Volume 2025, Issue 829

We show that, for any polynomial F ⁢ ( X , Y 0 , Y 1 , Y 2 ) ∈ C ⁢ [ X , Y 0 , Y 1 , Y 2 ] F(X,Y_{0},Y_{1},Y_{2})\in\mathbb{C}[X,Y_{0},Y_{1},Y_{2}] , the equation F ⁢ ( z , j ⁢ ( z ) , j ′ ⁢ ( z ) , j ′′ ⁢ ( z ) ) = 0 F(z,j(z),j^{\prime}(z),j^{\prime\prime}(z))=0 has a Zariski dense set of solutions in the hypersurface F ⁢ ( X , Y 0 , Y 1 , Y 2 ) = 0 F(X,Y_{0},Y_{1},Y_{2})=0 , unless 𝐹 is in C ⁢ [ X ] \mathbb{C}[X] or it is divisible by Y 0 Y_{0} , Y 0 − 1728 Y_{0}-1728 , or Y 1 Y_{1} . Our methods establish criteria for finding solutions to more general equations involving periodic functions. Furthermore, they produce a qualitative description of the distribution of these solutions.

We introduce an equivalence relation for Lagrangians in a symplectic manifold known as algebraic Lagrangian cobordism , which is meant to mirror algebraic equivalence of cycles. From this, we prove a symplectic, mirror-symmetric analogue of the statement “the Ceresa cycle is non-torsion in the Griffiths group of the Jacobian of a generic genus 3 curve”. Namely, we show that, for a family of tropical curves, the Lagrangian Ceresa cycle , which is the Lagrangian lift of their tropical Ceresa cycle to the corresponding Lagrangian torus fibration, is non-torsion in its oriented algebraic Lagrangian cobordism group. We proceed by developing the notions of tropical (resp. symplectic) flux, which are morphisms from the tropical Griffiths (resp. algebraic Lagrangian cobordism) groups.

We prove the rigidity of rectifiable boundaries with constant distributional mean curvature in the Brendle class of warped product manifolds (which includes important models in general relativity, like the de Sitter–Schwarzschild and Reissner–Nordstrom manifolds). As a corollary, we characterize limits of rectifiable boundaries whose mean curvatures converge, as distributions, to a constant. The latter result is new, and requires the full strength of distributional CMC-rigidity, even when one considers smooth boundaries whose mean curvature oscillations vanish in arbitrarily strong C k C^{k} -norms. Our method also establishes that rectifiable boundaries of sets of finite perimeter in the hyperbolic space with constant distributional mean curvature are finite unions of possibly mutually tangent geodesic spheres.

Let 𝐸 be an elliptic curve over a quartic field 𝐾. By the Mordell–Weil theorem, E ⁢ ( K ) E(K) is a finitely generated group. We determine all the possibilities for the torsion group E ⁢ ( K ) tors E(K)_{\mathrm{tors}} , where 𝐾 ranges over all quartic fields 𝐾 and 𝐸 ranges over all elliptic curves over 𝐾. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves 𝐸. Proving this requires showing that numerous modular curves X 1 ⁢ ( m , n ) X_{1}(m,n) have no non-cuspidal degree 4 points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.

A new monotone quantity in graphical mean curvature flows of higher codimensions is identified in this work. The submanifold deformed by the mean curvature flow is the graph of a map between Riemannian manifolds, and the quantity is monotone increasing under the area-decreasing condition of the map. The flow provides a natural homotopy of the corresponding map and leads to sharp criteria regarding the homotopic class of maps between complex projective spaces, and maps from spheres to complex projective spaces, among others.

Let X = Γ ∖ B n X=\Gamma\setminus\mathbb{B}^{n} be an 𝑛-dimensional complex ball quotient by a torsion-free non-uniform lattice Γ whose parabolic subgroups are unipotent. We prove that the volumes of subvarieties of 𝑋 are controlled by the systole of 𝑋, which is the length of a shortest closed geodesic of 𝑋. There are a number of arithmetic and geometric consequences: the systole of 𝑋 controls the growth rate of rational points on 𝑋, uniformly in the field of definition. Also, we obtain effective global generation and very ampleness results for multiples of the canonical bundle K X ̄ K_{\overline{X}} , where X ̄ \overline{X} is the toroidal compactification of 𝑋. These follow from the bound we find for the Seshadri constant of K X ̄ K_{\overline{X}} in terms of the systole.

The classical Minkowski inequality implies that the volume of a bounded convex domain in the Euclidean space is controlled from above by the integral of the mean curvature of its boundary. In this note, an analogous inequality is established without assuming convexity, valid for all bounded smooth domains in a complete manifold whose bottom spectrum is suitably large relative to its Ricci curvature lower bound. An immediate consequence is the nonexistence of embedded closed minimal hypersurfaces in such manifolds. The same nonexistence issue is also addressed for steady and expanding Ricci solitons. The proofs are very much inspired by a sharp monotonicity formula, derived for positive harmonic functions on manifolds with positive spectrum.

We give obstructions for a noncompact manifold to admit a complete Riemannian metric with (nonuniformly) positive scalar curvature. We treat both the finite volume and infinite volume cases.