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Register a new userThe Distribution Relation and Inverse Function Theorem in Arithmetic Geometry
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We study arithmetic distribution relations and the inverse function theorem in algebraic and arithmetic geometry, with an emphasis
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We study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^{r-1}(x + 1)(x + t)$ over the function field $mathbb{F}_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, we compute the $L$-function of $J$ over $mathbb{F}_q(t^{1/d})$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb{F}_q(t^{1/d})$. When $d$ is divisible by $r$ and of the form $p^ u +1$, and $K_d := mathbb{F}_p(mu_d,t^{1/d})$, we write down explicit points in $J(K_d)$, show that they generate a subgroup $V$ of rank $(r-1)(d-2)$ whose index in $J(K_d)$ is finite and a power of $p$, and show that the order of the Tate-Shafarevich group of $J$ over $K_d$ is $[J(K_d):V]^2$. When $r>2$, we prove that the new part of $J$ is isogenous over $overline{mathbb{F}_p(t)}$ to the square of a simple abelian variety of dimension $phi(r)/2$ with endomorphism algebra $mathbb{Z}[mu_r]^+$. For a prime $ell$ with $ell mid pr$, we prove that $J[ell](L)={0}$ for any abelian extension $L$ of $overline{mathbb{F}}_p(t)$.
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.
Let X and Y be curves over a finite field. In this article we explore methods to determine whether there is a rational map from Y to X by considering L-functions of certain covers of X and Y and propose a specific family of covers to address the special case of determining when X and Y are isomorphic. We also discuss an application to factoring polynomials over finite fields.
Given a newform f, we extend Howards results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovav{r}s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures `a la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.
In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Starks conjecture. We also discuss connections with 1-cocycles for the modular group $mathrm{PSL}(2,mathbb{Z})$.
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