q(ζpq) or Q(ζpα ). This result generalizes previous work of Brumer, Mestre, and Tautz-Top-Verberkmoes. Our curves are constructed as branched covers of the projective line, and the endomorphisms arise as quotients of double coset algebras of the Galois groups of these coverings. In ...

Endomorphism Algebras of Superelliptic Jacobians

As usual, we write Z,Q,Fp,C for the ring of integers, the field of rational numbers, the finite field with p elements and the field of complex numbers respectively. If Z is a smooth algebraic variety over an algebraically closed field then we write Ω(Z) for the space of differentials of the first kind on Z. If Z is an abelian variety then we write End(Z) for its ring of (absolute) endomorphisms...

The Endomorphism Rings of Jacobians of Cyclic Covers of the Projective Line

Suppose K is a eld of characteristic 0, Ka is its algebraic closure, p is an odd prime. Suppose, f(x) 2 K[x] is a polynomial of degree n 5 without multiple roots. Let us consider a curve C : y = f(x) and its jacobian J(C). It is known that the ring End(J(C)) of all Ka-endomorphisms of J(C) contains the ring Z[ p] of integers in the pth cyclotomic eld (generated by obvious automorphisms of C). W...

Computing endomorphism rings of Jacobians of genus 2 curves over finite fields

We present algorithms which, given a genus 2 curve C defined over a finite field and a quartic CM field K, determine whether the endomorphism ring of the Jacobian J of C is the full ring of integers in K. In particular, we present probabilistic algorithms for computing the field of definition of, and the action of Frobenius on, the subgroups J [l] for prime powers l. We use these algorithms to ...

Endomorphism rings of bimodules