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.
We find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type $D_{4n}$ and the cosets in their integral duals, in terms of the well-known Essenstein series $E_4(z)$ and Ramanujan series $\Dlt_{24}(z)$. Using these two identities, we determine the theta series of certain infinite families of postive definite even unimodular lattices obtained by gluing finite copies of the root lattices of the finite-dimensional simple Lie algebras of type $D_{2n}$. It turns out that these theta series are weighted symmetric polynomials of two…
On the degree-$2$ Siegel theta series of extremal even unimodular lattices of ranks $48$, $72$, $96$, and $120$
Harvard Business School（哈佛商学院）；Department of Economics and Center of Mathematical Sciences and Applications, Harvard University（哈佛大学经济学系与数学科学及应用中心）；a16z crypto
We look at genera of even unimodular lattices of rank $12$ over the ring of integers of $\mathbb{Q}(\sqrt{5})$ and of rank $8$ over the ring of integers of $\mathbb{Q}(\sqrt{3})$, using 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 instances 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 $\mathbb…
We give a classification of the lattices of rank r=4, r=8 and r=12 over \Q(\sqrt{-3}), which are even and unimodular \Z-lattices. Using this classification we construct the associated theta series, which are Hermitian modular forms, and compute the filtration of cusp forms.
Let $\vartheta := \frac{-1+\sqrt{5}}{2}$ be the golden ratio. A golden lattice is an even unimodular $\Z[\vartheta ]$-lattice of which the Hilbert theta series is an extremal Hilbert modular form. We construct golden lattices from extremal even unimodular lattices and obtain families of dense modular lattices.
We show that there is a unique extremal even unimodular lattice of dimension 48 which has an automorphism of order 5 of type 5-(8,16)-8. Since the three known extremal lattices do not admit such an automorphism, this provides a new example of an extremal even unimodular lattice in dimension 48.
We derive a mass formula for n-dimensional unimodular lattices having any prescribed root system. We use Katsurada's formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimension n < 31. In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimension n < 31, verifying Bacher and Venkov's enumerations in dimensions 27 and 28. We also compute better lower bounds on the numbe…
The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in {\sc Magma} using the very sophisticated algorithm for computing class groups of algebraic number fields written by Steve Donnelly it is shown that the extremal even unimodular lattice $Γ_{72}$ from \cite{dim72} is the unique extremal even unimodular lattice of dimension 72 that admits a large automorphism, where a $d\times d$ matrix is called large, if its minimal polynomial has an irreducible factor of degree $> d/2$.
Given a unimodular lattice $Λ\subseteq \mathbb{R}^2$ consider the counting function $\mathcal{N}_Λ(T)$ counting the number of lattice points of norm less than $T$, and the remainder $\mathcal{R}_Λ(T)=\mathcal{N}(T)-πT^2$. We give an elementary proof that the mean square of the remainder over the set of all shears of a unimodular lattice is bounded by $O(T\log^2(T))$.
It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.
We extend the results of Ozeki on the configurations of extremal even unimodular lattices. Specifically, we show that if L is such a lattice of rank 56, 72, or 96, then L is generated by its minimal-norm vectors.
The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are computed using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order $m$ with $φ(m) > 24$. In particular the lattice $P_{48n}$ is the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.
In this article, we study the relation between lattice basis and successive minima and give an estimate for the measure-theoretical distribution of successive minima. As consequences, we also discuss some logarithm laws associated to higher sucessive minima.
We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1.
In this paper, we construct odd unimodular lattices in dimensions n=36,37 having minimum norm 3 and 4s=n-16, where s is the minimum norm of the shadow. We also construct odd unimodular lattices in dimensions n=41,43,44 having minimum norm 4 and 4s=n-24.
Lattice coding over a Gaussian wiretap channel, where an eavesdropper listens to transmissions between a transmitter and a legitimate receiver, is considered. A new lattice invariant called the secrecy gain is used as a code design criterion for wiretap lattice codes since it was shown to characterize the confusion that a chosen lattice can cause at the eavesdropper: the higher the secrecy gain of the lattice, the more confusion. In this paper, a formula for the secrecy gain of unimodular lattices is derived. Secrecy gains of extremal odd unimodular lattices as well as unimodular lattices in d…
We consider even unimodular lattices which are critical for potential energy with respect to Gaussian potential functions in the manifold of lattices having point density 1. All even unimodular lattices up to dimension 24 are critical. We show how to determine the Morse index in these cases. While all these lattices are either local minima or saddle points, we find lattices in dimension 32 which are local maxima. Also starting from dimension 32 there are non-critical even unimodular lattices.
We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the $E_8$ lattice (resp. $E_8^2$ and $Λ_{16}$) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for co…
We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan's fifth order mock theta functions appear as components.
We characterize conjugacy classes of isometries of odd prime order in unimodular Z-lattices. This is applied to give a complete classification of odd prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds up to deformation and birational conjugacy.
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.
An even unimodular 72-dimensional lattice $Γ$ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary quadratic number field with discriminant $-7$. The automorphism group of $Γ$ contains the absolutely irreducible rational matrix group $(\PSL_2(7) \times \SL _2(25)) : 2$.
An $s$-extremal optimal unimodular lattice in dimension $52$ is constructed for the first time. This lattice is constructed from a certain self-dual $\mathbb{F}_5$-code by Construction A. In addition, as neighbors of the lattice, two more $s$-extremal optimal unimodular lattices are constructed.
If $P$ is a lattice polytope (that is, the convex hull of a finite set of lattice points in $\mathbf{R}^n$), then every sum of $h$ lattice points in $P$ is a lattice point in the $h$-fold sumset $hP$. However, a lattice point in the $h$-fold sumset $hP$ is not necessarily the sum of $h$ lattice points in $P$. It is proved that if the polytope $P$ is a union of unimodular simplices, then every lattice point in the $h$-fold sumset $hP$ is the sum of $h$ lattice points in $P$.
We characterize the irreducible polynomials that occur as a characteristic polynomial of an automorphism of an even unimodular lattice of given signature, generalizing a theorem of Gross and McMullen. As part of the proof, we give a general criterion in terms of Witt groups for a bilinear form equipped with an action of a group G over a discretely valued field to contain a unimodular G-stable lattice.
A recent line of work on lattice codes for Gaussian wiretap channels introduced a new lattice invariant called secrecy gain as a code design criterion which captures the confusion that lattice coding produces at an eavesdropper. Following up the study of unimodular lattice wiretap codes [1], this paper investigates 2- and 3-modular lattices and compares them with unimodular lattices. Most even 2- and 3-modular lattices are found to have better performance, that is, a higher secrecy gain than the best unimodular lattices in dimension n, n is between 2 and 23. Odd 2-modular lattices are consider…
In this paper we first of all determine all possible genera of (odd and even) definite unimodular lattices over an imaginary-quadratic field. The main questions are whether the partial class numbers of lattices with given Steinitz class within one genus are equal for all occuring Steinitz classes and whether the partial masses of those partial genera are equal. We show that the answer to the first question in general is "no" by giving a counter example, while the answer to the second question is "yes" by proving a mass formula for partial masses. Finally, we determine a list of all single-clas…
The purpose of this paper is to show how to obtain the mass of a unimodular lattice from the point of view of the Bruhat-Tits theory. This is achieved by relating the local stabilizer of the lattice to a maximal parahoric subgroup of the special orthogonal group, and appealing to an explicit mass formula for parahoric subgroups developed by Gan, Hanke and Yu. Of course, the exact mass formula for positive defined unimodular lattices is well-known. Moreover, the exact formula for lattices of signature (1,n) (which give rise to hyperbolic orbifolds) was obtained by Ratcliffe and Tschantz, starti…
Irreducible rootless unimodular lattices generated by vectors of norm $3$
University of Science and Technology of China（中国科学技术大学）；CAS Wu Wen-Tsun Key Laboratory of Mathematics, University of Science and Technology of China（中国科学院吴文俊数学重点实验室，中国科学技术大学）；Toho University（东海大学）
In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40.
For some extremal (optimal) odd unimodular lattices L in dimensions n=12,16,20,32,36,40 and 44, we determine all positive integers k such that L contains a k-frame. This result yields the existence of an extremal Type I Zk-code of lengths 12,16,20,32,36,40 and 44 and a near-extremal Type I Zk-code of length 28 for positive integers k with only a few exceptions.
Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2,2}$ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2,2}$, a new interpretation of the Gauss product of binary forms brings new light to a number of related o…
For lengths up to 47 except 37, we determine the largest minimum Euclidean weight among all Type I Z4-codes of that length. We also give the first example of an optimal odd unimodular lattice in dimension 41 explicitly, which is constructed from some Type I Z4-code of length 41.
New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over $\mathbb{F}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over $\mathbb{F}_5$. In addition, the code implies a $[43,22,13]$ code over $\mathbb{F}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.
Let L be the even unimodular lattice of signature (2,10), In the paper [FS] we considered the subgroup O(L)^+ of index two in the orthogonal group. It acts biholomorphically on a ten dimensional tube domain H_{10}. We found a 715 dimensional space of modular forms with respect to the principal congruence subgroup of level two O^+(L)[2]. It defines an everywhere regular birational embedding of the related modular variety into the 714 dimensional projective space. In this paper, we prove that this space of orthogonal modular forms is related to a space of theta series. The main tool is a modular…
Following the paper "Note on theta series for Niemeier lattices", we study some congruence properties satisfied by the theta series associated with Niemeier lattices.
A Sharp Reverse Minkowski Inequality for the Gaussian Mass of Integral Unimodular Lattices Through Rank $32$
Harvard Business School（哈佛商学院）；Department of Economics and Center of Mathematical Sciences and Applications, Harvard University（哈佛大学经济与数学科学应用中心）；a16z crypto（a16z 加密）
We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different evaluations, we derive recurrences for the coefficients of mock theta functions, such as Hurwitz class numbers, Andrews' spt-function, and Ramanujan's mock theta functions.
This is an unpublished manuscript written in 1983-4. It contains several results about lattices (=integral quadratic forms) including the classification of the unimodular lattices in dimensions up to 25 and the construction of unimodular lattices with no roots in dimensions 26 and 27.
Given a polarization of an even unimodular lattice and integer $k\ge 1$, we define a family of unimodular lattices $L(M,N,k)$. Of special interest are certain $L(M,N,3)$ of rank 72. Their minimum norms lie in $\{4, 6, 8\}$. Norms 4 and 6 do occur. Consequently, 6 becomes the highest known minimum norm for rank 72 even unimodular lattices. We discuss how norm 8 might occur for such a $L(M,N,3)$. We note a few $L(M,N,k)$ in dimensions 96, 120 and 128 with moderately high minimum norms.
The space spanned by theta series of adjoints of maximal even lattices of exact level $N$ and determinant $N^2$ has the Weierstrass property and hence allows to define extremality for arbitrary squarefree level $N$. We find examples of such dual extremal lattices.
Let $n \in \mathbb{Z}_{\geq 3}.$ Given any Borel subset $A$ of $\mathbb{R}^n$ with finite and nonzero measure, we prove that the probability that the set of primitive points of a random full-rank unimodular lattice in $\mathbb{R}^n$ does not contain any $\mathbb{R}$-linearly independent subset of $A$ of cardinality $(n-2)$ is bounded from above by a constant multiple, which depends only on $n$, of $\left(\mathrm{vol}(A)\right)^{-1}.$ This generalizes a result that is jointly due to J. S. Athreya and G. A. Margulis (see \cite[Theorem 2.2]{Log}). We also generalize independent results of C. A. R…
In an earlier paper (math.NT/9906019) we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that 2|(v,w-v) for all v in L; it is known that the characteristic vectors all have norm congruent to n mod 8 and comprise a coset of 2L in L.] Here we use modular forms and the classification of unimodular lattices of rank <24 to find all L whose minimal characteristic vectors have norm n-8. Along the way we also obtain congruences and a lower bound on the kissin…
We present two families of lattice theta functions accompanying the family of lattice theta functions studied by Montgomery in [H.~Montgomery. Minimal theta functions. \textit{Glasgow Mathematical Journal}, 30(1):75--85, 1988]. The studied theta functions are generalizations of the Jacobi theta-2 and theta-4 functions. Contrary to Montgomery's result, we show that, among lattices, the hexagonal lattice is the unique maximizer of both families of theta functions. As an immediate consequence, we obtain a new universal optimality result for the hexagonal lattice among two-dimensional alternating …
We discuss relations between two different representations of hypothetical holomorphic NSR measures, based on two different ways of constructing the semi-modular forms of weight 8. One of these ways is to build forms from the ordinary Riemann theta constants and another -- from the lattice theta constants. We discuss unexpectedly elegant relations between lattice theta constants, corresponding to 16-dimensional self-dual lattices, and Riemann theta constants and present explicit formulae expressing the former ones through the latter. Starting from genus 5 the modular-form approach to construct…
We study the theta decomposition of Jacobi forms of nonintegral lattice index for a representation that arises in the theory of Weil representations associated to even lattices, and suggest possible applications.
We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated…
