A regular dessin d'enfant, in this paper, will be a pair (S,β), where S is a closed Riemann surface and β: S → ℂ̂ is a regular branched cover whose branch values are contained in the set {∞,0,1}. Let Aut(S,β) be the group of automorphisms of (S,β), that is, the deck group of β. If Aut(S,β) is Abelian, then it is known that (S,β) can be defined over ℚ. We prove that, if A is an Abelian group and Aut(S,β) ≅ A ⋊ ℤ₂, then (S,β) is also definable over ℚ. Moreover, if A ≅ ℤₙ, then we provide explicitly...

Lax operator algebras constitute a new class of infinite dimensional Lie algebras of geometric origin. More precisely, they are algebras of matrices whose entries are meromorphic functions on a compact Riemann surface. They generalize classical current algebras and current algebras of Krichever-Novikov type. Lax operators for 𝔤𝔩(n), with the spectral parameter on a Riemann surface, were introduced by Krichever. In joint works of Krichever and Sheinman their algebraic structure was revealed and...

The Generalized Elliptic Curves $(GECs)$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of “points” from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $GEC$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(CML)$. If in addition $x^2=x$, we have Hall $GECs$ and $(Q,*)$ is an exponent $3$

Let $E/Q$ be a modular elliptic curve, and let $K$ be an imaginary quadratic field. We show that the $p$-Selmer group of $E$ over certain finite anticyclotomic extensions of $K$, modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$. This refines in the current contest a result of [1].

Let $(𝒪,\sum ,F_{\infty })$ be an arithmetic ring of Krull dimension at most $1,S=\text{Spec}\left(𝒪\right)$ and $(𝒳\to S;{\sigma }_1,...,{\sigma }_n)$ a pointed stable curve. Write $𝒰=𝒳\setminus {\cup }_j{\sigma }_j\left(S\right)$. For every integer $k>0$, the invertible sheaf ${\omega }_{𝒳/S}^{k+1}(k{\sigma }_1+...+k{\sigma }_n)$ inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface ${𝒰}_{\infty }$. In this article we define a Quillen type metric ${∥·∥}_Q$ on the determinant line ${\lambda }_{k+1}=\lambda {\omega }_{𝒳/S}^{k+1}$$(k...$

We prove that for any $t\in 𝐐$, the curve$$5x^3+9y^3+10z^3+12{\left(\frac{t^2+82}{t^2+22}\right)}^3{(x+y+z)}^3=0$$in ${𝐏}^2$ is a genus $1$ curve violating the Hasse principle. An explicit Weierstrass model for its jacobian $E_t$ is given. The Shafarevich-Tate group of each $E_t$ contains a subgroup isomorphic to $𝐙/3\times 𝐙/3$.