Related papers: Explicit CM-theory for level 2-structures on abeli…

Let $A$ be an abelian variety with commutative endomorphism algebra over a finite field $k$. The $k$-isogeny class of $A$ is uniquely determined by a Weil polynomial $f_A$ without multiple roots. We give a classification of the groups of…

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action…

We discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety,…

In this manuscript we consider a special complex torus, denoted $S_{\Delta_{2k}}$ (for each $k \in \mathbb{N},\, k \geq 1$) and called the Dirac spinor torus. It is an Abelian variety of complex dimension $2^{k}$ whose covering space is the…

The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary…

For $J$ an abelian surface, the Galois representation $\varrho_{J, \ell} : {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow {\rm Aut}(J[\ell]) \simeq {\rm GSp}_4(\mathbb{F}_\ell)$ is typically surjective, with smaller images…

We consider generalized $\Lambda$-structures on algebras and schemes over the ring of integers $\mathit{O}_K$ of a number field $K$. When $K=\mathbb{Q}$, these agree with the $\lambda$-ring structures of algebraic K-theory. We then study…

In this article we perform an extensive study of the spaces of automorphic forms for GL(2) of weight two and level N, for N an ideal in the ring of integers of the quartic CM field generated by the twelfth roots of unity. This study is…

The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched…

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan's decomposition theorem. We show that they are essentially the same.…

Let $G$ be a finite group. To every smooth $G$-action on a compact, connected and oriented Riemann surface we can associate its data of singular orbits. The set of such data becomes an Abelian group $B_G$ under the $G$-equivariant connected…

We present an explicit and computationally actionable blueprint for constructing vector-valued Siegel modular forms associated to real multiplication (RM) abelian surfaces, leveraging the theta correspondence for the unitary dual pair…

The moduli space of principally polarized abelian surfaces is parametrized by three Igusa functions. In this article we investigate a new way to evaluate these functions by using Siegel Eisenstein series. We explain how to compute the…

Given a number field $K$ and an integer $m\geq 0$, let $K_m$ denote the maximal $m$-step solvable Galois extension of $K$ and write $G_K^m$ for the maximal $m$-step solvable Galois group Gal$(K_m/K)$ of $K$. In this paper, we prove that the…

Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$…

We develop connections between the qualitative dynamics of Hamiltonian isotopies on a surface $\Sigma$ and their chain-level Floer theory using ideas drawn from Hofer-Wysocki-Zehnder's theory of finite energy foliations. We associate to…

We determine the number of Del Pezzo surfaces of degree 2 over finite fields of odd characteristic with specified action of the Frobenius endomorphism, i.e. we solve the "quantitative inverse Galois problem". As applications we determine…

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the…

For each prime number $\ell$ and for each imaginary quadratic order of class number one or two, we determine all the possible $\ell$-adic Galois representations that occur for any elliptic curve with complex multiplication by such an order…

Let $k=k_0(\sqrt[3]{d})$ be a cubic Kummer extension of $k_0=\mathbb{Q}(\zeta_3)$ with $d>1$ a cube-free integer and $\zeta_3$ a primitive third root of unity. Denote by $C_{k,3}^{(\sigma)}$ the $3$-group of ambiguous classes of the…