Let $A$ be a principally polarized abelian variety of dimension $g$ over a number field $K$. Assume that the image of the adelic Galois representation of $A$ is an open subgroup of $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. Then there exists a positive integer $m$ so that the Galois image of $A$ is the full preimage of its reduction modulo $m$. The least $m$ with this property, denoted $m_A$, is called the image conductor (also called the level) of $A$. Jones recently established an upper bound for $m_A$, in terms of standard invariants of $A$, in the case that $A$ is an elliptic curve without complex multiplication. In this paper, we generalize the aforementioned result to provide an analogous bound in arbitrary dimension.
Given a rational variety $V$ defined over $K$, we consider a principally polarized abelian variety $A$ of dimension $g$ defined over $V$. For each prime l we then consider the galois representation on the $l$-torsion of $A_t$, where $t$ is a $K$-rational point of $V$. The largest possible image is $GSp_{2g}(\mathbb{F}_l)$ and in the cases $g=1$ and 2, we are able to get surjectivity for all $l$ and almost all $t$. In the case $g=1$ this recovers a theorem originally proven by William Duke.
In this paper, we use a series of targeted Hunter searches to prove that the minimal prime conductor of an even icosahedral Galois representation is 1951. In addition, we give a complete list of all even icosahedral Galois representations of prime conductor less than 10,000.
Let F denote an unramified extension of the cyclotomic extension of Q_p by (p^n)th roots of unity, for an odd prime p. We determine the conductors of those Kummer extensions of F of degree dividing p^n which are Galois over the maximal unramified subfield of F. This follows from a comparison of the Galois module structure of the multiplicative group of F with the unit filtration of F.
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and get much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.
For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.
Let $\overlineρ: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the $3$-torsion of a principally polarized abelian surface $A/\mathbf{Q}$. We prove that the moduli space $\mathcal{A}_2(\overlineρ)$ of principally polarized abelian surfaces $B/\mathbf{Q}$ admitting a symplectic isomorphism $B[3] \simeq \overlineρ$ of Galois representations is never rational over $\mathbf{Q}$ when $\overlineρ$ is surjective, even though it is both ratio…
Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$. Let $σ$ be an irreducible smooth representation of the absolute Weil group $\Cal W_F$ of $F$ and $\sw(σ)$ the Swan exponent of $σ$. Assume $\sw(σ) \ge1$. Let $\Cal I_F$ be the inertia subgroup of $\Cal W_F$ and $\Cal P_F$ the wild inertia subgroup. There is an essentially unique, finite, cyclic group $\varSigma$, of order prime to $p$, so that $σ(\Cal I_F) = σ(\Cal P_F)\varSigma$. In response to a query of Mark Reeder, we show that the multiplicity in $σ$ of any character of $\varSigma$ is bounded by $\sw(σ)$.
Using Shimura's reciprocity law, we calculate the Galois action on the special values of theta functions associated to primitive Dirichlet characters of odd conductor, normalised by the Dedekind eta function, at the point i. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime conductor.
Given an elliptic curve $E$ without complex multiplication defined over a number field $K$, consider the image of the Galois representation defined by letting Galois act on the torsion of $E$. Serre's open image theorem implies that there is a positive integer $m$ for which the Galois image is completely determined by its reduction modulo $m$. In this note, we prove a bound on the smallest such $m$ in terms of standard invariants associated with $E$. The bound is sharp and improves upon previous results.
Up to isomorphism over C, every simple principally polarized abelian variety of dimension 3 is the Jacobian of a smooth projective curve of genus 3. Furthermore, this curve is either a hyperelliptic curve or a plane quartic. Given a sextic CM field K, we show that if there exists a hyperelliptic Jacobian with CM by K, then all principally polarized abelian varieties that are Galois conjugated to it are hyperelliptic. Using Shimura's reciprocity law, we give an algorithm for computing approximations of the invariants of the initial curve, as well as their Galois conjugates. This allows us ton d…
Let $k$ be a complete discretely valued field of equal characteristic $p > 0$ with possibly imperfect residue field and let $G_k$ be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on $G_k$ coincide with the differential Artin conductors and Swan conductors of Galois representations of $G_k$. As a consequence, we give a Hasse-Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse-Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors a…
In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre weight $k$ satisfies $l\textgreater{}k+1$, we give a precise characterization of strongly modular representations, hence generalizing a classical theorem of Ribet pertaining to the case of conductor $1$.When the representation $ρ$ is not strongly modular, we give a necessary and sufficient condition on the primes $p$ not dividing $Nl$ for which it arises in level…
Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting \ell-torsion representation ρ_{A,\ell} from G_F to GSp(A[\ell](\bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(\mathbb{F}_\ell) as the Galois group of a finite tame extension of number fields F'/F such that F is unramified above \ell.
We define a numerical invariant, the differential Swan conductor, for certain differential modules on a rigid analytic annulus over a p-adic field. This gives a definition of a conductor for p-adic Galois representations with finite local monodromy over an equal characteristic discretely valued field, which agrees with the usual Swan conductor when the residue field is perfect. We also establish analogues of some key properties of the usual Swan conductor, such as integrality (the Hasse-Arf theorem), and the fact that the graded pieces of the associated ramification filtration on Galois groups…
We study the arithmetic of division fields of semistable abelian varieties A over the rationals. The Galois group of the 2-division field of A is analyzed when the conductor is odd and squarefree. The irreducible semistable mod 2 representations of small conductor are determined under GRH. These results are used in "Paramodular abelian varieties of odd conductor," arXiv:1004.4699.
Given a non-CM elliptic curve E over Q, define the ``torsion conductor'' m_E to be the smallest positive integer so that the Galois representation on the torsion of E has image Pi^{-1}(Gal(Q(E[m_E])/Q), where Pi denotes the natural projection GL_2(\hat{Z}) onto GL_2(Z/m_E Z). We show that, uniformly for semi-stable non-CM elliptic curves E over Q, m_E is less than a constant times the 5th power of the conductor of E.
Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two general results: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any information about a Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space.
We study quotients of principally polarized abelian varieties with real multiplication by Galois-stable finite subgroups and describe when these quotients are principally polarizable. We use this characterization to provide an algorithm to compute explicit cyclic isogenies from kernel for abelian varieties with real multiplication over finite fields. Our algorithm is polynomial in the size of the finite field as well as the degree of the isogeny and is based on Mumford's theory of theta functions and theta embeddings. Recently, the algorithm has been successfully applied to obtain new results …
We prove the following theorem: Let $\bar\F_p$ be an algebraic closure of a finite field of characteristic $p$. Let $ρ$ be a continuous homomorphism from the absolute Galois group of $\Q$ to $\GL(3,\bar\F_p)$ which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. Under the condition that the conductor of $ρ$ is squarefree, we prove that $ρ$ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup $Γ$ of $\GL(3,\Z)$. In addition, we prove that the coefficient module needed is, in fact, predicted by a conjecture of Ash, Doud, P…
Let A_2 be the moduli stack of principally polarized abelian surfaces and V a smooth l-adic sheaf on A_2 associated to an irreducible rational finite dimensional representation of Sp(4). We give an explicit expression for the cohomology of V in any degree in terms of Tate type classes and Galois representations attached to elliptic and Siegel cusp forms. This confirms a conjecture of Faber and van der Geer. As an application we prove a dimension formula for vector-valued Siegel cusp forms for Sp(4,Z) of weight three, which had been conjectured by Ibukiyama.
Let F be a real quadratic field, and let R be an order in F. Suppose given a polarized abelian surface (A,λ) defined over a number field k with a symmetric action of R defined over k. This paper considers varying A within the k-isogeny class of A to reduce the degree of λand the conductor of R. It is proved, in particular, that there is a k-isogenous principally polarized abelian surface with an action of the full ring of integers of F when F has class number 1 and the degree of λand the conductor of R are odd and coprime.
Let A be a complete discrete valuation ring with possibly imperfect residue field, and let $χ$ be a one-dimensional Galois representation over A. I show that the non-logarithmic variant of Kato's Swan conductor is the same for $χ$ and the pullback of $χ$ to the generic residual perfection of A. This implies the conductor from "Conductors and the moduli of residual perfection" (math.NT/0112305) extends the non-logarithmic variant of Kato's.
We describe a procedure for determining the existence, or non-existence, of an algebraic variety of a given conductor via an analytic calculation involving L-functions. The procedure assumes that the Hasse-Weil L-function of the variety satisfies its conjectured functional equation, but there is no assumption of an associated automorphic object or Galois representation. We demonstrate the method by finding the Hasse-Weil L-functions of all hyperelliptic curves of conductor less than 500.
Using Galois Theory, we construct explicitly absolutely simple (principally polarized) Prym varieties that are not isomorphic to jacobians of curves even if we ignore the polarizations. Our approach is based on the previous papers math/0610138 [math.AG] and math/0605028 [math.AG] .
Let $χ$ denote a primitive, non-quadratic Dirichlet character with conductor $q$, and let $L(s, χ)$ denote its associated Dirichlet $L$-function. We show that $|L(1, χ)| \geq 1/(9.12255 \log(q/π))$ for sufficiently large $q$, and that $|L(1, χ)| \geq 1/(9.69030 \log(q/π))$ for all $q\geq2$, improving some results of Louboutin. The improvements stem principally from the construction, via simulated annealing, of some real trigonometric polynomials having particularly favorable properties.
The goal of this article is to give an explicit classification of the possible $p$-adic Galois representations that are attached to elliptic curves $E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by $\mathcal{O}_{K,f}$, such that $E$ is defined by a model over $\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the absolute Galois group of $\mathbb{Q}(j(E))$, and let $ρ_{E,p^\infty}\colon G_{\mathbb{Q}(j(E))}\t…
We prove that there is no geometric $p$-adic representation of the Galois group of $\Q$ which is irreducible, of dimension 2, of conductor 1 and low weight, according to a conjecture of Fontaine and Mazur.
The refined Swan conductor is defined by K.\ Kato \cite{KK2}, and generalized by T.\ Saito \cite{wild}. In this part, we consider some smooth $l$-adic étale sheaves of rank $p$ such that we can be define the $rsw$ following T.\ Saito, on some smooth dense open subscheme $U$ of a smooth separated scheme X of finite type over a perfect fields $κ$ of characteristic $p>0$. We give an explicit expression of $rsw(\mathcal{F})$ in some situation. As a consequence, we show that it is integral.
A finiteness theorem for mod $p$ Galois representations over global function fields
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）
Let E be an elliptic curved defined over $\Q$ and let $K/\Q$ be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for $h^1(E\times_{\Q} K)(1)$ viewed as a motive over $\Q$ with coefficients in $\Q[G]$ relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper we prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S_3-extension of $\Q$.
Let E be an elliptic curve without complex multiplication (CM) over a number field K, and let G_E(ell) be the image of the Galois representation induced by the action of the absolute Galois group of K on the ell-torsion subgroup of E. We present two probabilistic algorithms to simultaneously determine G_E(ell) up to local conjugacy for all primes ell by sampling images of Frobenius elements; one is of Las Vegas type and the other is a Monte Carlo algorithm. They determine G_E(ell) up to one of at most two isomorphic conjugacy classes of subgroups of GL_2(Z/ell Z) that have the same semisimplif…
Let k be a complete discrete valuation field of equal characteristic p>0. Using the tools of p-adic differential modules, we define refined Artin and Swan conductors for a representation of the absolute Galois group $G_k$ with finite local monodromy; this leads to a description of the subquotients of the ramification filtration on $G_k$. We prove that our definition of the refined Swan conductors coincide with that is given by Saito, which uses etale cohomology. We also study its relation with the toroidal variation of the Swan conductors.
Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$ without complex multiplication and with conductor $N$. For each positive integer $m$, the action of the absolute Galois group $G_{\mathbb{Q}}=\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on the torsion points over $\bar{\mathbb{Q}}$ gives rise to a representation of $G_\mathbb{Q}$. A celebrated paper of Serre shows that this representation is surjective for all sufficiently large primes; the other primes are termed \emph{exceptional}. Serre conjectures that there are no exceptional primes $\ell>37$ for any non CM elliptic curv…
Let $2^+S_4$ denote the double cover of $S_4$ corresponding to the element in $H^2(S_4,\Z/2\Z)$ where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4 (denoted $\tilde S_4$ in \cite{Serre}). Given an elliptic curve $E$, let $E[2]$ denote its 2-torsion points. Under some conditions on $E$ (as in \cite{Bayer}) elements in $H^1(\Gal_\Q,E[2])\backslash \{0 \}$ correspond to Galois extensions $N$ of $\Q$ with Galois group (isomorphic to) $S_4$. On this work we give an interpretation of the addition law on such fields, and prove that th…
Let $A$ be a semistable abelian variety defined over ${\bf Q}$ with bad reduction only at one prime $p$. Let $L= {\bf Q}(A[\ell])$ be the $\ell$-division field of $A$ for a prime $\ell$ not equal to $p$ and let $F={\bf Q}(μ_\ell)$ be the cyclotomic field generated by the group of $\ell^{th}$-roots of unity. We study the varieties $A$ for which $H={\rm Gal(L/F)}$ is "small" in the sense that $H$ is an $\ell$-group or, more generally, that $H$ is nilpotent. We show that if $\ell=2$ or 3 and $H$ is nilpotent then the reduction of $A$ at $p$ is totally toroidal, so its conductor is $p^{\dim A}$. T…
Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let $\mathcal E$ be an absolutely irreducible group scheme of rank $p^4$ over $\mathbb Z_p$. We provide a complete description of the Honda systems of $p$-divisible groups $\mathcal G$ such that $\mathcal G[p^{n+1}]/\mathcal G[p^n] \simeq \mathcal E$ for all $n$. Then we find a bound for the abelian conductor of the second layer $\mathbb Q_p(\mathcal G[p^2])/\mathbb Q_p(\mathcal G[p])$, stronger in our case than can be deduced from Fontaine's bound. Let $π\!: \, {\rm Sp}_{2g}(\mathbb Z_p) \t…
Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the ring class field of K of conductor c prime to ND with Galois group G over K. Fix a complex character χof G. Our main result is that if the special value of the χ-twisted L-function of E/K is non-zero then the tensor product (with respect to χ) of the p-Selmer group of E/H with W over Z[G] is 0 for all but finitely many primes p, where W is a suitable finite ext…
