In this note, we prove a formula for the cancellation exponent $k_{v,n}$ between division polynomials $\psi _n$ and $\phi _n$ associated with a sequence $\{nP\}_{n\in \mathbb {N}}$ of points on an elliptic curve $E$ defined over a discrete valuation field $K$. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.

We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$. We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$, the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$.

In this paper, we prove Kato’s main conjecture for $CM$ modular forms for primes of potentially ordinary reduction under certain hypotheses on the modular form.

In this paper, we investigate the theory of heights in a family of stacky curves following recent work of Ellenberg, Satriano, and Zureick-Brown. We first give an elementary construction of a height which is seen to be dual to theirs. We count rational points having bounded ESZ-B height on a particular stacky curve, answering a question of Ellenberg, Satriano, and Zureick-Brown. We also show that when the Euler characteristic of stacky curves is non-positive, the ESZ-B height coming from the anti-canonical divisor class fails to have the Northcott property. We prove that a stacky version of a conjecture of Vojta is equivalent to the $abc$-conjecture.

The Manin–Peyre conjecture is established for a class of smooth spherical Fano varieties of semisimple rank one. This includes all smooth spherical Fano threefolds of type T as well as some higher-dimensional smooth spherical Fano varieties.

This paper goes beyond Katz–Sarnak theory on the distribution of curves over finite fields according to their number of rational points, theoretically, experimentally, and conjecturally. In particular, we give a formula for the limits of the moments measuring the asymmetry of this distribution for (non-hyperelliptic) curves of genus $g\geq 3$. The experiments point to a stronger notion of convergence than the one provided by the Katz–Sarnak framework for all curves of genus $\geq 3$. However, for elliptic curves and for hyperelliptic curves of every genus, we prove that this stronger convergence cannot occur.

DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$, for each degree $d\geq 2$. The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.

Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree-$2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree-$2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

We prove the existence of $\mathrm {GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${\mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb {H}}$, arising from forms of ${\mathrm {GSO}}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${\mathrm {SO}}_{2n}$-valued Galois representations by removing the outer automorphism ambiguity.

For homogeneous polynomials $G_1,\ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork’s theory. In this article, we will construct an explicit cochain map from the Dwork complex of $G_1,\ldots ,G_k$ to the Monsky–Washnitzer complex associated with some affine bundle over the complement $\mathbb {P}^n\setminus X_G$ of the common zero $X_G$ of $G_1,\ldots ,G_k$, which computes the rigid cohomology of $\mathbb {P}^n\setminus X_G$. We verify that this cochain map realizes the rigid cohomology of $\mathbb {P}^n\setminus X_G$ as a direct summand of the Dwork cohomology of $G_1,\ldots ,G_k$. We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz’s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.

We study finite orbits of non-elementary groups of automorphisms of compact projective surfaces. We prove that if the surface and the group are defined over a number field $\mathbf {k}$ and the group contains parabolic elements, then the set of finite orbits is not Zariski dense, except in certain very rigid situations, known as Kummer examples. Related results are also established when $\mathbf {k} = \mathbf {C}$. An application is given to the description of ‘canonical vector heights’ associated to such automorphism groups.

The problem of classifying elliptic curves over $\mathbb Q$ with a given discriminant has received much attention. The analogous problem for genus $2$ curves has only been tackled when the absolute discriminant is a power of $2$. In this article, we classify genus $2$ curves C defined over ${\mathbb Q}$ with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many $1$-parameter families of genus $2$ curves. In particular, we provide genus $2$ analogues to Neumann–Setzer families of elliptic curves over the rationals.

In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $\overline{M}_{0,n}$.

Let $\Gamma \subset \overline {\mathbb {Q}}^*$ be a finitely generated subgroup. Denote by $\Gamma _{\mathrm {div}}$ its division group. A recent conjecture due to Rémond, related to the Zilber–Pink conjecture, predicts that the absolute logarithmic Weil height of an element of $\mathbb {Q}(\Gamma _{\mathrm {div}})^*\backslash \Gamma _{\mathrm {div}}$ is bounded from below by a positive constant depending only on $\Gamma $. In this paper, we propose a new way to tackle this problem.

We study for each fixed integer $g \ge 2$, for all primes $\ell $ and p with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell $-marked principally polarized superspecial abelian varieties of dimension g in characteristic p, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of p. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles–Goren–Lauter-type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$. As a byproduct, we also show that the finite regular directed graphs constructed by Jordan and Zaytman also has the same property.

We prove that there exist infinitely many coprime numbers a, b, c with $a+b=c$ and $c>\operatorname {\mathrm {rad}}(abc)\exp (6.563\sqrt {\log c}/\log \log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. Our work builds on that of van Frankenhuysen (J. Number Theory 82(2000), 91–95) who proved the existence of examples satisfying the above bound with the constant $6.068$ in place of $6.563$. We show that the constant $6.563$ may be replaced by $4\sqrt {2\delta /e}$ where $\delta $ is a constant such that all unimodular lattices of sufficiently large dimension n contain a nonzero vector with $\ell _1$-norm at most $n/\delta $.

In 2007 Chang and Yu determined all the algebraic relations among Goss’s zeta values for $A=\mathbb F_q[\theta ]$, also known as the Carlitz zeta values. Goss raised the problem of determining all algebraic relations among Goss’s zeta values at positive integers for a general base ring A, but very little is known. In this paper, we develop a general method, and we determine all algebraic relations among Goss’s zeta values for the base ring A which is the coordinate ring of an elliptic curve defined over $\mathbb F_q$. To our knowledge, this is the first work tackling Goss’s problem when the base ring has class number strictly greater than 1.

In this paper, together with the preceding Part I [10], we develop a framework for tame geometry on Henselian valued fields of characteristic zero, called Hensel minimality. It adds to [10] the treatment of the mixed characteristic case. Hensel minimality is inspired by o-minimality and its role in real geometry and diophantine applications. We develop geometric results and applications for Hensel minimal structures that were previously known only under stronger or less axiomatic assumptions, and which often have counterparts in o-minimal structures. We prove a Jacobian property, a strong form of Taylor approximations of definable functions, resplendency results and cell decomposition, all under Hensel minimality – more precisely, $1$-h-minimality. We obtain a diophantine application of counting rational points of bounded height on Hensel minimal curves.

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$, where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$-extension, then it holds for almost all $\mathbb{Z}_{p}$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus g over $\mathbb{F}_q$. As a consequence of Katz–Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all q large enough, the existence of a curve of genus g over $\mathbb{F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result: We show that the bound $1+q+4 \sqrt{q} -32$ is valid for all $g\ge 2$ and for all q.