Denominators of Igusa class polynomials

Kristin Lauter; Bianca Viray

Displaying similar documents to “Denominators of Igusa class polynomials”

On the error term of the logarithm of the lcm of a quadratic sequence

Juanjo Rué, Paulius Šarka, Ana Zumalacárregui (2013)

Journal de Théorie des Nombres de Bordeaux

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We study the logarithm of the least common multiple of the sequence of integers given by $1^{2} + 1, 2^{2} + 1, \dots, n^{2} + 1$ . Using a result of Homma [] on the distribution of roots of quadratic polynomials modulo primes we calculate the error term for the asymptotics obtained by Cilleruelo [].

$L^{2}$ well-posed Cauchy problems and symmetrizability of first order systems

Guy Métivier (2014)

Journal de l’École polytechnique — Mathématiques

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The Cauchy problem for first order system $L (t, x, \partial_{t}, \partial_{x})$ is known to be well-posed in $L^{2}$ when it admits a microlocal symmetrizer $S (t, x, ξ)$ which is smooth in $ξ$ and Lipschitz continuous in $(t, x)$ . This paper contains three main results. First we show that a Lipschitz smoothness globally in $(t, x, ξ)$ is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of having the same smoothness. This notion was first introduced in []. This is the key point to prove...

$L^{2}$ -type contraction for systems of conservation laws

Denis Serre, Alexis F. Vasseur (2014)

Journal de l’École polytechnique — Mathématiques

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The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in $L^{1}$ . However it is not a contraction in $L^{p}$ for any $p > 1$ . Leger showed in [] that for a convex flux, it is however a contraction in $L^{2}$ up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of systems. We treat in details...

Algebraic independence of the generating functions of Stern’s sequence and of its twist

Peter Bundschuh, Keijo Väänänen (2013)

Journal de Théorie des Nombres de Bordeaux

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Very recently, the generating function $A (z)$ of the Stern sequence ${(a_{n})}_{n \geq 0}$ , defined by $a_{0} : = 0, a_{1} : = 1,$ and $a_{2 n} : = a_{n}, a_{2 n + 1} : = a_{n} + a_{n + 1}$ for any integer $n > 0$ , has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A (α)$ for every algebraic $α$ with $0 < | α | < 1$ , and this result was generalized in [6] to the effect that, for the same $α$ ’s, all numbers $A (α), A^{'} (α), A^{''} (α), ...$ are algebraically independent. At about the same time, Bacher...

Hecke operators in half-integral weight

Soma Purkait (2014)

Journal de Théorie des Nombres de Bordeaux

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In [], Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm’s bounds give generators of the Hecke algebra as a module. We also have well-known recursion formulae for the operators $T_{p^{ℓ}}$ with $p$ prime. It is the purpose of this paper to prove analogous results in the half-integral weight setting. We also give an explicit formula for how operators...

Distributions of truncations of the heat kernel on the complex projective space

Nizar Demni (2014)

Annales mathématiques Blaise Pascal

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Let ${(U_{t})}_{t \geq 0}$ be a Brownian motion valued in the complex projective space $ℂ P^{N - 1}$ . Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $| U_{t}^{1} |^{2}$ and of $(| U_{t}^{1} |^{2}, | U_{t}^{2} |^{2})$ , and express them through Jacobi polynomials in the simplices of $ℝ$ and $ℝ^{2}$ respectively. More generally, the distribution of $(| U_{t}^{1} |^{2}, \dots, | U_{t}^{k} |^{2}), 2 \leq k \leq N - 1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $𝒰 (N - k + 1)$ yet computations become tedious. We also revisit the approach initiated in [] and...

The KSBA compactification for the moduli space of degree two $K 3$ pairs

Radu Laza (2016)

Journal of the European Mathematical Society

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Inspired by the ideas of the minimal model program, Shepherd-Barron, Kollár, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs $(X, H)$ consisting of a degree two $K 3$ surface $X$ and an ample divisor $H$ . Specifically, we construct and describe explicitly a geometric compactification ${\bar{P}}_{2}$ for the moduli of degree two $K 3$ pairs. This compactification...

Essential dimension of moduli of curves and other algebraic stacks

Patrick Brosnan, Zinovy Reichstein, Angelo Vistoli (2011)

Journal of the European Mathematical Society

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In this paper we consider questions of the following type. Let $k$ be a base field and $K / k$ be a field extension. Given a geometric object $X$ over a field $K$ (e.g. a smooth curve of genus $g$ ), what is the least transcendence degree of a field of definition of $X$ over the base field $k$ ? In other words, how many independent parameters are needed to define $X$ ? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete...

On $1$ -cocycles induced by a positive definite function on a locally compact abelian group

Jordan Franks, Alain Valette (2014)

Annales mathématiques Blaise Pascal

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For $ϕ$ a normalized positive definite function on a locally compact abelian group $G$ , let $π_{ϕ}$ be the unitary representation associated to $ϕ$ by the GNS construction. We give necessary and sufficient conditions for the vanishing of 1-cohomology $H^{1} (G, π_{ϕ})$ and reduced 1-cohomology ${\bar{H}}^{1} (G, π_{ϕ})$ . For example, ${\bar{H}}^{1} (G, π_{ϕ}) = 0$ if and only if either $Hom (G, ℂ) = 0$ or $μ_{ϕ} (1_{G}) = 0$ , where $1_{G}$ is the trivial character of $G$ and $μ_{ϕ}$ is the probability measure on the Pontryagin dual $\hat{G}$ associated to $ϕ$ by Bochner’s Theorem. This streamlines an argument of Guichardet...

Moduli of smoothness of functions and their derivatives

Z. Ditzian, S. Tikhonov (2007)

Studia Mathematica

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Relations between moduli of smoothness of the derivatives of a function and those of the function itself are investigated. The results are for $L_{p} (T)$ and $L_{p} [- 1, 1]$ for 0 < p < ∞ using the moduli of smoothness $ω^{r} {(f, t)}_{p}$ and $ω_{φ}^{r} {(f, t)}_{p}$ respectively.

On the best observation of wave and Schrödinger equations in quantum ergodic billiards

Yannick Privat, Emmanuel Trélat, Enrique Zuazua (2012)

Journées Équations aux dérivées partielles

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This paper is a proceedings version of the ongoing work [20], and has been the object of the talk of the second author at Journées EDP in 2012. In this work we investigate optimal observability properties for wave and Schrödinger equations considered in a bounded open set $Ω \subset ℝ^{n}$ , with Dirichlet boundary conditions. The observation is done on a subset $ω$ of Lebesgue measure $| ω | = L | Ω |$ , where $L \in (0, 1)$ is fixed. We denote...

Some decay properties for the damped wave equation on the torus

Nalini Anantharaman, Matthieu Léautaud (2012)

Journées Équations aux dérivées partielles

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This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012). We address the decay rates of the energy of the damped wave equation when the damping coefficient $b$ does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger...

Finitistic dimension and restricted injective dimension

Dejun Wu (2015)

Czechoslovak Mathematical Journal

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We study the relations between finitistic dimensions and restricted injective dimensions. Let $R$ be a ring and $T$ a left $R$ -module with $A = {End}_{R} T$ . If $_{R} T$ is selforthogonal, then we show that $rid (T_{A}) \leq findim (A_{A}) \leq findim (_{R} T) + rid (T_{A})$ . Moreover, if $R$ is a left noetherian ring and $T$ is a finitely generated left $R$ -module with finite injective dimension, then $rid (T_{A}) \leq findim (A_{A}) \leq fin . inj . \dim (_{R} R) + rid (T_{A})$ . Also we show by an example that the restricted injective dimensions of a module may be strictly smaller than the Gorenstein injective dimension.

Explicit Teichmüller curves with complementary series

Carlos Matheus, Gabriela Weitze-Schmithüsen (2013)

Bulletin de la Société Mathématique de France

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We construct an explicit family of arithmetic Teichmüller curves $𝒞_{2 k}$ , $k \in ℕ$ , supporting $SL (2, ℝ)$ -invariant probabilities $μ_{2 k}$ such that the associated $SL (2, ℝ)$ -representation on $L^{2} (𝒞_{2 k}, μ_{2 k})$ has complementary series for every $k \geq 3$ . Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves $𝒞_{2 k}$ has arbitrarily slow rate of exponential mixing.

The Kodaira dimension of the moduli space of Prym varieties

Gavril Farkas, Katharina Ludwig (2010)

Journal of the European Mathematical Society

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We study the enumerative geometry of the moduli space $ℛ_{g}$ of Prym varieties of dimension $g - 1$ . Our main result is that the compactication of $ℛ_{g}$ is of general type as soon as $g > 13$ and $g$ is different from 15. We achieve this by computing the class of two types of cycles on $ℛ_{g}$ : one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical...

On malnormal peripheral subgroups of the fundamental group of a $3$ -manifold

Pierre de la Harpe, Claude Weber (2014)

Confluentes Mathematici

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Let $K$ be a non-trivial knot in the $3$ -sphere, $E_{K}$ its exterior, $G_{K} = π_{1} (E_{K})$ its group, and $P_{K} = π_{1} (\partial E_{K}) \subset G_{K}$ its peripheral subgroup. We show that $P_{K}$ is malnormal in $G_{K}$ , namely that $g P_{K} g^{- 1} \cap P_{K} = {e}$ for any $g \in G_{K}$ with $g \notin P_{K}$ , unless $K$ is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in $E_{K}$ attached to $T_{K}$ which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental...