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A Course in Arithmetic
... which establishes the negative odd integer case of Theorem 2, (ii) if we specify a modular form whose q-expansion is (34). In slightly different notation from (p. 83, [23]), we introduce the Eisenstein series ...
... where E 2k (τ) is the (normalized) Eisenstein series of weight 2k ((34), p. 92, [23]). ...
... Proof. The first assertion is (Proposition 4, p. 83, [23]); (40) is given on (p. 92, [23]) and (41) is given by (16). ...
In this paper, we study the Bochner modular relation (Lambert series) for the kth power of the product of two Riemann zeta-functions with difference α, an integer with the Voronoĭ function weight Vk. In the case of V1(x)=e−x, the results reduce to Bochner modular relations, which include the Ramanujan formula, Wigert–Bellman approximate functional equation, and the Ewald expansion. The results abridge analytic number theory and the theory of modular forms in terms of the sum-of-divisor function. We pursue the problem of (approximate) automorphy of the associated Lambert series. The α=0 case is the divisor function, while the α=1 case would lead to a proof of automorphy of the Dedekind eta-function à la Ramanujan.
... is uniform if and only if I n is anisotropic as a σ -Hermitian form over Q( √ d), see Theorem 4.17 and Proposition 2.15 in [27]. However Meyer's Theorem implies that it is always isotropic, see Corollary 2 in §3 of Chapter 4 in [30]. Definition 1.8 A quaternion algebra A over a field k is a 4-dimensional unital associative algebra which admits a basis {1, i, j, i j} satisfying i 2 = a ∈ k, j 2 = b ∈ k and i j = − ji. ...
... If F = Q, Meyer's Theorem (Corollary 2 in §3 of Chapter 4 in [30]) implies that B is isotropic over Q. Thus SO(B, Z) is non-uniform, see Theorem 4.17 and Proposition 2.14 in [27]. ...
... Any -Hermitian form of rank at least 2 over a quaternion algebra is isotropic as can be deduced by Meyer's Theorem, see Corollary 2 in §3 of Chapter 4 in[30] ...
We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups , , , and . These subgroups are images of Hitchin representations. In particular, we show that every uniform lattice of , of with and of contains infinitely many mapping class group orbits of Zariski-dense Hitchin representations of fixed genus. Together with Long and Thistlethwaite (Exp. Math. 27(1):82–92 2018) and Audibert (2022) it implies that all lattices of contain a Zariski-dense surface subgroup. This paper follows Audibert (2022), where we constructed Zariski-dense Hitchin representations in non-uniform lattices.
... Take any prime power q ≡ 1 (mod v) which certainly exists by Dirichlet's theorem on arithmetic progressions (see, e.g., [32]). Now take a generator g of the subgroup G of F * q of order v and consider the map ...
... Now we have to consider the map φ + : Z + 20 −→ G + defined by φ + (x) = 2 x for every x ∈ Z 20 and φ + (∞) = 0. This map turns D into the isomorphic G + -rotational (K 21 , 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40} and the base cycles are 1,8,21,32,31,23,36,5,18,10,9,20,33,40), 21,16,32,8,10,20,5,37,33,39,18,36,9). ...
... Now we have to consider the map φ + : Z + 20 −→ G + defined by φ + (x) = 2 x for every x ∈ Z 20 and φ + (∞) = 0. This map turns D into the isomorphic G + -rotational (K 21 , 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40} and the base cycles are 1,8,21,32,31,23,36,5,18,10,9,20,33,40), 21,16,32,8,10,20,5,37,33,39,18,36,9). ...
A design is additive if, up to isomorphism, the point set is a subset of an abelian group G and every block is zero-sum. This definition was introduced in Caggegi et al. (J Algebr Comb 45:271-294, 2017) and was the starting point of an interesting new theory. Although many additive designs have been constructed and known designs have been shown to be additive, these structures seem quite hard to construct in general, particularly when we look for additive Steiner 2-designs. One might generalize additive Steiner 2-designs in a natural way to graph decompositions as follows: given a simple graph , an additive -design is a decomposition of the graph into subgraphs (blocks) all isomorphic to , such that the vertex set is a subset of an abelian group G, and the sets are zero-sum in G. In this work we begin the study of additive -designs: we develop different tools instrumental in constructing these structures, and apply them to obtain some infinite classes of designs and many sporadic examples. We will consider decompositions into various graphs , for instance cycles, paths, and k-matchings. Similar ideas will also allow us to present here a sporadic additive 2-(124, 4, 1) design.
... This is always possible since for a characteristic vector χ, a vector χ + 2λ o (λ o ∈ Λ δ-odd ) is also characteristic, but it has a different mod 2 value of the inner product with δ 2 . Then, we obtain the spin selection rule for each sector extended by the shift symmetry in lattice CFTs in table 3. Note that using the fact that a characteristic vector χ of a self-dual lattice Λ ⊂ n satisfies χ 2 ≡ n mod 8 [27,28], we obtain the spin of the R sector s ∈ n/8 + except for s ∈ 1 2 + n 8 + for the h-odd states in the twisted R sector. ...
... where Z T [s 0 , s 1 ] is given by (27). ...
We consider chiral fermionic conformal field theories (CFTs) constructed from lattices and investigate their orbifolds under reflection and shift \mathbb{Z}_2 ℤ 2 symmetries. For lattices based on binary error-correcting codes, we show the duality between reflection and shift orbifolds using a triality structure inherited from the binary codes. Additionally, we systematically compute the partition functions of the orbifold theories for both binary and nonbinary codes. Finally, we explore applications of this code-based construction in the search for supersymmetric CFTs and chiral fermionic CFTs without continuous symmetries.
... Some proofs are based on the Poisson summation formula: [9] contains one such proof, using the triple product formula and theta functions of Jacobi; [6] contains variant proofs, using instead the pentagonal number theorem of Euler. Other proofs rest on transformation properties of Eisenstein series: see [7] and [5] for such a proof involving the exceptional (or 'forbidden') Eisenstein series E 2 ; see [3] for a related proof, involving instead the Ramanujan-Eisenstein series Q and R, which satisfy Q 3 − R 2 = 1728 η 24 . A proof using the Weierstrass zeta function and simplifying an idea due to Petersson is presented in [4]. ...
... the resulting function of q without the leading power of q is holomorphic in the open unit disc, but c(q) itself is to be regarded as a holomorphic function of τ in H according to (7). In terms of these 'new' functions, we now have the following transformation laws. ...
... The lattice L is called even if for every lattice vector x ∈ L the inner product x · x is an even integer. It is well-known (see for instance Serre [28]) that in a given dimension the number of non-isometric even unimodular lattices is finite and that they exist only in dimensions which are divisible by 8. ...
... One particular easy way to justify this claim is to use that the dimension formula for Eis k (Γ 0 (ℓ)) to see that this is a 2-dimensional space for any prime ℓ and then check that both (obviously linearly independent) elements above are modular forms for Γ 0 (ℓ). From (28) we see that the coefficients of both basis elements can be trivially bounded by the coefficients of E k = n≥0 b n q n . For the latter we use the explicit form b n = − 2k B k σ k−1 (n). ...
We discuss the local analysis of Gaussian potential energy of modular lattices. We show for instance that the 3-modular 12-dimensional Coxeter-Todd lattice and the 2-modular 16-dimensional Barnes-Wall lattice, which both provide excellent sphere packings, are not, even locally, universally optimal (in the sense of Cohn and Kumar).
... Similarities are found with Majorana modes in superconducting nanowires and other quasiparticle systems [60][61][62]. The invariance under modular transformations and related symmetries play crucial roles in our formulation and defining topological quantum computing platforms [63][64][65][66][67] and Majorana zero modes [68,69]. ...
... Thus, the interplay between modular invariance and the spectral properties of the Hamiltonian provides a natural quantization condition that enforces real eigenvalues precisely at the nontrivial zeros of ζ and coincide when nontrivial zeros are on the CL. A deeper insight into boundary and Gorbachuk triplets can be found in [48,49,[57][58][59] and for modular forms in [63][64][65][66][67]. ...
We present a spectral realization of the nontrivial zeros of the Riemann zeta function ζ(z) from a Mellin-Barnes integral that explicitly contains it in the critical strip for which 0 < Re {z} < 1 for the Hilbert-Pólya approach to the Riemann Hypothesis. This spectrum corresponds to the real-valued energy eigenvalues of a Majorana particle with Hermitian Hamiltonian HM in a domain DM of the (1 + 1)-dimensional Rindler spacetime, or to equivalent Kaluza-Klein reductions of (n + 1)-dimensional geometries, corresponding to the critical strip. We show that, under these assumptions, all nontrivial zeros have real part Re {z} = 1/2, the critical line, and are infinite in number in agreement with Hardy-Littlewood's theorem from number theory. This result is consistent with the Hilbert-Pólya conjecture, which provides a framework for proving the Riemann Hypothesis through spectral analysis. We establish a spectral realization of the nontrivial zeta zeros using HM and confirm its essential self-adjointness also via deficiency index analysis, boundary triplet theory, and Krein's extension theorem. Extending this framework to noncommutative geometry, we interpret HM as a Dirac operator D in a spectral triple, linking our results to Connes' program and reinforcing our approach as another viable pathway to the Riemann Hypothesis from our premises. This formulation constructs a noncommutative spectral triple (A, H, D), where the algebra A encodes the modular symmetries underlying the spectral realization of ζ in the Hilbert space H of Majorana wavefunctions, integrating concepts from quantum mechanics, general relativity, and number theory.
... Recent advancements have seen π and Euler numbers interfacing with contemporary mathematical disciplines such as quantum computing, information theory, and algebraic topology. The connections between these constants and modular forms, for example, have opened new avenues to understanding symmetries and invariants in higher-dimensional spaces [7]. ...
... Another interesting case that warrants further research (although it has previously been commented on generally) concerns the theory of L-functions in the study of modular forms. In particular, Euler numbers appear in the coefficients of the Fourier expansions associated with the formulation of L-functions, which generalize the Riemann zeta function and are important for many conjectures and theorems in number theory [7]. The connection between Euler numbers and modular forms enhances the analytical tools available for probing the properties of L-functions and their zeros, which are deeply connected to the distribution of prime numbers. ...
The mathematical constant π and Euler numbers En have long been of relevance in various branches of mathematics, particularly in number theory, combinatorics, numerical analysis, and mathematical physics. This review article introduces an exploration of their historical evolution, theoretical foundations, and recent advancements. We examine how π and Euler numbers have facilitated advances in different scientific areas like quantum field theory and numerical algorithms. We introduce some emerging perspectives that highlight their interdisciplinary applications and potential future trajectories. Certainly, both numbers are of great relevance in current mathematical research, and their adaptability and ubiquity will continue to ensure their continuous appearance in future mathematical discoveries. The integration of π and Euler numbers into advanced computational techniques and fields like artificial intelligence exemplifies their potential in driving mathematical innovation.
... Summarizing [16], pp. 82-83 or [17], pp. ...
... Equation (1) is a typical example of the lattice function. Another is the classical Eisenstein series [16], p. 83, (12), given as follows: ...
Zeta-functions play a fundamental role in many fields where there is a norm or a means to measure distance. They are usually given in the forms of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain in question has a unique factorization property. In applied disciplines, those zeta-functions which satisfy the functional equation but do not have Euler products often appear as a lattice zeta-function or an Epstein zeta-function. In this paper, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta-functions by considering some generalizations of the Eisenstein series of level 2ϰ to the complex variable level s. Naturally, generalized Eisenstein series and Barnes multiple zeta-functions arise, which have affinity to dissections, as they are (semi-) lattice functions. The method of Lewittes (and Chapman) and Kurokawa leads to some limit formulas without absolute value due to Tsukada and others. On the other hand, Komori, Matsumoto and Tsumura make use of the Barnes multiple zeta-functions, proving their modular relation, and they give rise to generalizations of Ramanujan’s formula as the generating zeta-function of σs(n), the sum-of-divisors function. Lewittes proves similar results for the 2-dimensional case, which holds for all values of s. This in turn implies the eta-transformation formula as the extreme case, and most of the results of Chapman. We shall unify most of these as a tapestry of ideas arising from the merging of additive entity (Dirichlet series) and multiplicative entity (Euler product), especially in the case of limit formulas.
... are the orbifold points of orders 2 and 3 (fixed by S and ST ) and ′ indicates that the sum runs over all points not equal to i, ω or i∞. A standard theorem in the theory of modular forms resulting from integrating dW/W around the boundary of the fundamental domain of the modular group as in [28] then states that ...
... In this section we give a brief review of standard Hecke operators for Γ = SL(2, Z) and the principal congruence subgroup Γ(N ). For SL(2, Z) this is standard material and may be familiar from discussions in [28,35]. Hecke operators for congruence subgroups are probably less familiar to physicists. ...
We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their q expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.
... In 2019, Rupert Li proved the Serre's congruence for weighted hypersurfaces by a slight modification of the proof of the Chevalley-Warning Theorem by Serre [35,Thm. 3]. ...
This paper examines the arithmetic of the loci \cL_n, parameterizing genus 2 curves with -split Jacobians over finite fields \F_q. We compute rational points |\cL_n(\F_q)| over \F_3, \F_9, \F_{27}, \F_{81}, and \F_5, \F_{25}, \F_{125}, derive zeta functions Z(\cL_n, t) for . Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.
... Given a field K, let R K be a set of representatives for K × /K ×2 ; we choose 1 to represent the trivial class K ×2 . In particular, we make the following choices for Q and its completions (see [Ser73,§II.3.3]): ...
We prove the thin set version of Manin's conjecture for the chordal (or: determinantal) cubic fourfold, which is the secant variety of the Veronese surface. We reduce this counting problem to a result of Schmidt for quadratic points in the projective plane by showing that the chordal cubic fourfold is isomorphic to the symmetric square of the projective plane over the rational numbers.
... Let S and T be the elements of G = SL 2 (Z)/{±1} as S = 0 − 1 1 0 and T = 1 1 0 1 , therefore, S(z) = − 1 z and T (z) = z + 1. The group G is generated by S and T and the detailed proof could be found in [30]. Suppose H is the upper half plane H = {z ∈ C | Imz > 0} ⊂ C. ...
In this paper, we study linear codes over based on lattices and theta functions. We obtain the complete weight enumerators MacWilliams identity and the symmetrized weight enumerators MacWilliams identity based on the theory of theta function. We extend the main work by Bannai, Dougherty, Harada and Oura to the finite ring for any positive integer k and present the complete weight enumerators MacWilliams identity in genus g. When k=p is a prime number, we establish the relationship between the theta function of associated lattices over a cyclotomic field and the complete weight enumerators with Hamming weight of codes, which is an analogy of the results by G. Van der Geer and F. Hirzebruch since they showed the identity with the Lee weight enumerators.
... Since x and y are coprime, the conditions p ≡ 1 (mod x) and p ≡ 2 (mod y) are equivalent to p ≡ c (mod xy) for some c ∈ N by the Chinese remainder theorem. As c is coprime to xy, the proportion of primes congruent to c modulo xy is 1/φ(xy) > 0 (see [Ser12,Section VI.4]). For any prime p with this property, p − 1 is divisible by x and is coprime to y. ...
We study a random walk on the Lie algebra in which new elements are produced by randomly applying the adjoint operators corresponding to two generators. Focusing on the generic case where the generators are selected at random, we analyze both the limiting distribution of the random walk and the speed at which it converges to this distribution. These questions reduce to the study of a random walk on the cyclic group . We show that, with high probability, the walk exhibits a pre-cutoff phenomenon after roughly p steps. Notably, the limiting distribution need not be uniform; rather, it depends on the prime divisors of .
... In view of Remark 2 the proof of Lemma 4 is reduced to the diagonal case (see [14,Chapters IV] for the technique for reducing a quadratic form over a finite field to the diagonal form). The diagonal case is factorized to the onedimensional variant, which in turn is reduced to the calculation of the sum g 1 (q). ...
Consider a finite field , , where p is an odd number. Let M=(E,r) be a regular matroid; denote by the family of its bases, , where , . Let a subset in E have the maximal cardinality and satisfy the condition , while . Let us represent the value of the characteristic polynomial of the matroid M at the point q as the linear combination of Legendre symbols with respect to , whose coefficients are modulo equal to . This representation generalizes the formula for a flow polynomial of a graph which was obtained by us earlier. The latter formula is an analog of the so-called -representation of vacuum Feynman amplitudes in the case of a finite field, which has inspired the Kontsevich conjecture (1997). The -representation technique is also applicable for expressing the number of Tait colorings for a cubic biconnected planar graph in terms of principal minors of the matrix of faces of this graph.
... The issues deduced in these studies set up out significance in demonstrating the operation of ways and unveiling new perceptivity in Diophantine proposition within the literature. Covering computation from a broad perspective, the book from Serre (Serre, 1996) offers a comprehensive course, including number proposition, algebra and more. It aims to give a well-rounded understanding of computation, making it suitable for scholars and mathematicians at colorful levels. ...
Mathematicians have long been interested in Diophantine sets. They have good ways to analyze the calculations and results. The aim of this paper is to explore the enigmatic world of diophantine D( ∓3) set shapes, revealing a new emphasis on its complex specifications and deep correlations. The Diophantine D( ∓3) sets, defined as integer values in this work, represent significant domain ripe for examinations. Our study analyzes these sets in detail, ignoring their cardinals, and aims to reveal hidden patterns and unique characteristics. By scrutinizing their structure, our intention is to reveal the high mathematics content of these collections. In our discussion we highlight basic principles of basic algebraic number theory, invoking the law of quadratic reciprocity, Diophantine equations, and the enduring grace of major mathematicians like Gauss, Dirichlet and Fermat. These tools and logic serve as viewers of our discussion, ultimately Diophantine provides a deeper appreciation of the concepts in the D( ∓3) sets and their importance in the broader mathematical terrain.
... [TNN13,Ser96,EH91,Nea06,Cho03,Duf98,KS03,PB07]. ...
Bernhard Reimann proposed his hypothesis in 1859. Until now, after 165 years of existence, no one has yet proven this hypothesis. Due to the nature of the hypothesis’s contribution to modern mathematics. It is necessary to prove the hypothesis. The approach to solving the problem is using classical physics and mathematical knowledge to find a value that satisfies the constraint. Then, prove that this value is unique. To do this, the author uses MATLAB (2023a) calculation software. The results of the research process show the correctness of the hypothesis. However, the specific characteristics of the current calculation tool still need to be met, so more appropriate programs are needed in the future to solve the problem. Many theories have been proposed to solve the Reimann hypothesis but have failed. This article has resolved and ended the lack of proof for the Reimann hypothesis
... where 2 is the 2 × 2 identity matrix (Serre, 1973;Katok, 1992). The action of the group P L 2 (ℝ) on ℋ is as follows: ...
is in the PDF file. J. Bangladesh Acad. Sci. 48(2); 207-215: December 2024
... article (see also [2]), an even indefinite unimodular form over Z is uniquely defined by its rank r and signature τ and is equal to ...
We construct a canonical basis of two-cycles, on a K3 surface, in which the intersection form takes the canonical form . The basic elements are realized by formal sums of smooth submanifolds.
... It is well known (see [9] ch. II for example) that an element 2 r u (where u is a unit) of Z 2 is a square if and only if r is even and u ≡ 1 (mod 8). ...
We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry, and an application of Weil's theorem. We relate the weights appearing in the dual codes to the number of rational points on a family of genus 2 curves over a finite field.
... Consider n a odd number bigger than 1. Due to Dirichlet Theorem ( [9], Chapter 3, Lemma 3), there exists a prime number p such that p ≡ 1 (mod n). Denote the p-th primitive root of unity e 2iπ p by ζ p . ...
In this paper we construct some families of rotated -lattices with full diversity for any n. These lattices can be good for signal transmission over both Gaussian and Rayleigh fading channels. In order to get bounds for their minimum product distances, we show that the Z-modules used in \cite{sethoggier} to obtain rotated -lattices with n odd are ideals and find a sufficient condition for such ideals being principal ideals.
... The proof we provide seems to be new, simpler, and more direct. Theorem 2.6 is devoted to show that the complex Hermite-Gauss coefficients a p,q m,n ( |ν, χ) are lattice's functions in the sense of Serre [18,Chapter VII]. ...
We consider the reproducing kernel function of the theta Bargmann-Fock Hilbert space associated to given full-rank lattice and pseudo-character, and we deal with some of its analytical and arithmetical properties. Specially, the distribution and discreteness of its zeros are examined and analytic sets inside a product of fundamental cells is characterized and shown to be finite and of cardinal less or equal to the dimension of the theta Bargmann-Fock Hilbert space. Moreover, we obtain some remarkable lattice sums by evaluating the so-called complex Hermite-Taylor coefficients. Some of them generalize some of the arithmetic identities established by Perelomov in the framework of coherent states for the specific case of von Neumann lattice. Such complex Hermite-Taylor coefficients are nontrivial examples of the so-called lattice's functions according the Serre terminology. The perfect use of the basic properties of the complex Hermite polynomials is crucial in this framework.
We investigate Manin's conjecture for del Pezzo surfaces of degree five with a conic bundle structure, proving matching upper and lower bounds, and the full conjecture in the Galois general case.
We produce lattice extensions of a dense family of classical Schottky subgroups of the isometry group of d-dimensional hyperbolic space. The extensions produced are said to be systolic, since all loxodromic elements with short translation length are conjugate into the Schottky groups. Various corollaries are obtained, in particular showing that for all , the set of complex translation lengths realized by systoles of closed hyperbolic d-manifolds is dense inside the set of all possible complex translation lengths. We also prove an analogous result for arithmetic hyperbolic d-manifolds.
A positive integer m is called a Hall number if any finite group of order precisely divisible by m has a Hall subgroup of order m. We prove that, except for the obvious examples, the three integers 12, 24 and 60 are the only Hall numbers, solving a problem proposed by Jiping Zhang.
In this paper a generalisation of Legendre’s three-square theorem to representations of two positive integers as sums of three squares for which the first square of each representation is the same is presented.
In this note, we introduce the first basics on Grothendieck rings for incidence geometries as a new motivic way and tool to study synthetic geometry. In this first instance, we concentrate on generalized quadrangles and related geometries. Many questions, new properties and insights arise along the way.
A desmic quartic surface is a birational model of the Kummer surface of the self-product of an elliptic curve. We recall the classical geometry of these surfaces and study their analogs in arbitrary characteristic. Moreover, we discuss the cubic line complex \frakG associated with the desmic tetrahedra introduced by G. Humbert. We prove that \frakG is a rational \bbQ-Fano threefold with 34 nodes and the group of projective automorphisms isomorphic to (\frakS_4\times \frakS_4)\rtimes 2.
This paper explores the algebraic conditions under which a cellular automaton with a non-linear local rule exhibits surjectivity and reversibility. We also analyze the role of permutivity as a key factor influencing these properties and provide conditions that determine whether a non-linear CA is (bi)permutive. Through theoretical results and illustrative examples, we characterize the relationships between these fundamental properties, offering new insights into the dynamical behavior of non-linear CA.
We develop a version of the Hardy-Littlewood circle method to obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments in several variables. As an application, we count the number of fibers with a rational point in families of high-dimensional Ch\^atelet varieties, allowing for arbitrarily large subordinate Brauer groups.
The mirror symmetry among Calabi-Yau manifolds is mysterious, however, the mirror operation in 2d N=(2,2) supersymmetric conformal field theory (SCFT) is an elementary operation. In this paper, we mathematically formulate SCFTs using unitary full vertex operator superalgebras (full VOAs) and develop a cohomology theory of unitary SCFTs (aka holomorphic / topological twists). In particular, we introduce cohomology rings, Hodge numbers, and the Witten index of a unitary N=(2,2) full VOA, and prove that the cohomology rings determine 2d topological field theories and give relations between them (Hodge duality and T-duality). Based on this, we propose a possible approach to prove the existence of mirror Calabi-Yau manifolds for the Hodge numbers using SCFTs. For the proof, one need a construction of sigma models connecting Calabi-Yau manifolds and SCFTs which is still not rigorous, but expected properties are tested for the case of Abelian varieties and a special K3 surface based on some unitary N=(2,2) full VOAs.
Using computations related to vertex operator algebra characters we derive the simplest example of almost holomorphic modular forms.
Let p be an odd prime. In this paper, we write down the functional equations of Siegel Eisenstein series of degree n, level p with quadratic or trivial characters. First, we study the action of U(p)-operator in the space of Siegel Eisenstein series to get U(p)-eigenfunctions. Second, we show that the functional equations for such U(p)-eigenfunctions are quite simple and easy to write down by studying the constant term of the Fourier expansion, together with the functional equations of Eisenstein series due to Langlands. Consequently, we show that the matrix that represents functional equations of Siegel Eisenstein series are given by the products of the strict upper triangular matrices and the anti-diagonal matrix.
In this note we show how to use the determinant representations for correlation functions in CFT to derive new determinant formulas for powers of the modular discriminant expressed via deformed elliptic functions with parameters. In particular, we obtain counterparts of Garvan's formulas for the modular discriminant corresponding to the genus two Riemann surface case.
In this paper, we classify and study commutative algebras having a one-dimensional square. In finite dimension (see Theorem 3.9) besides some cases (which are all associative and nilpotent with nilpotency index 3), the algebras with zero annihilator are either of symplectic type (appearing only in characteristic 2), or evolution algebras. In infinite dimension, ruling out the associative case, we prove that our algebras are either of symplectic type or evolution algebras provided some technical conditions are satisfied (see Theorem 3.12). Our main tool is the theory of inner product spaces and quadratic forms. More precisely, if A denotes an evolution algebra with dim ( A 2 ) = 1 and a a generator of A 2 , then A admits an inner product ⟨ · , · ⟩ such that the product of A is given by x y = ⟨ x , y ⟩ a . There are three classes to consider: a ∈ Ann ( A ) ;
a ∉ Ann ( A ) and a is isotropic relative to ⟨ · , · ⟩ ;
a ∉ Ann ( A ) and a is anisotropic relative to ⟨ · , · ⟩ .
The isomorphism problem among these objects is investigated. For some of these algebras, we have also determined the existence of faithful associative representations in certain Clifford algebras.
We study the monodromy of the following third order linear differential equation where is a parameter, is the Weierstrass -function with periods 1 and , and are constants such that the local exponents at the singularity 0 are three distinct integers, which can always be written as after a dual transformation, where . This ODE can be seen as the third order version of the well-known Lamé equation . We say that the monodromy is unitary if the monodromy group is conjugate to a subgroup of the unitary group. We show that if n, l are both odd, then the monodromy can not be unitary;
if n is odd and l is even, then there exist finite values of B such that the monodromy is the Klein four-group and hence unitary;
if n is even, then whether there exists B such that the monodromy is unitary depends on the choice of the period .
The methods of studying the second order Lamé equation cannot work here, and we need to develop different approaches to treat these different cases separately. These results have interesting applications to the integrable SU(3) Toda system in another work (Chen and Lin in J Differ Geom 127:899–943, 2024).
We study the Siegel modular variety A g ⊗ F ¯ p \mathscr {A}_g\otimes \overline {\mathbb {F}}_p of genus g g and its supersingular locus S g \mathscr {S}_g . As our main result we determine precisely when S g \mathscr {S}_g is irreducible, and we list all x x in A g ⊗ F ¯ p \mathscr {A}_g\otimes \overline {\mathbb {F}}_p for which the corresponding central leaf C ( x ) \mathscr {C}(x) consists of one point, that is, for which x x corresponds to a polarised abelian variety which is uniquely determined by its associated polarised p p -divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus g = 4 g=4 .
The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real place. On the other hand, we also study the growth of integral points on the three-dimensional punctured affine cone, as a quantitative version of strong approximation with Brauer--Manin obstruction for this quasi-affine variety.
We construct an example of a field and a del Pezzo surface of degree 2 over this field without points such that its automorphism group is isomorphic to which is the largest possible automorphism group of del Pezzo surface of degree 2 over an algebraically closed field of characteristic zero.
We prove that a geometrically integral smooth projective 3‐fold XX with nef anti‐canonical class and negative Kodaira dimension over a finite field Fq of characteristic p>5 and cardinality q=pe>19 has a rational point. Additionally, under the same assumptions on pp and qq, we show that a smooth projective 3‐fold XX with trivial canonical class and non‐zero first Betti number b1(X)≠0 has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi–Yau 3‐fold pairs over perfect fields.
Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by non-principal ideals yields simple constructions of further lattices including the Leech lattice.
We use theta series and modular forms to prove that Z^n is the only integral unimodular lattice of rank n without characteristic vectors of norm <n, i.e. the only integral unimodular lattice not containing a vector w such that (w,w)<n and 2|(v,v+w) for all lattice vectors v. By the work of Kronheimer and others on the Seiberg-Witten equation this yields an alternative proof of a theorem of Donaldson on the geometry of 4-manifolds.
Suppose k is a positive integer. In this work, we establish formulas for for the number of representations of integers by the quadratic forms for .
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