Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and consider an E-compatible system of lisse sheaves on the curve X. For each place lambda of E not lying over p, the lambda-component of the system is a lisse E_lambda-sheaf on X, whose associated arithmetic monodromy group is an algebraic group over the local field E_lambda. We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the E-compatible system is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of lambda''. More precisely: after replacing E by a finite extension, there exists a connected split reductive algebraic group G_0 over the number field E such that for every place lambda of E not lying over p, the identity component of the arithmetic monodromy group of the lambda-component of the system is isomorphic to the group G_0 with coefficients extended to the local field E_lambda.
For a family of varieties, we prove that the alternating sum of the traces of "local" monodromy acting on the $\ell$-adic étale cohomology groups of the generic fiber is an integer which is independent of $\ell$.
Using the sieve for Frobenius, we show that, in a certain sense, the roots of the L-functions of "most" algebraic curves over finite fields do not satisfy any non-trivial (linear or multiplicative) rational dependency relations. This can be seen as an analogue of conjectures of linear independence among ordinates of zeros of L-functions over number fields. As a corollary, we find, for "most" pairs of distinct algebraic curves over a finite field, the limiting distribution of the (suitably normalized) difference between the number of rational points over extensions of the ground field. The meth…
In this paper we develop a general method to prove independence of algebraic monodromy groups in compatible systems of representations, and we apply it to deduce independence results for compatible systems both in automorphic and in positive characteristic settings. In the abstract case, we prove an independence result for compatible systems of Lie-irreducible representations, from which we deduce an independence result for compatible systems admitting what we call a Lie-irreducible decomposition. In the case of geometric compatible systems of Galois representations arising from certain classe…
We provide an effective version of Katz' criterion for finiteness of the monodromy group of a lisse, pure of weight zero, $\ell$-adic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf
We compute the $\integ/\ell$ and $\integ_\ell$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the $\integ/\ell$ monodromy of the moduli space of hyperelliptic curves of genus $g$ is the symplectic group $\sp_{2g}(\integ/\ell)$. We prove that the $\integ/\ell$ monodromy of the moduli space of trielliptic curves with signature $(r,s)$ is the special unitary group $\su_{(r,s)}(\integ/\ell\tensor\integ[ζ_3])$.
Let k be a field not of characteristic two and L be a set of almost all rational primes invertible in k. Suppose we have a variety X/k and strictly compatible system {M_ell -> X : ell in L} of constructible F_ell-sheaves. If the system is orthogonally or symplectically self-dual, then the geometric monodromy group of M_ell is a subgroup of a corresponding isometry group G_ell over F_ell, and we say it has big monodromy if it contains the derived subgroup DG_ell=[G_ell,G_ell]. We prove a theorem which gives sufficient conditions for M_ell to have big monodromy. We apply the theorem to explicit …
Let $E$ be an abelian scheme over a geometrically connected variety $X$ defined over $k$, a finitely generated field over $\mathbb{Q}$. Let $η$ be the generic point of $X$ and $x\in X$ a closed point. If $\mathfrak{g}_l$ and $(\mathfrak{g}_l)_x$ are the Lie algebras of the $l$-adic Galois representations for abelian varieties $E_η$ and $E_x$, then $(\mathfrak{g}_l)_x$ is embedded in $\mathfrak{g}_l$ by specialization. We prove that the set $\{x\in X$ closed point $| (\mathfrak{g}_l)_x\subsetneq \mathfrak{g}_l\}$ is independent of $l$ and confirm Conjecture 5.5 in [2].
Let $X$ be a smooth connected algebraic curve over an algebraically closed field $k$. We study the deformation of $\ell$-adic Galois representations of the function field of $X$ while keeping the local Galois representations at all places undeformed.
We compute the Z/\ell and \ell-adic monodromy of every irreducible component of the moduli space M_g^f of curves of genus and and p-rank f. In particular, we prove that the Z/\ell-monodromy of every component of M_g^f is the symplectic group Sp_{2g}(Z/\ell) if g>=3 and \ell is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.
For any simple algebraic group $G$ of exceptional type, we construct geometric $\ell$-adic Galois representations with algebraic monodromy group equal to $G$, in particular producing the first such examples in types $\mathrm{F}_4$ and $\mathrm{E}_6$. To do this, we extend to general reductive groups Ravi Ramakrishna's techniques for lifting odd two-dimensional Galois representations to geometric $\ell$-adic representations.
Given a complex quasiprojective curve $B$ and a non-isotrivial family $\mathcal{E}$ of elliptic curves over $B$, the $p$-torsion $\mathcal{E}[p]$ yields a monodromy representation $ρ_\mathcal{E}[p]:π_1(B)\rightarrow \mathrm{GL}_2(\mathbb{F}_p)$. We prove that if $ρ_{\mathcal E}[p]\cong ρ_{\mathcal E'}[p]$ then $\mathcal{E}$ and $\mathcal E'$ are isogenous, provided $p$ is larger than a constant depending only on the gonality of $B$. This can be viewed as a function field analog of the Frey--Mazur conjecture, which states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by i…
We study l-independence of monodromy groups G_l of any compatible system of l-adic representations (in the sense of Serre) of number field K assuming semisimplicity. We prove that the formal character of the derived group of the identity component of G_l is independent of l and the (complexified) Lie algebra g_l of G_l satisfies an equal-rank subalgebra equivalence for all l. This equivalence is equivalent to the l-independence of the number of A_n factors for all n belonging to {6,9,10,11,...} and the parity of A_4 factors in g_l.
Let K be a number field and {V_l} be a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let G_l and V_l^ab be respectively the algebraic monodromy group and the maximal abelian subrepresentation of V_l for all l. We prove that the system {V_l^ab} is also a rational strictly compatible system under some group theoretic conditions, e.g., when G_l' is connected and satisfies Hypothesis A for some prime l'. As an application, we prove that the Tate conjecture for abelian variety X/K is independent of l if the algebraic monodromy groups of the Gal…
Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli…
Let $f(z)$ be in $1+z\mathbb{Q}[[z]]$ and $\mathcal{S}$ be an infinite set of prime numbers such that, for all $p\in\mathcal{S}$, we can reduce $f(z)$ modulo $p$. We let $f(z)_{\mid p}$ denote the reduction of $f(z)$ modulo $p$. Generally, when $f(z)$ is D-finite, $f(z)_{\mid p}$ is algebraic over $\mathbb{F}_p(z)$. It turns out that if $f(z)$ is a solution of a polynomial of the form $X-A_p(z)X^{p^l}$, we can use this type of equations to obtain results of transcendence and algebraic independence over $\mathbb{Q}(z)$. In the present paper, we look for conditions on the differential operators …
This paper is complementary to the work Rosen-Silverman, which derives a criteria on the number fields for the independence of Heegner points associated to them on non-CM elliptic curves. We show that the same criteria holds for CM elliptic curves. A generalisation of the independence criteria for the Heegner points associated to an order of fixed conductor of quadratic imaginary fields can also be found in this paper.
We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "…
We first show joint uniform distribution of values of Kloosterman sums or Birch sums among all extensions of a finite field $\mathbb{F}_q$, for almost all couples of arguments in $\mathbb{F}_q^\times$, as well as lower bounds on differences. Using similar ideas, we then study the biases in the distribution of generalized angles of Gaussian primes over function fields and primes in short intervals over function fields, following recent works of Rudnick--Waxman and Keating--Rudnick, building on cohomological interpretations and determinations of monodromy groups by Katz. Our results are based on…
Let $X$ be a connected scheme, smooth and separated over an algebraically closed field $k$ of characteristic $p\geq 0$, let $f:Y\rightarrow X$ be a smooth proper morphism and $x$ a geometric point on $X$. We prove that the tensor invariants of bounded length $\leq d$ of $π_1(X,x)$ acting on the étale cohomology groups $H^*(Y_x,F_\ell)$ are the reduction modulo-$\ell$ of those of $π_1(X,x)$ acting on $H^*(Y_x,Z_\ell)$ for $\ell$ greater than a constant depending only on $f:Y\rightarrow X$, $d$. We apply this result to show that the geometric variant with $F_\ell$-coefficients of the Grothendiec…
Slopes and weights of $\ell$-adic cohomology of rigid spaces
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；University of the Chinese Academy of Sciences（中国科学院大学）
We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\ge 2$ on $\mathbb{G}_m/\mathbb{F}_q$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz's results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_q)$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz's ideas, in particular the classification of maxim…
Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ …
We prove a result which describes, for each $n\ge 1$, all linear dependencies among $n$ images in elliptic curves of special points in modular or Shimura curves under parameterizations (or correspondences). Our result unifies and improves in certain aspects previous work of Rosen-Silverman--Kühne and Buium-Poonen.
We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a functional equation and discriminant considerations. As an application, we consider the Frobenius polynomials arising from the middle étale cohomology of hypersurfaces in $\mathbb{P}_{\mathbb{F}_q}^{2n+1}$ of degree at least $3$. We also consider the $L$-functions of quadratic twists of fixed degree of an elliptic curve over a function field $\mathbb{F}_q(t)$. To de…
Let $K$ be a global field of characteristic different from 2 and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group over $K$ is either the full symmetric group $S_{2g}$ or the alternating group $A_{2g}$. We describe explicitly how to choose (infinitely many) pairs of distinct elements $t_1, t_2$ of $K$ such that the $g$-dimensional jacobian of a hyperelliptic curve $y^2=(x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms over an algebraic closure of $K$ and has big $\ell$-adic monodromy.
Let $X$ be a complete smooth variety defined over number field $K$ and $i$ an integer. The absolute Galois group of $K$ acts on the $i$th $l$-adic etale cohomology of $X$ for all $l$, producing a system of $l$-adic representations $\{Φ_l\}$. The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of $Φ_\ell$ admits a common reductive $Q$-form for all $l$ if $X$ is projective. Denote by $Γ_l$ and $G_l$ respectively the monodromy group and the algebraic monodromy group of $Φ_l^{ss}$, the semisimplification of $Φ_\ell$. Assuming…
Let E/Q be an elliptic curve with a fixed modular parametrization F : X_0(N) --> E and let P_1,...,P_r be Heegner points on E attached to the rings of integers of distinct quadratic imaginary field k_1,...,k_r. We prove that if the odd parts of the class numbers of k_1,...,k_r are larger than a constant C=C(E,F) depending only on E and F, then the points P_1,...,P_r are independent in E/(torsion). We also discuss a possible application to the elliptic curve discrete logarithm problem.
The Linear Independence hypothesis (LI), which states roughly that the imaginary parts of the critical zeros of Dirichlet L-functions are linearly independent over the rationals, is known to have interesting consequences in the study of prime number races, as was pointed out by Rubinstein and Sarnak. In this paper, we prove that a function field analogue of LI holds generically within certain families of elliptic curve L-functions and their symmetric powers. More precisely, for certain algebro-geometric families of elliptic curves defined over the function field of a fixed curve over a finite …
Let $F$ be a totally real field and $n\leq 4$ a natural number. We study the monodromy groups of any $n$-dimensional strictly compatible system $\{ρ_λ\}_λ$ of $λ$-adic representations of $F$ with distinct Hodge-Tate numbers such that $ρ_{λ_0}$ is irreducible for some $λ_0$. When $F=\mathbb{Q}$, $n=4$, and $ρ_{λ_0}$ is fully symplectic, the following assertions are obtained. (i) The representation $ρ_λ$ is fully symplectic for almost all $λ$. (ii) If in addition the similitude character $μ_{λ_0}$ of $ρ_{λ_0}$ is odd, then the system $\{ρ_λ\}_λ$ is potentially automorphic and the residual image …
For a lattice Λin the complex plane, let K_Λ be the field of Λ-elliptic functions. For two relatively prime integers p (respectively q) greater than 1, consider the endomorphisms ψ(resp. ϕ) of K_Λ given by multiplication by p (resp. q) on the elliptic curve \mathbb{C}/Λ. We prove that if f (resp. g) are complex Laurent power series that satisfy linear difference equations over K_Λ with respect to ϕ(resp. ψ) then there is a dichotomy. Either, for some sublattice Λ' of Λ, one of f or g belongs to the ring K_{Λ'}[z,z^{-1},ζ(z,Λ')], where ζ(z,Λ') is the Weierstrass zeta function, or f and g are al…
Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the group…
Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the ell-adic etale cohomology modules of X. We prove that this family of representations varying over ell is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ell become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting f…
We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(π)$ denote an $n$-dimensional $p$-adic representation of the Galois group of a CM field $F$ attached to a regular algebraic cuspidal automorphic representation $π$ of $GL_n(\mathbb{A}_F)$. We show that the restriction of $r_p(π)$ to the decomposition group of a place $v\nmid p$ of $F$ corresponds up to semisimplification to $rec(π_v)$, the image of $π_v$ under the local Langlands…
The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.
We study the de Rham fundamental group of the configuration space $E^{(n)}$ of $n+1$ marked points on an elliptic curve $E$, and define multiple elliptic polylogarithms. These are multivalued functions on $E^{(n)}$ with unipotent monodromy, and are constructed by a general averaging procedure. We show that all iterated integrals on $E^{(n)}$, and in particular the periods of the unipotent fundamental group of the punctured curve $E \backslash \{0\}$, can be expressed in terms of these functions.
We partially answer, in terms of monodromy, Murty and Patankar's question: Given an absolutely simple abelian variety over a number field, does it have simple specializations at a set of places of positive Dirichlet density? The answer is based on the classification of pairs (G,V) consisting of a semi-simple algebraic group G over a non-archimedean local field and an absolutely irreducible representation V of G such that G admits a maximal torus acting irreducibly on V.
It is an open question of Baker whether the numbers $L(1, χ)$ for non-trivial Dirichlet characters $χ$ with period $q$ are linearly independent over $\mathbb{Q}$. The best known result is due to Baker, Birch and Wirsing which affirms this when $q$ is co-prime to $φ(q)$. In this article, we extend their result to any arbitrary family of moduli. More precisely, for a positive integer $q$, let $X_q$ denote the set of all $L(1,χ)$ values as $χ$ varies over non-trivial Dirichlet characters with period $q$. Then for any finite set of pairwise co-prime natural numbers $q_i, 1\le i \le \ell$ with $(q_…
Let $\{ρ_\ell\}_\ell$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a complete smooth variety $X$ defined over a number field $K$. Let $Γ_\ell$ and $\mathbf{G}_\ell$ be respectively the image and the algebraic monodromy group of $ρ_\ell$. We prove that the reductive quotient of $\mathbf{G}_\ell^\circ$ is unramified over every degree 12 totally ramified extension of $\mathbb{Q}_\ell$ for all sufficiently large $\ell$. We give a necessary and sufficient condition $(\ast)$ on $\{ρ_\ell\}_\ell$ such that for all sufficiently large $\ell$, the subgrou…
