Matthew Baker
We investigate the interplay between linear systems on curves and graphs in the context of specialization of divisors on an arithmetic surface. We also provide some applications of our results to graph theory, arithmetic geometry, and tropical geometry.
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Using the Shioda-Tate theorem and an adaptation of Silverman's specialization theorem, we reduce the specialization of Mordell-Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields $k$ to the the specialization theorem for Néron-Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface $S$, for all vertical curves $S_x$ of a fibration $S \to U \subseteq \mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme o…
Let $E:y^2=(x-e_1)(x-e_2)(x-e_3)$ be a nonconstant elliptic curve over $\mathbb{Q}(t)$, where $e_j\in \mathbb{Z}[t]$. We describe a method for finding a specialization $t\mapsto t_0\in\mathbb{Q}$ such that the specialization homomorphism is injective. The method can be directly extended to elliptic curves with $e_j\in \mathcal{R}_K[t]$ where $K$ is a number field and $\mathcal{R}_K$ is some UFD such that $\mathcal O_K\subset\mathcal R_K\subset K$. Further, we make a simplification of the method for a special case of quadratic twists. The method is applied to obtain exactly the rank and prove t…
Let $E:y^2=x^3+Ax^2+Bx+C$ be a nonconstant elliptic curve over $\mathbb{Q}(t)$ with at least one nontrivial $\mathbb{Q}(t)$-rational $2$-torsion point. We describe a method for finding $t_0\in\mathbb Q$ for which the corresponding specialization homomorphism $t\mapsto t_0\in\mathbb{Q}$ is injective. The method can be directly extended to elliptic curves over $K(t)$ for a number field $K$ of class number $1$, and in principal for arbitrary number field $K$. One can use this method to calculate the rank of elliptic curves over $\mathbb Q(t)$ of the form as above, and to prove that given points a…
Let $(G, ω)$ be a hyperelliptic vertex-weighted graph of genus $g \geq 2$. We give a characterization of $(G, ω)$ for which there exists a smooth projective curve $X$ of genus $g$ over a complete discrete valuation field with reduction graph $(G, ω)$ such that the ranks of any divisors are preserved under specialization. We explain, for a given vertex-weighted graph $(G, ω)$ in general, how the existence of such $X$ relates the Riemann--Roch formulae for $X$ and $(G, ω)$, and also how the existence of such $X$ is related to a conjecture of Caporaso.
Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. In 2015, Gusić and Tadić gave an easy-to-check criterion, based only on a Weierstrass equation for $E/\mathbb Q(t)$, that is sufficient to conclude that the specialization map at $t_0$ is injective. The criterion critically requires that $E$ has nontrivial $\mathbb Q(t)$-rational 2-torsion points. In this article, we explain how the criterion can be used in some cases where this requirement is not satisfied and provide some examples.
A metrized complex of algebraic curves is a finite metric graph together with a collection of marked complete nonsingular algebraic curves, one for each vertex, the marked points being in bijection with incident edges. We establish a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical Riemann-Roch theorem and its graph-theoretic and tropical analogues due to Baker-Norine, Gathmann-Kerber, and Mikhalkin-Zharkov. We also establish generalizations of the second author's specialization lemma and its weighted graph analogue due to Caporaso and the first author…
We describe the specialization of Heegner points on Shimura curves at primes of bad reduction. Moreover, we give some reciprocity laws relating the Galois action on these points to natural actions on the set of singular points and the set of connected components of the fiber.
Given a family of abelian varieties over a quasiprojective smooth curve $T^0$ over a global field and a point $P$ on the generic fiber, we show that the Néron-Tate canonical height $h_{X_t}(P_t)$ of $P_t$ along each fiber is exactly equal to a Weil height $h_{\overline M}(t)$ given by an adelic metrized line bundle $\overline M$ on the unique smooth projective curve $T$ containing $T^0$. As a consequence, we show that a conjecture of Zhang on the finiteness of small-height specializations of $P$ is equivalent to $\overline M$ being big.
Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. Given a subgroup $M$ of $E(\mathbb Q(t))$ with mild conditions and $t_0 \in \mathbb Q$ coming from a relatively large subset $S_M \subset \mathbb Q$, we provide an algorithm that can show that the specialization map $σ_{t_0} : E(\mathbb Q(t)) \to E_{t_0}(\mathbb Q)$ is injective when restricted to $M$. The set $S_M$ is effectively computable in certain cases, and we carry out this computation for some explicit examples where $E$ is given by a W…
Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes
本文将 Genus Theory 扩展至系数在任意 PID 上的扭曲二次型，证明其与超椭圆曲线上 2-descent 映射的关联，并应用于解决除子类特化问题，证明非平凡特化集合密度为 1。
Realizing Galois representations in abelian varieties by specialization
本文研究在阿贝尔簇中实现给定伽罗瓦表示的问题，证明了在特征不为2的希尔伯特域上，对足够大的维数g存在无穷多个绝对简单阿贝尔簇实现该表示，并利用Néron特化定理的推广统一处理了相关曲线构造。
Low rank specializations of elliptic surfaces
针对带2阶挠点的非同构椭圆曲线 $E/\mathbb{Q}(T)$，在判别式假设下证明无穷多有理特化的秩具有显式上界，并给出满足 $r_t\leq r+1$ 的例子。
Tropical Weierstrass points and Weierstrass weights
本文研究热带Weierstrass点，通过分析热带曲线上的理函数斜率，为连通分支赋予内在权重，证明了权重总和与曲线的亏格、度数和秩的关系，并探讨了热带线性序列的类比结果。
There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular curves. It includes a length two complex on every modular curve. Their second cohomology groups recover the Beilinson-Kato Euler system in K_2 of modular curves. We show that the above connection in the double logarithm case is provided by the specialization at a cusp of the Euler complex data on the modular curve Y_1(N). Furthermore, specializing the Euler com…
We study the formal properties of correspondences of curves without a core, focusing on the case of étale correspondences. The motivating examples come from Hecke correspondences of Shimura curves. Given a correspondence without a core, we construct an infinite graph $\mathcal{G}_{gen}$ together with a large group of "algebraic" automorphisms $A$. The graph $\mathcal{G}_{gen}$ measures the "generic dynamics" of the correspondence. We construct specialization maps $\mathcal{G}_{gen}\rightarrow\mathcal{G}_{phys}$ to the "physical dynamics" of the correspondence. We also prove results on the numb…
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