arXiv subjects

Dimitrios Noulas

Publications and source records attributed to Dimitrios Noulas.

4 recordsLinked to original sources

arXiv2026

On periods and Jacobians of Heisenberg curves

Heisenberg curves are cyclic covers of Fermat curves that also arise as non-abelian covers of the projective line, branched over three points by the discrete Heisenberg group modulo an integer. As normal Belyi covers, these are curves with many automorphisms in the sense of Oort, who questioned whether such curves have CM Jacobians. In 1986, Ihara proposed using towers of curves to study the pro-$\ell$ Galois representation associated with the thrice-punctured projective line. To study the kernel of this representation, he suggested using Heisenberg curves, but it was unknown to him at the time whether their Jacobians lacked complex multiplication. In this paper, for any odd prime $\ell$, we prove that Heisenberg curves of level $\ell^n\neq 3$ do not have CM Jacobians. Thus, we resolve the missing part of Ihara's original argument and in doing so we provide an infinite family of new counterexamples to Oort's question.

math.AG↗

arXiv2026

Distortion maps for elliptic curves over finite fields

The Weil pairing on elliptic curves has deep links with discrete logarithm problems. In practice, to better suit the functionalities of cryptosystems, one often needs to modify the original Weil pairing via what is called a distortion map. We propose a study on the question of the existence of distortion maps for elliptic curves over finite fields. We revisit results from the literature and provide detailed proofs. We also propose new perspectives at times.

math.NT↗

arXiv2025

An Arithmetic Topology viewpoint on Descent theory and Equivariant Categories

We establish a unified group-theoretic framework bridging the arithmetic homotopy exact sequence of a variety and the Birman exact sequence of a surface. Within this framework, we reinterpret classical arithmetic notions - such as the descent of varieties and of covers - and construct their topological analogues. We formalize the parallel setting between closed subgroups of the absolute Galois group and subgroups of the Mapping Class Group of a base space and their actions on fundamental groups. This provides an analogy between arithmetic and topological invariants, allowing us to define the groups of moduli, definition, and invariance in both settings. Using this unified perspective, some purely group-theoretic proofs provide results in both settings simultaneously. Applications include a topological analogue of Weil's Descent Theorem for mapping class groups and an adaptation of Débes and Douai's cohomological obstructions regarding descent of algebraic covers to the topological setting. Finally, we elevate these results to the categorical level. We demonstrate that the classical Weil cocycle condition is equivalent to the existence of a linearization in the language of equivariant categories. Applying this perspective to the bounded derived category of coherent sheaves $\mathsf{D^b}(X)$, we show that the equivariant derived category $\mathsf{D^b}(X)^G$, under the action induced by a Weil descent datum, recovers the derived category of the descended variety.

math.AG↗

arXiv2024

Galois Action on Homology of the Heisenberg Curve

The Heisenberg curve is defined topologically as a cover of the Fermat curve and corresponds to an extension of the projective line minus three points by the non-abelian Heisenberg group modulo n. We compute its fundamental group and investigate an action from Artin's Braid group to the curve itself and its homology. We also provide a description of the homology in terms of irreducible representations of the Heisenberg group over a field of characteristic $0$.

math.NT↗