arXiv subjects

Hazem Hassan

Publications and source records attributed to Hazem Hassan.

2 recordsLinked to original sources

arXiv2025

$p$-adic Higher Green's Functions for Stark-Heegner Cycles

Heegner cycles are higher weight analogues of Heegner points. Their arithmetic intersection numbers also appear as Fourier coefficients of modular forms and often belong to abelian extensions of imaginary-quadratic fields. Rotger and Seveso propose a precise recipe for the $p$-adic Abel-Jacobi images of cycle classes whose existence is predicted by conjectures of Bloch and Beilinson and which would be a real-quadratic analogue to Heegner cycles: the Stark-Heegner cycles of the title. In this paper, we generalize Darmon-Vonk's theory of rigid meromorphic cocycles to higher weight, producing a higher Green's pairing of real-quadratic divisors on the $p$-adic upper half-plane, which seems to be the real-quadratic analogue of the pairing of Heegner cycles. Computation of these values for "principal cycles'' gives evidence that they lie in abelian extensions of real-quadratic fields. The algebraicity of certain values of the higher Green's function is indirect evidence for the existence of algebraic Stark-Heegner cycles.

math.NT↗

arXiv2024

Integral aspects of Fourier duality for abelian varieties

We prove several results about integral versions of Fourier duality for abelian schemes, making use of Pappas's work on integral Grothendieck-Riemann-Roch. If $S$ is smooth quasi-projective of dimension $d$ over a field and $π\colon X\to S$ is a $g$-dimensional abelian scheme, we prove, under very mild assumptions on $X/S$, that all classical results about Fourier duality, including the existence of a Beauville decomposition, are valid for the Chow ring $\mathrm{CH}(X;Λ)$ with coefficients in the ring $Λ= \mathbb{Z}[1/(2g+d+1)!]$. If $X$ admits a polarization $θ$ of degree $ν(θ)^2$ we further construct an $\mathfrak{sl}_2$-action on $\mathrm{CH}(X;Λ_θ)$ with $Λ_θ= Λ[1/ν(θ)]$, and we show that $\mathrm{CH}(X;Λ_θ)$ is a sum of copies of the symmetric powers $\mathrm{Sym}^n(\mathrm{St})$ of the $2$-dimensional standard representation, for $n=0,\ldots,g$. For an abelian variety over an algebraically closed field, we use our results to produce torsion classes in $\mathrm{CH}^i(X;Λ_θ)$ for every $i\in \{1,\ldots,g\}$.

math.AG↗