For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of our functions f(X) is indecomposable, which means that it cannot be written as the composition of lower-degree rational functions in F_q(X). In case q is not a power of 3, these are the first known examples of indecomposable exceptional rational functions f(X) over F_q which have non-solvable monodromy groups and have arbitrarily large degree. These are also the first known examples of wildly ramified indecomposable exceptional rational functions f(X), other than linear changes of polynomials.
We show that every exceptional Lie algebra over a number field can be obtained by Tits' construction from an octonion algebra O and a cubic Jordan algebra J. In particular, the exceptional Lie algebra contains a dual pair which is the direct sum of the derivation algebras of O and J. We determine rational forms of this dual pair.
If $α$ is a non-zero algebraic number, we let $m(α)$ denote the Mahler measure of the minimal polynomial of $α$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version of the Mahler measure called the $t$-metric Mahler measure, denoted $m_t(α)$. For fixed $α\in \bar{\mathbb Q}$, the map $t\mapsto m_t(α)$ is continuous, and moreover, is infinitely differentiable at all but finitely many points, called {\it exceptional points} for $α$. It remains open to determine whether there is a sequence of elements $α_n\in \bar{\mathbb Q}$ such that …
The existence of a Landau-Siegel zero leads to the Deuring-Heilbronn phenomenon, here appearing in the 1-level density in a family of quadratic twists of a fixed genus character L-function. We obtain explicit lower order terms describing the vertical distribution of the zeros, and realize the influence of the Landau-Siegel zero as a resonance phenomenon.
Let $k$ be a field of characteristic not equal to $2,3$, $\mathbb{O}$ an octonion over $k$ and $\mathcal{J}$ the exceptional Jordan algebra defined by $\mathbb{O}$. We consider the prehomogeneous vector space $(G,V)$ where $G=GE_6\times \mathrm{GL}(2)$ and $V=\mathcal{J}\oplus \mathcal{J}$. We prove that generic rational orbits of this prehomogeneous vector space are in bijective correspondence with $k$-isomorphism classes of pairs $(\mathcal{M},\mathfrak{n})$ where $\mathcal{M}$'s are isotopes of $\mathcal{J}$ and $\mathfrak{n}$'s are cubic étale subalgebras of $\mathcal{M}$. Also we prove th…
The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals …
Let $d\ge 2$ be an integer, let $c(t)$ be any rational map, and let $f_t(z) := (z^d+t)/z$ be a family of rational maps indexed by t. For each algebraic number $t$, we let $h_{f_t}(c(t))$ be the canonical height of $c(t)$ with respect to the rational map $f_t$. We prove that the map $H(t):=h_{f_t}(c(t))$ (as $t$ varies among the algebraic numbers) is a Weil height.
A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic …
We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively onto Y(F_q) if and only if f maps X(F_q) injectively into Y(F_q). Surprisingly, the bounds on q for these two implications have different orders of magnitude. The main tools used in our proof are the Chebotarev density theorem for covers of curves over finite fields, the Castelnuovo genus inequality, and ideas from Galois theory.
We use the method of quadratic Chabauty on the quotients $X_0^+(N)$ of modular curves $X_0(N)$ by their Fricke involutions to provably compute all the rational points of these curves for prime levels $N$ of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels $137$ and $311$. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves $X_0^+(N)$ for $N=137,173,199,251,311,157,181,227,263,163,197,211,223,269,271,359$.
In this note we shall study the Witten multiple zeta function associated to the exceptional Lie algebra g_2. Our main result shows that its special values at nonnegative integers can always be expressed as rational linear combinations of the multi-polylogs evaluated at 12th roots of unity, except for two irregular cases.
In this paper we consider Diophantine equation x4 + y4 = z4 + w4 (1)We construct some family of cubic curves.We prove that every rational point on Quar- tica x4 + y4 = z4 + w4 can be mapped to a point on some curve of this family. We also prove the opposite: each rational point belonging to our family of curves can be mapped to a rational point on the Quartica. (2) We find the point on our family of curves corresponding to a parametric solution of Leonard Euler. We construct several new parametric solutions of our Quartica, using a parametric solution of Leonard Euler and the algebraic operati…
In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of linear and Dickson polynomials, was proved by M. Fried in 1970. Here we investigate the analogous question for rational functions, also we allow the base field to be any number field. As a result, there are many more rational functions for which the analogous property holds. Some infinite series come from rational isogenies of elliptic curves and deformations…
Manin's Conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin's Conjecture is a thin set.
We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. T…
We construct a concrete example of a 1-parameter family of smooth projective geometrically integral varieties over an open subscheme of P^1_Q such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.
We consider the problem of counting the number of varieties in a family over a number field which contain a rational point. In particular, for products of Brauer-Severi varieties and closely related counting functions associated to Brauer group elements. Using harmonic analysis on toric varieties, we provide a positive answer to a question of Serre on such counting functions in some cases. We also formulate some conjectures on generalisations of Serre's problem.
For rational numbers $c$, we present a trichotomy of the set of totally real (totally $p$-adic, respectively) preperiodic points for maps in the quadratic unicritical family $f_c(x)=x^2+c$. As a consequence, we classify quadratic polynomials $f_c$ with rational parameters $c\in\mathbb{Q}$ so that $f_c$ has only finitely many totally real (totally $p$-adic, respectively) preperiodic points. These results rely on an adelic Fekete-type theorem and dynamics of the filled Julia set of $f_c$. Moreover, using a numerical criterion introduced in [NP], we make explicit calculations of the set of totall…
A powerful new perspective in the analysis of absolute Galois groups has recently emerged from the study of Galois modules related to classical parameterizing spaces of certain Galois extensions. The recurring trend in these decompositions is their stunning simplicity: almost all summands are free over some quotient ring. The non-free summands which appear are exceptional not only because they are different in form, but because they play the key role in controlling arithmetic conditions that allow the remaining summands to be easily described. In this way, these exceptional summands are the ly…
