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Aristides Kontogeorgis

Publications and source records attributed to Aristides Kontogeorgis.

48 records · Page 3Linked to original sources
arXiv2009

Arithmetic equivalence for function fields, the Goss zeta function and a generalization

A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this note, we investigate what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss. We prove that for function fields whose characteristic exceeds their degree, equality of the Goss zeta function is the same as Gassmann-equivalence (a purely group theoretical property), but this statement fails if the degree exceeds the characteristic. We introduce a `Teichmueller lift' of the Goss zeta function and show that equality of such is always the same as Gassmann equivalence.

math.NT↗
arXiv2008

Three examples of the relation between rigid-analytic and algebraic deformation parameters

We consider three examples of families of curves over a non-archimedean valued field which admit a non-trivial group action. These equivariant deformation spaces can be described by algebraic parameters (in the equation of the curve), or by rigid-analytic parameters (in the Schottky group of the curve). We study the relation between these parameters as rigid-analytic self-maps of the disk.

math.AG↗
arXiv2008

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves $X_0(D, N)$ attached to an Eichler order of square-free level $N$ in an indefinite rational quaternion algebra of discriminant $D>1$. We prove that, when the genus $g$ of the curve is greater than or equal to 2, $\Aut (X_0(D, N))$ is a 2-elementary abelian group which contains the group of Atkin-Lehner involutions $W_0(D, N)$ as a subgroup of index 1 or 2. It is conjectured that $\Aut (X_0(D, N)) = W_0(D, N)$ except for finitely many values of $(D, N)$ and we provide criteria that allow us to show that this is indeed often the case. Our methods are based on the theory of complex multiplication of Shimura curves and the Cerednik-Drinfeld theory on their rigid analytic uniformization at primes $p\mid D$.

math.NT↗
arXiv2008

Deformation of Curves with Automorphisms and representations on Riemann-Roch spaces

We study the deformation theory of nonsigular projective curves defined over algebraic closed fields of positive characteristic. We show that under some assumptions the local deformation problem for automorphisms of powerseries can be reduced to a deformation problem for matrix representations. We study both equicharacteristic and mixed deformations in the case of two dimensional representations.

math.AG↗
arXiv2008

Introducing Ramanujan's Class Polynomials in the Generation of Prime Order Elliptic Curves

In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method. We compare (theoretically and experimentally) the efficiency of using this new class against the use of the Weber, $M_{D,l}(x)$ and $M_{D,p_1,p_2}(x)$ polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves.

math.NT↗
arXiv2007

On the tangent space of the deformation functor of curves with automorphisms

We provide a method to compute the dimension of the tangent space to the global infinitesimal deformation functor of a curve together with a subgroup of the group of automorphisms. The computational techniques we developed are applied to several examples including Fermat curves, $p$-cyclic covers of the affine line and to Lehr-Matignon curves.

math.AG↗
arXiv2006

Computing polynomials of the Ramanujan $\mathbf{t_n}$ class invariants

We compute the minimal polynomials of the Ramanujan values $t_n$, where $n\equiv 11 \mod 24$, using Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$, and have much smaller coefficients than the Hilbert polynomials.

math.NT↗
arXiv2003

A homological approach for computing the tangent space of the deformation functor of curves with automorphisms

We give an alternative approach to the computation of the dimension of the tangent space of the deformation space of curves with automorphisms. A homological version of the local-global principle similar to the one of J.Bertin, A. Mézard is proved, and a computation in the case of ordinary curves is obtained, by application of the results of S. Nakajima for the Galois module structure of the space of 2-holomorphic differentials on them.

math.AG↗