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1. Shadow line distributions

   https://doi.org/10.1090/mcom/4110  

   Balakrishnan, Jennifer; Çiperiani, Mirela; Mazur, Barry; Rubin, Karl (July 2025, Mathematics of Computation)

   Let $E$be an elliptic curve over $\mathbb{Q}$with Mordell–Weil rank $2$and $p$be an odd prime of good ordinary reduction. For every imaginary quadratic field $K$satisfying the Heegner hypothesis, there is (subject to the Shafarevich–Tate conjecture) a line, i.e., a free ${\mathbb{Z}}_p$-submodule of rank $1$, in $E\left(K\right)\otimes <\#comment/>{\mathbb{Z}}_p$given by universal norms coming from the Mordell–Weil groups of subfields of the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$; we call it theshadow line. When the twist of $E$by $K$has analytic rank $1$, the shadow line is conjectured to lie in $E\left(\mathbb{Q}\right)\otimes <\#comment/>{\mathbb{Z}}_p$; we verify this computationally in all our examples. We study the distribution of shadow lines in $E\left(\mathbb{Q}\right)\otimes <\#comment/>{\mathbb{Z}}_p$as $K$varies, framing conjectures based on the computations we have made. 

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2. Integral models for spaces via the higher Frobenius

   https://doi.org/10.1090/jams/998  

   Yuan, Allen (January 2023, Journal of the American Mathematical Society)

   We give a fully faithful integral model for simply connected finite complexes in terms of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring spectra and the Nikolaus–Scholze Frobenius. The key technical input is the development of a homotopy coherent Frobenius action on a certain subcategory of $p$-complete ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings for each prime $p$. Using this, we show that the data of a simply connected finite complex $X$is the data of its Spanier-Whitehead dual, as an ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-ring, together with a trivialization of the Frobenius action after completion at each prime. In producing the above Frobenius action, we explore two ideas which may be of independent interest. The first is a more general action of Frobenius in equivariant homotopy theory; we show that a version of Quillen’s $Q$-construction acts on the $\mathrm{\infty <\#comment/>}$-category of ${\mathbb{E}}_{\mathrm{\infty <\#comment/>}}$-rings with “genuine equivariant multiplication,” which we call global algebras. The second is a “pre-group-completed” variant of algebraic $K$-theory which we callpartial $K$-theory. We develop the notion of partial $K$-theory and give a computation of the partial $K$-theory of ${\mathbb{F}}_p$up to $p$-completion. 

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3. Type-I permanence

   https://doi.org/10.1090/ert/648  

   Chirvasitu, Alexandru (July 2023, Representation Theory of the American Mathematical Society)

   We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb{N}⊴<\#comment/>\mathbb{E}$of locally compact groups and a twisted action $(\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>})$thereof on a (post)liminal $C^{\ast <\#comment/>}$-algebra $A$the twisted crossed product $A{⋊<\#comment/>}_{\mathrm{\alpha <\#comment/>},\mathrm{\tau <\#comment/>}}\mathbb{E}$is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb{N}⊴<\#comment/>\mathbb{E}$is type-I as soon as $\mathbb{E}$is. This happens for instance if $\mathbb{N}$is discrete and $\mathbb{E}$is Lie, or if $\mathbb{N}$is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions. In the same spirit, call a locally compact group $\mathbb{G}$type-I-preserving if all semidirect products $\mathbb{N}⋊<\#comment/>\mathbb{G}$are type-I as soon as $\mathbb{N}$is, andlinearlytype-I-preserving if the same conclusion holds for semidirect products $V⋊<\#comment/>\mathbb{G}$arising from finite-dimensional $\mathbb{G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie. 

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4. 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  ${\mathbb{F}}_2$. 

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5. On the Boundary Behavior of Mass-Minimizing Integral Currents

   https://doi.org/10.1090/memo/1446  

   De_Lellis, Camillo; De_Philippis, Guido; Hirsch, Jonas; Massaccesi, Annalisa (November 2023, Memoirs of the American Mathematical Society)

   Let $\mathrm{\Sigma <\#comment/>}$be a smooth Riemannian manifold, $\mathrm{\Gamma <\#comment/>}\subset <\#comment/>\mathrm{\Sigma <\#comment/>}$a smooth closed oriented submanifold of codimension higher than $2$and $T$an integral area-minimizing current in $\mathrm{\Sigma <\#comment/>}$which bounds $\mathrm{\Gamma <\#comment/>}$. We prove that the set of regular points of $T$at the boundary is dense in $\mathrm{\Gamma <\#comment/>}$. Prior to our theorem the existence of any regular point was not known, except for some special choice of $\mathrm{\Sigma <\#comment/>}$and $\mathrm{\Gamma <\#comment/>}$. As a corollary of our theorem we answer to a question in Almgren’sAlmgren’s big regularity paperfrom 2000 showing that, if $\mathrm{\Gamma <\#comment/>}$is connected, then $T$has at least one point $p$of multiplicity $\frac{1}{2}$, namely there is a neighborhood of the point $p$where $T$is a classical submanifold with boundary $\mathrm{\Gamma <\#comment/>}$; we generalize Almgren’s connectivity theorem showing that the support of $T$is always connected if $\mathrm{\Gamma <\#comment/>}$is connected; we conclude a structural result on $T$when $\mathrm{\Gamma <\#comment/>}$consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when $\mathrm{\Sigma <\#comment/>}={\mathbb{R}}^{m+1}$and $T$is $m$-dimensional. 

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