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1. Newton Polygon Stratification of the Torelli Locus in Unitary Shimura Varieties

   https://doi.org/10.1093/imrn/rnaa306  

   Li, Wanlin; Mantovan, Elena; Pries, Rachel; Tang, Yunqing (December 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $$p$$ reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $$p$$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen’s work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases). 

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2. Expressive curves

   https://doi.org/10.1090/cams/12  

   Fomin, Sergey; Shustin, Eugenii (August 2023, Communications of the American Mathematical Society)

   We initiate the study of a class of real plane algebraic curves which we callexpressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve  $C$ is expressive if (a) each irreducible component of  $C$ can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of $C$ in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of $C$ in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more. 

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3. Triangular modular curves of small genus

   https://doi.org/10.1007/s40993-022-00395-w  

   Duque-Rosero, Juanita; Voight, John (March 2023, Research in Number Theory)

   Abstract Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy $$\text {PGL}_2(\mathbb {F}_q)$$ PGL 2 ( F q ) or $$\text {PSL}_2(\mathbb {F}_q)$$ PSL 2 ( F q ) . We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus. 

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4. A CLASSIFICATION OF LAGRANGIAN PLANES IN HOLOMORPHIC SYMPLECTIC VARIETIES

   https://doi.org/10.1017/S1474748015000328  

   Bakker, Benjamin (September 2017, Journal of the Institute of Mathematics of Jussieu)

   Classically, an indecomposable class $$R$$ in the cone of effective curves on a K3 surface $$X$$ is representable by a smooth rational curve if and only if $$R^{2}=-2$$ . We prove a higher-dimensional generalization conjectured by Hassett and Tschinkel: for a holomorphic symplectic variety $$M$$ deformation equivalent to a Hilbert scheme of $$n$$ points on a K3 surface, an extremal curve class $$R\in H_{2}(M,\mathbb{Z})$$ in the Mori cone is the line in a Lagrangian $$n$$ -plane $$\mathbb{P}^{n}\subset M$$ if and only if certain intersection-theoretic criteria are met. In particular, any such class satisfies $$(R,R)=-\frac{n+3}{2}$$ , and the primitive such classes are all contained in a single monodromy orbit. 

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5. Entire Theta Operators at Unramified Primes

   https://doi.org/10.1093/imrn/rnab190  

   Eischen, E; Mantovan, E (July 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $$p$$-adic and $$\textrm{mod}\; p$$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes $$p$$ to the whole Shimura variety of the $$\textrm{mod}\; p$$ reduction of $$p$$-adic Maass–Shimura operators a priori defined only over the $$\mu $$-ordinary locus, and (2) the construction of new $$\textrm{mod}\; p$$ theta operators that do not arise as the $$\textrm{mod}\; p$$ reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree. 

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