More Like this

1. Every finite abelian group is the group of rational points of an ordinary abelian variety over 𝔽₂, 𝔽₃ and 𝔽₅

   https://doi.org/10.1090/proc/16127  

   Marseglia, Stefano; Springer, Caleb (February 2023, Proceedings of the American Mathematical Society)

   We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over ${\mathbb{F}}_2$, ${\mathbb{F}}_3$and ${\mathbb{F}}_5$. We produce partial results for abelian varieties over a general finite field  ${\mathbb{F}}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over ${\mathbb{F}}_q$when $q$is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over  ${\mathbb{F}}_2$. 

   more » « less
2. Staircase patterns in Hirzebruch surfaces

   https://doi.org/10.4171/cmh/571  

   Magill, Nicki; McDuff, Dusa; Weiler, Morgan (July 2024, Commentarii Mathematici Helvetici)

   The ellipsoidal capacity function of a symplectic four manifoldXmeasures how much the form onXmust be dilated in order for it to admit an embedded ellipsoid of eccentricityz. In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions,Xis said to have a staircase. This paper gives an almost complete description of the staircases in the ellipsoidal capacity functions of the family of symplectic Hirzebruch surfacesH_{b}formed by blowing up the projective plane with weightb. We describe an interweaving, recursively defined, family of obstructions to symplectic embeddings of ellipsoids that show there is an open dense set of shape parametersbthat are blocked, i.e. have no staircase, and an uncountable number of other values ofbthat do admit staircases. The remainingb-values form a countable sequence of special rational numbers that are closely related to the symmetries discussed in Magill–McDuff (arXiv:2106.09143). We show that none of them admit ascending staircases. Conjecturally, none admit descending staircases. Finally, we show that, as long asbis not one of these special rational values, any staircase inH_{b}has irrational accumulation point. A crucial ingredient of our proofs is the new, more indirect approach to using almost toric fibrations in the analysis of staircases by Magill (arXiv:2204.12460). In particular, the structure of the relevant mutations of the set of almost toric fibrations onH_{b}is echoed in the structure of the set of blockedb-intervals. 

   more » « less
3. The Supersingular Locus of the Shimura Variety of $$\textrm{GU}(2,n-2)$$

   https://doi.org/10.1007/s42543-025-00102-5  

   Fox, Maria; Imai, Naoki (July 2025, Peking Mathematical Journal)

   Abstract We study the supersingular locus of a reduction at an inert prime of the Shimura variety attached to$$\textrm{GU}(2,n-2)$$ $\text{GU}(2,n-2)$. More concretely, we realize irreducible components of the supersingular locus as closed subschemes of flag schemes over Deligne–Lusztig varieties defined by explicit conditions after taking perfections. Moreover, we study the intersections of the irreducible components. Stratifications of Deligne–Lusztig varieties defined using powers of Frobenius action appear in the description of the intersections. 

   more » « less
4. The distribution of negative eigenvalues of Schrödinger operators on asymptotically hyperbolic manifolds

   https://doi.org/10.4171/JST/561  

   Sá_Barreto, Antônio; Wang, Yiran (May 2025, Journal of Spectral Theory)

   We study the asymptotic behavior of the counting function of negative eigenvalues of Schrödinger operators with real valued potentials which decay at infinity on asymptotically hyperbolic manifolds. We establish conditions on the rate of decay of the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval(-\infty, -E)asE\rightarrow 0. 

   more » « less
5. Multiplicative dependence of rational values modulo approximate finitely generated groups

   https://doi.org/10.1017/S0305004124000173  

   BÉRCZES, ATTILA; BUGEAUD, YANN; GYŐRY, KÁLMÁN; MELLO, JORGE; OSTAFE, ALINA; SHA, MIN (July 2024, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are ‘close’ (with respect to the Weil height) to division groups of finitely generated multiplicative groups of a number fieldK. For example, we show that under some conditions on rational functions$$f_1, \ldots, f_n\in K(X)$$, there are only finitely many elements$$\alpha \in K$$such that$$f_1(\alpha),\ldots,f_n(\alpha)$$are multiplicatively dependent modulo such sets. 

   more » « less