Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the Neron-Severi torus. Let
$mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $mathcal{T}$ over $S_L$. We prove that $mathcal{G}$ does not descend to S unless $mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $mathcal{G}$ always descend to singular del Pezzo surfaces over $mathbb{C}$ from their desingularizations.
We give an effective proof of Faltings theorem for curves mapping to Hilbert modular stacks over odd-degree totally real fields. We do this by giving an effective proof of the Shafarevich conjecture for abelian varieties of $mathrm{GL}_2$-type over a
n odd-degree totally real field. We deduce for example an effective height bound for $K$-points on the curves $C_a : x^6 + 4y^3 = a^2$ ($ain K^times$) when $K$ is odd-degree totally real. (Over $overline{mathbb{Q}}$ all hyperbolic hyperelliptic curves admit an {e}tale cover dominating $C_1$.)
We show how the minimal free resolution of a set of $n$ points in general position in projective space of dimension $n-2$ explicitly determines structure constants for a ring of rank $n$. This generalises previously known constructions of Levi-Delone-Faddeev and Bhargava in the cases $n=3,4,5$.
Let $G$ be a connected reductive group over the non-archime-dean local field $F$ and let $pi$ be a supercuspidal representation of $G(F)$. The local Langlands conjecture posits that to such a $pi$ can be attached a parameter $L(pi)$, which is an equi
valence class of homomorphisms from the Weil-Deligne group with values in the Langlands $L$-group ${}^LG$ over an appropriate algebraically closed field $C$ of characteristic $0$. When $F$ is of positive characteristic $p$ then Genestier and Lafforgue have defined a parameter, $L^{ss}(pi)$, which is a homomorphism $W_F ra {}^LG(C)$ that is {it semisimple} in the sense that, if the image of $L^{ss}(pi)$, intersected with the Langlands dual group $hat{G}(C)$, is contained in a parabolic subgroup $P subset hat{G}(C)$, then it is contained in a Levi subgroup of $P$. If the Frobenius eigenvalues of $L^{ss}(pi)$ are pure in an appropriate sense, then the local Langlands conjecture asserts that the image of $L^{ss}(pi)$ is in fact {it irreducible} -- its image is contained in no proper parabolic $P$. In particular, unless $G = GL(1)$, $L^{ss}(pi)$ is ramified: it is non-trivial on the inertia subgroup $I_F subset W_F$. In this paper we prove, at least when $G$ is split and semisimple, that this is the case provided $pi$ can be obtained as the induction of a representation of a compact open subgroup $U subset G(F)$, and provided the constant field of $F$ is of order greater than $3$. Conjecturally every $pi$ is compactly induced in this sense, and the property was recently proved by Fintzen to be true as long as $p$ does not divide the order of the Weyl group of $G$. The proof is an adaptation of the globalization method of cite{GLo} when the base curve is $PP^1$, and a simple application of Delignes Weil II.
Let $f : X to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $mathbb{V} = R^{2k} f_{*} mathbb{Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology i
t induces. Associated to $mathbb{V}$ one has the so-called Hodge locus $textrm{HL}(S) subset S$, which is a countable union of special algebraic subvarieties of $S$ parametrizing those fibres of $mathbb{V}$ possessing extra Hodge tensors (and so conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S subset overline{S}$ and very ample line bundle $mathcal{L}$ on $overline{S}$). As a corollary of our algorithm we prove conjectures of Daw-Ren and Daw-Javanpeykar-Kuhne on the finiteness of sets of special and weakly special subvarieties of bounded degree.
We carry out Hecke summation for the classical Eisenstein series $E_k$ in an adelic setting. The connection between classical and adelic functions is made by explicit calculations of local and global intertwining operators and Whittaker functions. In
the process we determine the automorphic representations generated by the $E_k$, in particular for $k=2$, where the representation is neither a pure tensor nor has finite length. We also consider Eisenstein series of weight $2$ with level, and Eisenstein series with character.
Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees the vector
of coefficients of the polynomial as a word on a ternary alphabet ${-1,0 ,+1}$. It designs an efficient algorithm that computes a compact representation of this word. This algorithm is of linear time with respect to the size of the output, and, thus, optimal. This approach allows to recover known properties of coefficients of binary cyclotomic polynomials, and extends to the case of polynomials associated with numerical semi-groups of dimension 2.
Let $W$ be a smooth test function with compact support in $(0,infty)$. Conditional on the Generalized Riemann Hypothesis for Hecke $L$-functions over $mathbb{Q}(omega)$, we prove that $$sum_{p equiv 1 pmod{3}} frac{1}{2 sqrt{p}} cdot Big ( sum_{x pmo
d{p}} e^{2pi i x^3 / p} Big ) W Big ( frac{p}{X} Big ) sim frac{(2pi)^{2/3}}{3 Gamma(tfrac 23)} int_{0}^{infty} W(x) x^{-1/6} dx cdot frac{X^{5/6}}{log X},$$ as $X rightarrow infty$ and $p$ runs over primes. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846 and confirms (conditionally on the Generalized Riemann Hypothesis) a conjecture of Patterson from 1978. There are two important byproducts of our proof. The first is an explicit level aspect Voronoi summation formula for cubic Gauss sums, extending computations of Patterson and Yoshimoto. Secondly, we show that Heath-Browns cubic large sieve is sharp up to factors of $X^{o(1)}$ under the Generalized Riemann Hypothesis. This disproves the popular belief that the cubic large sieve can be improved. An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term. This estimate relies on the Generalized Riemann Hypothesis, and is one of the fundamental reasons why our result is conditional.
Let $E$ be an elliptic curve defined over a number field $F$ with good ordinary reduction at all primes above $p$, and let $F_infty$ be a finitely ramified uniform pro-$p$ extension of $F$ containing the cyclotomic $mathbb{Z}_p$-extension $F_{cyc}$.
Set $F^{(n)}$ be the $n$-th layer of the tower, and $F^{(n)}_{cyc}$ the cyclotomic $mathbb{Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $lambda$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $nrightarrow infty$. This method has its origins in work of A.Cuoco, who studied $mathbb{Z}_p^2$-extensions. Refined estimates for growth are proved that are close to conjectured estimates. The results are illustrated in special cases.
For distinct odd primes $p$ and $q$, we define the Catalan curve $C_{p,q}$ by the affine equation $y^q=x^p-1$. In this article we construct the Sato-Tate groups of the Jacobians in order to study the limiting distributions of coefficients of their no
rmalized L-polynomials.Catalan Jacobians are nondegenerate and simple with noncyclic Galois groups (of the endomorphism fields over $mathbb Q$), thus making them interesting varieties to study in the context of Sato-Tate groups. We compute both statistical and numerical moments for the limiting distributions. Lastly, we determine the Galois endomorphism types of the Jacobians using both old and new techniques.
