More Like this

1. Modular curves and Néron models of generalized Jacobians

   https://doi.org/10.1112/S0010437X23007662  

   Jordan, Bruce W; Ribet, Kenneth A; Scholl, Anthony J (May 2024, Compositio Mathematica)

   Let$$X$$be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring$$R$$, and$$\mathfrak {m}$$a modulus on$$X$$, given by a closed subscheme of$$X$$which is geometrically reduced. The generalized Jacobian$$J_\mathfrak {m}$$of$$X$$with respect to$$\mathfrak {m}$$is then an extension of the Jacobian of$$X$$by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of$$X$$over$$R$$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves$$X_0(N)$$with moduli supported on the cusps. 

   more » « less
2. PL-Genus of surfaces in homology balls

   https://doi.org/10.1017/fms.2023.126  

   Hom, Jennifer; Stoffregen, Matthew; Zhou, Hugo (January 2024, Forum of Mathematics, Sigma)

   Abstract We consider manifold-knot pairs$$(Y,K)$$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$$\Sigma $$in a homology ballX, such that$$\partial (X, \Sigma ) = (Y, K)$$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$$(Y, K)$$to any knot in$$S^3$$can be arbitrarily large. The proof relies on Heegaard Floer homology. 

   more » « less
3. Quadratic Chabauty for modular curves: algorithms and examples

   https://doi.org/10.1112/S0010437X23007170  

   Balakrishnan, Jennifer S.; Dogra, Netan; Müller, J. Steffen; Tuitman, Jan; Vonk, Jan (June 2023, Compositio Mathematica)

   We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$$g>1$$whose Jacobians have Mordell–Weil rank$$g$$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$$X_0^+(N)$$of prime level$$N$$, the curve$$X_{S_4}(13)$$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$$X_{\scriptstyle \mathrm { ns}} ^+ (17)$$. 

   more » « less
4. The definable content of homological invariants II: Čech cohomology and homotopy classification

   https://doi.org/10.1017/fmp.2024.7  

   Bergfalk, Jeffrey; Lupini, Martino; Panagiotopoulos, Aristotelis (January 2024, Forum of Mathematics, Pi)

   Abstract This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functorson the category of locally compact separable metric spaces each factor into (i) what we term theirdefinable version, a functortaking values in the category$$\mathsf {GPC}$$ofgroups with a Polish cover(a category first introduced in this work’s predecessor), followed by (ii) a forgetful functor from$$\mathsf {GPC}$$to the category of groups. These definable cohomology functors powerfully refine their classical counterparts: we show that they are complete invariants, for example, of the homotopy types of mapping telescopes ofd-spheres ord-tori for any$$d\geq 1$$, and, in contrast, that there exist uncountable families of pairwise homotopy inequivalent mapping telescopes of either sort on which the classical cohomology functors are constant. We then apply the functorsto show that a seminal problem in the development of algebraic topology – namely, Borsuk and Eilenberg’s 1936 problem of classifying, up to homotopy, the maps from a solenoid complement$$S^3\backslash \Sigma $$to the$$2$$-sphere – is essentially hyperfinite but not smooth. Fundamental to our analysis is the fact that the Čech cohomology functorsadmit two main formulations: a more combinatorial one and a more homotopical formulation as the group$$[X,P]$$of homotopy classes of maps fromXto a polyhedral$$K(G,n)$$spaceP. We describe the Borel-definable content of each of these formulations and prove a definable version of Huber’s theorem reconciling the two. In the course of this work, we record definable versions of Urysohn’s Lemma and the simplicial approximation and homotopy extension theorems, along with a definable Milnor-type short exact sequence decomposition of the space$$\mathrm {Map}(X,P)$$in terms of its subset ofphantom maps; relatedly, we provide a topological characterization of this set for any locally compact Polish spaceXand polyhedronP. In aggregate, this work may be more broadly construed as laying foundations for the descriptive set theoretic study of the homotopy relation on such spaces$$\mathrm {Map}(X,P)$$, a relation which, together with the more combinatorial incarnation of, embodies a substantial variety of classification problems arising throughout mathematics. We show, in particular, that ifPis a polyhedralH-group, then this relation is both Borel and idealistic. In consequence,$$[X,P]$$falls in the category ofdefinable groups, an extension of the category$$\mathsf {GPC}$$introduced herein for its regularity properties, which facilitate several of the aforementioned computations. 

   more » « less
5. Eliminating Thurston obstructions and controlling dynamics on curves

   https://doi.org/10.1017/etds.2023.114  

   BONK, MARIO; HLUSHCHANKA, MIKHAIL; ISELI, ANNINA (September 2024, Ergodic Theory and Dynamical Systems)

   Abstract Every Thurston map$$f\colon S^2\rightarrow S^2$$on a$$2$$-sphere$$S^2$$induces a pull-back operation on Jordan curves$$\alpha \subset S^2\smallsetminus {P_f}$$, where$${P_f}$$is the postcritical set off. Here the isotopy class$$[f^{-1}(\alpha )]$$(relative to$${P_f}$$) only depends on the isotopy class$$[\alpha ]$$. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the mapfcan be seen as a fixed point of the pull-back operation. We show that if a Thurston mapfwith a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying$$2$$-sphere and construct a new Thurston map$$\widehat f$$for which this obstruction is eliminated. We prove that no other obstruction arises and so$$\widehat f$$is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem. 

   more » « less