Dynamics on K3 Surfaces: Salem Numbers and Siegel Disks
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Automorphic Forms for Some Even Unimodular Lattices
AUTOMORPHIC FORMS FOR SOME EVEN UNIMODULAR LATTICES NEIL DUMMIGAN AND DAN FRETWELL Abstract. We lookp at genera of even unimodular lattices of rank 12p over the ring of integers of Q( 5) and of rank 8 over the ring of integers of Q( 3), us- ing Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find in- stances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over Z, we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift. 1. Introduction Nebe and Venkov [54] looked at formal linear combinations of the 24 Niemeier lattices, which represent classes in the genus of even, unimodular, Euclidean lattices of rank 24. They found a set of 24 eigenvectors for the action of an adjacency oper- ator for Kneser 2-neighbours, with distinct integer eigenvalues. This is equivalent to computing a set of Hecke eigenforms in a space of scalar-valued modular forms for a definite orthogonal group O24. They conjectured the degrees gi in which the (gi) Siegel theta series Θ (vi) of these eigenvectors are first non-vanishing, and proved them in 22 out of the 24 cases. (gi) Ikeda [37, §7] identified Θ (vi) in terms of Ikeda lifts and Miyawaki lifts, in 20 out of the 24 cases, exploiting his integral construction of Miyawaki lifts. -
Quartic Salem Numbers Which Are Mahler Measures of Non-Reciprocal 2-Pisot Numbers Tome 32, No 3 (2020), P
Toufik ZAÏMI Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers Tome 32, no 3 (2020), p. 877-889. <http://jtnb.centre-mersenne.org/item?id=JTNB_2020__32_3_877_0> © Société Arithmétique de Bordeaux, 2020, tous droits réservés. L’accès aux articles de la revue « Journal de Théorie des Nom- bres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement personnelle du copiste est con- stitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 32 (2020), 877–889 Quartic Salem numbers which are Mahler measures of non-reciprocal 2-Pisot numbers par Toufik ZAÏMI Résumé. Motivé par une question de M. J. Bertin, on obtient des paramé- trisations des polynômes minimaux des nombres de Salem quartiques, disons α, qui sont des mesures de Mahler des 2 -nombres de Pisot non-réciproques. Cela nous permet de déterminer de tels nombres α, de trace donnée, et de déduire que pour tout entier naturel t (resp. t ≥ 2), il y a un nombre de Salem quartique, de trace t, qui est (resp. qui n’est pas) une mesure de Mahler d’un 2 -nombre de Pisot non-réciproque. -
Genus of Vertex Algebras and Mass Formula
Genus of vertex algebras and mass formula Yuto Moriwaki * Graduate School of Mathematical Science, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan Abstract. We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which play an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras. Introduction Vertex operator algebras (VOAs) are algebraic structures that play an important role in two-dimensional conformal field theory. The classification of conformal field theories is an extremely interesting problem, of importance in mathematics, statistical mechanics, and string theory; Mathematically, it is related to the classification of vertex operator algebras, which is a main theme of this work. An important class of vertex operator algebras can be constructed from positive-definite even integral lattices, called lattice vertex algebras [Bo1, LL, FLM]. In this paper, we propose a new method to construct and classify vertex algebras by using a method developed in the study of lattices. A lattice is a finite rank free abelian group equipped with a Z-valued symmetric bilinear form. Two such lattices are equivalent (or in the same genus) if they are isomorphic over the ring of p-adic integers Zp for each prime p and also equivalent over the real number arXiv:2004.01441v2 [math.QA] 5 Mar 2021 field R. Lattices in the same genus are not always isomorphic over Z. -
The Smallest Perron Numbers 1
MATHEMATICS OF COMPUTATION Volume 79, Number 272, October 2010, Pages 2387–2394 S 0025-5718(10)02345-8 Article electronically published on April 26, 2010 THE SMALLEST PERRON NUMBERS QIANG WU Abstract. A Perron number is a real algebraic integer α of degree d ≥ 2, whose conjugates are αi, such that α>max2≤i≤d |αi|. In this paper we com- pute the smallest Perron numbers of degree d ≤ 24 and verify that they all satisfy the Lind-Boyd conjecture. Moreover, the smallest Perron numbers of degree 17 and 23 give the smallest house for these degrees. The computa- tions use a family of explicit auxiliary functions. These functions depend on generalizations of the integer transfinite diameter of some compact sets in C 1. Introduction Let α be an algebraic integer of degree d, whose conjugates are α1 = α, α2,...,αd and d d−1 P = X + b1X + ···+ bd−1X + bd, its minimal polynomial. A Perron number, which was defined by Lind [LN1], is a real algebraic integer α of degree d ≥ 2 such that α > max2≤i≤d |αi|.Any Pisot number or Salem number is a Perron number. From the Perron-Frobenius theorem, if A is a nonnegative integral matrix which is aperiodic, i.e. some power of A has strictly positive entries, then its spectral radius α is a Perron number. Lind has proved the converse, that is to say, if α is a Perron number, then there is a nonnegative aperiodic integral matrix whose spectral radius is α.Lind[LN2] has investigated the arithmetic of the Perron numbers. -
1 Positive Definite Even Unimodular Lattices
Algebraic automorphic forms and Hilbert-Siegel modular forms Tamotsu Ikeda (Kyoto university) Sep. 17, 2018, at Hakuba 1 Positive definite even unimodular lattices Let F be a totally real number field of degree d. Let n > 0 be an integer such that dn is even. Then, by Minkowski-Hasse theorem, there exists a quadratic space (Vn;Q) of rank 4n with the following properties: (1) (V; Q) is unramified at any non-archimedean place. (2) (V; Q) is positive definite ai any archimedean place. Definition 1. An algebraic automorphic form on the orthogonal group OQ is a locally constant function on OQ(F )nOQ(A). Put 1 (x; y) = (Q(x + y) − Q(x) − Q(y)) x; y 2 V: Q 2 Let L be a o-lattie in V (F ). The dual lattice L∗ is defined by ∗ L = fx 2 Vn(F ) j (x; L)Q ⊂ og: Then L is an integral lattice if and only if L ⊂ L∗. An integral lattice L is called an even lattice if Q(x) ⊂ 2o 8x 2 L: Moreover, an integral lattice L is unimodular if L = L∗. By the assumption (1) and (2), there exists a positive definite even unimodular latice L0 in V (F ). Two integral lattices L1 and L2 are equivalent if there exists an element g 2 OQ(F ) such that g · L1 = L2. Two integral lattices L1 and L2 are in the same genus if there 1 exists an element gv 2 OQ(Fv) such that g · L1;v = L2;v for any finite place v. The set of positive definite even unimodular lattices in V (F ) form a genus. -
Lattice Theory, K3 Surfaces, and the Torelli Theorem
UMass, Reading Seminar in Algebraic Geometry, Lattice theory, K3 surfaces, and the Torelli theorem Luca Schaffler October 4, 2019 1 What is a lattice? Definition 1.1. A lattice is a pair (L; ·), where L is a finitely generated free abelian group and ·: L × L ! Z such that (v; w) 7! v · w is a symmetric bilinear form. • L is even if v2 := v · v is an even number for all v 2 L. • The Gram matrix of L with respect to a basis fe1; : : : ; eng for L is the matrix of intersection (ei · ej)1≤i;j≤n. • The rank of L is defined to be the rank of one of its Gram matrices. • A lattice L is unimodular if the determinant of one of its Gram matrices is ±1. Equivalently, L of maximal rank is unimodular provided the discriminant group ∗ ∗ L =L is trivial (we let L = HomZ(L; Z)). • We can define the signature of L as the signature of one of its Gram matrices. By saying signature (a; b), we mean that a (resp. b) is the number of positive (resp. negative) eigenvalues. If L has signature (a; b), then we say that L is indefinite provided a; b > 0. 2 Example 1.2. U = (Z ; ·) with (v1; v2) · (w1; w2) = v1w2 + v2w1 is called the hyperbolic plane. Check that this is even, unimodular, and of signature (1; 1). 1 8 Example 1.3. E8 = (Z ;B), where 0 −2 1 0 0 0 0 0 0 1 B 1 −2 1 0 0 0 0 0 C B C B 0 1 −2 1 0 0 0 0 C B C B 0 0 1 −2 1 0 0 0 C B = B C : B 0 0 0 1 −2 1 0 1 C B C B 0 0 0 0 1 −2 1 0 C B C @ 0 0 0 0 0 1 −2 0 A 0 0 0 0 1 0 0 −2 This matrix can be reconstructed from the E8 Dynkin diagram. -
Arxiv:2001.07094V3 [Math.NT] 10 May 2021 Aeo Nirdcbeplnma)Alclgoa Principl a Local-Global Conditions, a Local Polynomial) I Are Irreducible Mcmullen 20]
ISOMETRIES OF LATTICES AND HASSE PRINCIPLES EVA BAYER-FLUCKIGER Abstract. We give necessary and sufficient conditions for an integral polynomial without linear factors to be the characteristic polynomial of an isometry of some even, unimodular lattice of given signature. This gives rise to Hasse principle questions, which we answer in a more general setting. As an application, we prove a Hasse principle for signatures of knots. 0. Introduction In [GM 02], Gross and McMullen give necessary conditions for a monic, irreducible polynomial to be the characteristic polynomial of an isometry of some even, unimodular lattice of prescribed signature. They speculate that these conditions may be sufficient; this is proved in [BT 20]. It turns out that the conditions of Gross and McMullen are local conditions, and that (in the case of an irreducible polynomial) a local-global principle holds. More generally, if F Z[X] is a monic polynomial without linear factors, the conditions of Gross∈ and McMullen are still necessary. Moreover, they are also sufficient everywhere locally (see Theorem 25.6). However, when F is reducible, the local-global principle no longer holds in general, as shown by the following example Example. Let F be the polynomial (X6 + X5 + X4 + X3 + X2 + X + 1)2(X6 X5 + X4 X3 + X2 X + 1)2, arXiv:2001.07094v3 [math.NT] 10 May 2021 − − − and let L be an even, unimodular, positive definite lattice of rank 24. The lattice L has an isometry with characteristic polynomial F everywhere locally, but not globally (see Example 25.17). In particular, the Leech lattice has no isometry with characteristic polynomial F , in spite of having such an isometry everywhere locally. -
Notes on 4-Manifolds
ALGEBRAIC SURFACES AND FOUR-MANIFOLDS PHILIP ENGEL Contents 1. Structures on manifolds 1 2. Sheaves, bundles, and connections 4 3. Cohomology 8 4. Characteristic classes 12 5. The Riemann-Roch theorem and its generalizations 16 6. The Hodge theorems 19 7. Introduction to algebraic surfaces 25 8. Blow-ups and blow-downs 30 9. Spin and Spinc groups 33 10. Spin bundles and Dirac operators 38 11. Rokhlin's theorem 42 12. Freedman's theorems 45 13. Classification of unimodular lattices 49 4 14. An exotic R 49 15. The Seiberg-Witten equations 49 1. Structures on manifolds Definition 1.1. A topological or C0-manifold of dimension n is a Hausdorff n topological space X, which admits local charts to φU : U ! R . It follows that the transition functions −1 tUV := φV ◦ φU : φU (U \ V ) ! φV (U \ V ) are homeomorphisms. Definition 1.2. By requiring tUV to respect extra geometric structures on n R , we produce many other classes of manifolds. In increasing order of restrictiveness, we say X is additionally a (1) PL manifold if tUV are piecewise-linear. k 1 k (2) C - or C -manifold if tUV 2 C are k-differentiable or smooth. (3) real-analytic manifold if tUV are real-analytic. (4) complex-analytic manifold if tUV is holomorphic: i.e. has a complex- n n=2 linear differential with respect the identification R = C . 1 2 PHILIP ENGEL (5) projective variety if there is a homeomorphism X ! V (f1; : : : ; fm) ⊂ N CP to a smooth vanishing locus of homogenous polynomials. N N+1 ∗ CP := (C n f0g)= ∼ with x ∼ λx for all λ 2 C : There is a natural notion of isomorphism for each class of manifold/variety, by declaring that a homeomorphism f : X ! Y defines an isomorphism if it preserves the given structure. -
Weil's Conjecture for Function Fields
Weil’s Conjecture for Function Fields February 12, 2019 1 First Lecture: The Mass Formula and Weil’s Conjecture Let R be a commutative ring and let V be an R-module. A quadratic form on V is a map q : V Ñ R satisfying the following conditions: paq The construction pv, wq ÞÑ qpv ` wq ´ qpvq ´ qpwq determines an R-bilinear map V ˆ V Ñ R. pbq For every element λ P R and every v P V , we have qpλvq “ λ2qpvq. A quadratic space over R is a pair pV, qq, where V is a finitely generated projective R-module and q is a quadratic form on V . One of the basic problems in the theory of quadratic forms can be formulated as follows: Question 1.1. Let R be a commutative ring. Can one classify quadratic spaces over R (up to isomorphism)? Let us begin by describing some cases where Question 1.1 admits a complete answer. Example 1.2 (Quadratic Forms over C). Let R “ C be the field of complex numbers (or, more generally, any algebraically closed field of characteristic different from 2). Then every quadratic space over R is isomorphic to pCn, qq, where q is given by the formula 2 2 qpx1, . , xnq “ x1 ` ¨ ¨ ¨ ` xr for some 0 ¤ r ¤ n. Moreover, the integer r is uniquely determined: it is an isomorphism- invariant called the rank of the quadratic form q. Example 1.3 (Quadratic Forms over the Real Numbers). Let R “ R be the field of real numbers. Then every quadratic space over R is isomorphic to pRn, qq, where the quadratic form q is given by 2 2 2 2 2 qpx1, . -
Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems
Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems Dissertation zur Erlangung des akademischen Grades eines Doktors der Mathematik (Dr. math.) vorgelegt von Chrizaldy Neil Ma~nibo Fakult¨atf¨urMathematik Universit¨atBielefeld April 2019 Gedruckt auf alterungsbest¨andigem Papier ◦◦ISO 9706 1. Berichterstatter: Prof. Dr. Michael Baake Universit¨atBielefeld, Germany 2. Berichterstatter: A/Prof. Dr. Michael Coons University of Newcastle, Australia 3. Berichterstatter: Prof. Dr. Uwe Grimm The Open University, Milton Keynes, UK Datum der m¨undlichen Pr¨ufung:04 Juni 2019 i Contents Acknowledgementsv Introduction viii 1. Prerequisites 1 d 1.1. Point sets in R .....................................1 1.2. Symbolic dynamics and inflation rules........................1 1.2.1. Substitutions..................................1 1.2.2. Perron{Frobenius theory............................3 1.2.3. The symbolic hull...............................4 1.2.4. Inflation systems and the geometric hull...................5 1.3. Harmonic analysis and diffraction...........................7 1.3.1. Fourier transformation of functions......................7 1.3.2. Measures....................................7 1.3.3. Decomposition of positive measures.....................9 1.3.4. Autocorrelation and diffraction measure...................9 1.4. Lyapunov exponents.................................. 11 1.4.1. Lyapunov exponents for sequences of matrices................ 11 1.4.2. Matrix cocycles................................. 12 1.4.3. Ergodic theorems............................... -
A Systematic Construction of Almost Integers Maysum Panju University of Waterloo [email protected]
The Waterloo Mathematics Review 35 A Systematic Construction of Almost Integers Maysum Panju University of Waterloo [email protected] Abstract: Motivated by the search for “almost integers”, we describe the algebraic integers known as Pisot numbers, and explain how they can be used to easily find irrational values that can be arbitrarily close to whole numbers. Some properties of the set of Pisot numbers are briefly discussed, as well as some applications of these numbers to other areas of mathematics. 1 Introduction It is a curious occurrence when an expression that is known to be a non-integer ends up having a value surprisingly close to a whole number. Some examples of this phenomenon include: eπ π = 19.9990999791 ... − 23 5 = 109.0000338701 ... 9 88 ln 89 = 395.0000005364 ... These peculiar numbers are often referred to as “almost integers”, and there are many known examples. Almost integers have attracted considerable interest among recreational mathematicians, who not only try to generate elegant examples, but also try to justify the unusual behaviour of these numbers. In most cases, almost integers exist merely as numerical coincidences, where the value of some expression just happens to be very close to an integer. However, sometimes there actually is a clear, mathematical reason why certain irrational numbers should be very close to whole numbers. In this paper, we’ll look at the a set of numbers called the Pisot numbers, and how they can be used to systematically construct infinitely many examples of almost integers. In Section 2, we will prove a result about powers of roots of polynomials, and use this as motivation to define the Pisot numbers. -
Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds
COUNTING SALEM NUMBERS OF ARITHMETIC HYPERBOLIC 3{ORBIFOLDS MIKHAIL BELOLIPETSKY, MATILDE LAL´IN, PLINIO G. P. MURILLO, AND LOLA THOMPSON Abstract. It is known that the lengths of closed geodesics of an arithmetic hyperbolic orbifold are related to Salem numbers. We initiate a quantitative study of this phenomenon. We show that any non-compact arithmetic 3-dimensional orbifold defines cQ1=2 +O(Q1=4) square-rootable Salem numbers of degree 4 which are less than or equal to Q. This quantity can be compared to the total number of such Salem numbers, 4 3=2 which is shown to be asymptotic to 3 Q + O(Q). Assuming the gap conjecture of Marklof, we can extend these results to compact arithmetic 3-orbifolds. As an application, we obtain lower bounds for the strong exponential growth of mean multiplicities in the geodesic spectrum of non-compact even dimensional arithmetic orbifolds. Previously, such lower bounds had only been obtained in dimensions 2 and 3. 1. Introduction A Salem number is a real algebraic integer λ > 1 such that all of its Galois conjugates except λ−1 have absolute value equal to 1. Salem numbers appear in many areas of mathematics including algebra, geometry, dynamical systems, and number theory. They are closely related to the celebrated Lehmer's problem about the smallest Mahler measure of a non-cyclotomic polynomial. We refer to [Smy15] for a survey of research on Salem numbers. It has been known for some time that the exponential lengths of the closed geodesics of an arithmetic hyperbolic n-dimensional manifold or orbifold are given by Salem numbers.