Answer to a question of Hung and Tiep on conductors of cyclotomic integers

We consider ℓ \ell -adic trace functions over finite fields taking values in cyclotomic integers, such as characters and exponential sums. Through ideas of Deligne and Katz, we explore probabilistic properties of the reductions modulo a prime ideal, exploiting especially the determination of their integral monodromy groups. In particular, this gives a generalization of a result of Lamzouri-Zaharescu on the distribution of short sums of the Legendre symbol reduced modulo an integer to all multiplicative characters and to hyper-Kloosterman sums.

Let $\mathscr{C}$ be a modular tensor category over an algebraically closed field $k$ of characteristic 0. Then there is the ubiquitous notion of the S-matrix $S(\mathscr{C})$ associated with the modular category. The matrix $S(\mathscr{C})$ is a symmetric matrix, its entries are cyclotomic integers and the matrix $(\dim \mathscr{C})^{-\frac{1}{2}}\cdot S(\mathscr{C})$ is a unitary matrix. Here $\dim \mathscr{C}\in k$ denotes the categorical dimension of $\mathscr{C}$ and it is a totally positive cyclotomic integer. Now suppose that we also have a modular autoequivalence $F:\mathscr{C}\to \mathscr{C}$. In this paper, we will define and study the notion of a crossed S-matrix associated with the modular autoequivalence $F$. We will see that the crossed S-matrix occurs as a submatrix of the usual S-matrix of some bigger modular category and hence the entries of a crossed S-matrix are also cyclotomic integers. We will prove that the crossed S-matrix (normalized by the factor $(\dim \mathscr{C})^{-\frac{1}{2}}$) associated with any modular autoequivalence is a unitary matrix. We will also prove that the crossed S-matrix is essentially the table of a certain semisimple commutative Frobenius $k$-algebra associated with the modular autoequivalence $F$. The motivation for most of our results comes from the theory of character sheaves on algebraic groups, where we expect that the transition matrices between irreducible characters and character sheaves can be obtained as certain crossed S-matrices. In the character theory of algebraic groups defined over finite fields, there is the notion of Shintani descent of Frobenius stable characters. We will define and study a categorical analogue of this notion of Shintani descent in the setting of modular categories.

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A_n or D_n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less then 5.

Many applications of fast fourier transforms (FFT's), such as computer- tomography, geophysical signal processing, high resolution imaging radars, and prediction filters, require high precision output. The usual method of fixed point computation of FFT's of vectors of length 2l leads to an average loss of l/2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real time applications. Several researchers have noted that calculation of FFT's with algebraic integers avoids computational noise entirely, see, e.g., [3]. We will show that complex numbers can be approximated accurately by cyclotomic integers, and combine this idea with Chinese remaindering strategies in the cyclotomic integers to, roughly, give a O(b1+? L log (L)) algorithm to compute b-bit precision FFT's of length L. The first part of the paper will describe the FFT strategy, assuming good app..

In this chapter, we return to one of Gauss’s favourite themes, cyclotomic integers, and look at how they were used by Kummer, one of the leaders of the next generation of German number theorists. French and German mathematicians did not keep up-to-date with each other’s work, and for a brief, exciting moment in Paris in 1847 it looked as if the cyclotomic integers offered a chance to prove Fermat’s last theorem, only for Kummer to report, via Liouville, that problems with the concept of a prime cyclotomic integer wrecked that hope. Primality, however, turned out to be a much more interesting concept, and one of the roots of the concept of an ideal.

Lattice-based hard problems are a leading candidate for implementation in future public key cryptographic schemes due to their conjectured quantum resilience. Lattice-based problems offer certain advantages over non-lattice-based cryptosystems, such as a relatively short key length [3] and versatility, since lattice cryptosystems can offer both encryption schemes (to securely transmit data from sender to receiver) and signature schemes (used for a receiver to verify that information actually originated from the claimed sender) [2]. Notably they are also the only known class of problems which give rise to fully homomorphic encryption schemes, in which computations can be securely performed on encrypted data [1]. Many of the 2017 submissions to the NIST Post-Quantum Cryptography challenge are based on lattice problems [4].Some lattice cryptosystems, most notably the ring-LWE [7] scheme proposed in 2013, rely on solving a hard problem over a subclass of lattices known as ideal lattices. However, it is currently unknown whether the additional algebraic structure found in ideal lattices make them less secure for cryptographic purposes, although it is widely conjectured that the ideal case is as secure as the general case [1].In this poster presentation, we give an overview of past attempts to approach a lattice hard problem known as the shortest vector problem (SVP) in a class of ideal lattices generated using the cyclotomic integers, which is a type of mathematical object known as a ring. The cyclotomic integers have a lot of algebraic structure, and some researchers have speculated that this structure could potentially make these lattices less secure [6]. The discovery that the ideal-lattice based cryptosystem Soliloquy is not quantum-secure [5] has motivated cryptographers to examine the feasibility of using new types of rings to generate lattices, such as the variant of the crytosystem NTRU proposed in [6]. This poster will include our preliminary results on the security of the SVP in ideal lattices generated in a ring which has been previously unstudied and discuss the practically of using the ring in place of the cyclotomic integers in some lattice cryptosystems.

The anti-field-descent method

Many applications of fast Fourier transforms (FFTs), such as computer tomography, geophysical signal processing, high-resolution imaging radars, and prediction filters, require high-precision output. An error analysis reveals that the usual method of fixed-point computation of FFTs of vectors of length 2/sup l/ leads to an average loss of l/2 bits of precision. This phenomenon, often referred to as computational noise, causes major problems for arithmetic units with limited precision which are often used for real-time applications. Several researchers have noted that calculation of FFTs with algebraic integers avoids computational noise entirely. We combine a new algorithm for approximating complex numbers by cyclotomic integers with Chinese remaindering strategies to give an efficient algorithm to compute b-bit precision FFTs of length L. More precisely, we approximate complex numbers by cyclotomic integers in Z[e(2/spl pi/i/2/sup n/)] whose coefficients, when expressed as polynomials in e(2/spl pi/i/2/sup n/), are bounded in absolute value by some integer M. For fixed n our algorithm runs in time O(log(M)), and produces an approximation with worst case error of O(1/M(2/sup n-2/-1)). We prove that this algorithm has optimal worst case error by proving a corresponding lower bound on the worst case error of any approximation algorithm for this task. The main tool for designing the algorithms is the use of the cyclotomic units, a subgroup of finite index in the unit group of the cyclotomic field. First implementations of our algorithms indicate that they are fast enough to be used for the design of low-cost high-speed/high-precision FFT chips.

According to a theorem of Masbaum and Wenzl [11], the Turaev-Viro invariant of a 3-manifold associated with the modular categories constructed from UqslN is a cyclotomic integer when q is of prime order. We extend this result to premodular categories (where the S-matrix need not be invertible) of 'type A of a more general kind. One defines premodular categories of type A, rank N and level K over C depending on two complex parameters a and u, with q=s2 a root of unity of order l=N+k and u a N-th root of a , in the case SL N or more generally any root of unity (a variant introduced by Blanchet, [2]). There are essentially two methods for constructing these premodular categories: through quantum groups, and through Hecke algebras. We carry out both constructions, and check that they yield the same result. A 'local handle-slide property' for a premodular category (1.6.4.) characterizes those for which the Turaev-Viro invariant TV is defined. This applies to premodular categories of type A. It turns out (3.3.6.) that TV is defined precisely when the premodular category is modularizable in the sense of [4] (assuming 5 is of order 2l) We show that the Turaev-Viro invariant associated with a such a premodular category, when denned, is a cyclotomic integer if l=N+K is prime; this holds also for PGL. More precisely, we show that such a category is 'defined' over . Extending a criterion of [11] to the premodular case, we conclude that the Turaev-Viro invariant lies in k.

We obtain an upper bound for the absolute value of cyclotomic integers which has strong implications on several combinatorial structures including (relative) difference sets, quasiregular projective planes, planar functions, and group invariant weighing matrices. Our results are of broader applicability than all previously known nonexistence theorems for these combinatorial objects. We will show that the exponent of an abelian group G containing a (v,k,ג,n)-difference set cannot exceed ((2^(s-1).F(v,n))/n)^0.5where is the number of odd prime divisors of v and F(v,n) is a number-theoretic parameter whose order of magnitude usually is the squarefree part of . One of the consequences is that for any finite set P of primes there is a constant C such that exp(G) ≤ C|G|^0.5for any abelian group G containing a Hadamard difference set whose order is a product of powers of primes in P. Furthermore, we are able to verify Ryser's conjecture for most parameter series of known difference sets. This includes a striking progress towards the circulant Hadamard matrix conjecture. A computer search shows that there is no Barker sequence of length l with 13< l <4x10^12. Finally, we obtain new necessary conditions for the existence of quasiregular projective planes and group invariant weighing matrices including asymptotic exponent bounds for cases which previously had been completely intractable.

The character values of commutative quasi-thin schemes

We define higher Frobenius-Schur indicators for objects in linear pivotal monoidal categories. We prove that they are category invariants, and take values in the cyclotomic integers. We also define a family of natural endomorphisms of the identity endofunctor on a $k$-linear semisimple rigid monoidal category, which we call the Frobenius-Schur endomorphisms. For a $k$-linear semisimple pivotal monoidal category -- where both notions are defined --, the Frobenius-Schur indicators can be computed as traces of the Frobenius-Schur endomorphisms.

Some conjectures about cyclotomic integers

We prove the last of five outstanding conjectures made by R.M. Robinson from 1965 concerning small cyclotomic integers. In particular, given any cyclotomic integer $\beta$ all of whose conjugates have absolute value at most 5, we prove that the largest such conjugate has absolute value one of four explicit types given by two infinite classes and two exceptional cases. We also extend this result by showing that with the addition of one form, the conjecture is true for $\beta$ with magnitudes up to 5+1/25.

Integrality and gauge dependence of Hennings TQFTs