Search

In this paper we mainly consider the notion of Drinfeld double for two weak multiplier Hopf (⁎-)algebras which are paired with each other. Then we show that the Drinfeld double is again a weak multiplier Hopf (⁎-)algebra. Furthermore, we study integrals on the Drinfeld double. Finally, we establish the correspondence between modules over a Drinfeld double D (A) and Yetter-Drinfeld modules over a weak algebraic quantum group A. [ABSTRACT FROM AUTHOR]

We study isogeny classes of Drinfeld A -modules over finite fields k with commutative endomorphism algebra D , in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order A [ π ] of D generated by the Frobenius π occurs as an endomorphism ring by proving when it is locally maximal at π , and show that this happens if and only if the isogeny class is ordinary or k is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring E of a Drinfeld module ϕ up to D -linear equivalence acts on the isomorphism classes in the isogeny class of ϕ , in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases, which can be implemented as a computer algorithm. [ABSTRACT FROM AUTHOR]

We prove a result (see Theorem 1.1) regarding unlikely intersections of orbits for a given 1-parameter family of Drinfeld modules. We also advance a couple of general conjectures regarding unlikely intersections for algebraic families of Drinfeld modules (see Conjectures 1.3 and 2.3). [ABSTRACT FROM AUTHOR]

Let A = F q [ t ] be the polynomial ring over a finite field F q and let ϕ and ψ be A -Drinfeld modules. In this paper we consider the group Ext 1 (ϕ , ψ) with the Baer addition. We show that if rank ϕ > rank ψ then Ex t 1 (ϕ , ψ) has the structure of a t -module. We give complete algorithm describing this structure. We generalize this to the cases: Ex t 1 (Φ , ψ) where Φ is a t -module and ψ is a Drinfeld module and Ex t 1 (Φ , C ⊗ e) where Φ is a t -module and C ⊗ e is the e -th tensor product of Carlitz module. We also establish duality between Ext groups for t -modules and the corresponding adjoint t σ -modules. Finally, we prove the existence of " Hom − Ext " six-term exact sequences for t -modules and dual t -motives. As the category of t -modules is only additive (not abelian) this result is nontrivial. [ABSTRACT FROM AUTHOR]

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite field whose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; we obtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbits of irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods. We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSL n (q) (n ≥ 4) , PSL 3 (q) (q > 2) , or PSp 2 n (q) (n ≥ 3) , is isomorphic to a group algebra. [ABSTRACT FROM AUTHOR]

Let L (x) be any q -linearized polynomial with coefficients in F q , of degree q n. We consider the Galois group of L (x) + t x over F q (t) , where t is transcendental over F q. We prove that when n is a prime, the Galois group is always G L (n , q) , except when L (x) = x q n . Equivalently, we prove that the arithmetic monodromy group of L (x) / x is G L (n , q) , except when L (x) = x q n , and also equivalently, we prove that the image of the mod-(t) Galois representation of the Drinfeld module arising from L (x) is all of G L (n , q). [ABSTRACT FROM AUTHOR]

This paper focuses on the rank varieties for modules over a group algebra F E where E is an elementary abelian p -group and p is the characteristic of an algebraically closed field F. In the first part, we give a sufficient condition for a Green vertex of an indecomposable module to contain an elementary abelian p -group E in terms of the rank variety of the module restricted to E. In the second part, given a homogeneous algebraic variety V , we explore the problem on finding a small module with rank variety V. In particular, we examine the simple module D (k p − p + 1 , 1 p − 1) for the symmetric group S k p. [ABSTRACT FROM AUTHOR]

Let X be an elliptic curve or a ramifying hyperelliptic curve over F q. We will discuss how to factorize the coefficients of the exponential and logarithm series for a Hayes module over such a curve. This allows us to obtain v -adic convergence results for such exponential and logarithm series, for v a "finite" prime. As an application, we can show that the v -adic Goss L -value L v (1 , Ψ) is log-algebraic for suitable characters Ψ. • Bounds for radius for convergence for Hayes exponential and logarithm series v -adically • Factorization of integral functions over an integral ring can be done by breaking into their zeros • Special value of v -adic L -functions for Hayes modules over elliptic curves and hypereliptic curves [ABSTRACT FROM AUTHOR]

To study the set of torsion classes of a finite dimensional basic algebra over a field, we use a decomposition, called sign-decomposition, parameterized by elements of { ± 1 } n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ϵ ∈ { ± 1 } n there is a hereditary algebra A ϵ ! with radical square zero and a bijection between the set of torsion classes of A associated to ϵ and the set of faithful torsion classes of A ϵ !. Furthermore, this bijection preserves the property of being functorially finite. From a point of view of tilting theory, it implies that there is a bijection between the set of isomorphism classes of basic two-term silting complexes for A associated to ϵ and the set of isomorphism classes of basic tilting A ϵ ! -modules. As an application, we prove that the number of two-term tilting complexes over Brauer line algebras (respectively, Brauer cycle algebras) having n edges is ( 2 n n ) (respectively, 2 2 n − 1 if n is odd, and ∞ if n is even). [ABSTRACT FROM AUTHOR]

This paper deals with the fundamental period π ˜ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field F q (θ). Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in π ˜ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here. [ABSTRACT FROM AUTHOR]

For q an odd prime power, A = F q [ T ] , and F = F q (T) , let ψ : A → F { τ } be a Drinfeld A -module over F of rank 2 and without complex multiplication, and let p = p A be a prime of A of good reduction for ψ , with residue field F p. We study the growth of the absolute value | Δ p | of the discriminant of the F p -endomorphism ring of the reduction of ψ modulo p and prove that, for all p , | Δ p | grows with | p |. Moreover, we prove that, for a density 1 of primes p , | Δ p | is as close as possible to its upper bound | a p 2 − 4 μ p p | , where X 2 + a p X + μ p p ∈ A [ X ] is the characteristic polynomial of τ deg p. [ABSTRACT FROM AUTHOR]

The modular invariant of rank 1 Drinfeld modules is introduced and used to prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is the global function field version of Shimura's Main Theorem of Complex Multiplication. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the adelic Galois representation ρ φ : Gal (F q (T) sep / F q (T)) ⟶ lim ← a Aut (φ [ a ]) ≅ G L 3 (A ˆ) associated to the Drinfeld module φ over F q (T) of rank 3, φ defined by φ T = T + τ 2 + T q − 1 τ 3 , is surjective. [ABSTRACT FROM AUTHOR]

Let F q [ T ] be the polynomial ring over a finite field F q. We study the endomorphism rings of Drinfeld F q [ T ] -modules of arbitrary rank over finite fields. We compare the endomorphism rings to their subrings generated by the Frobenius endomorphism and deduce from this a refinement of a reciprocity law for division fields of Drinfeld modules proved in our earlier paper. We then use these results to give an efficient algorithm for computing the endomorphism rings and discuss some interesting examples produced by our algorithm. [ABSTRACT FROM AUTHOR]

Let q be a power of the prime number p , let K = F q (t) , and let r ⩾ 2 be an integer. For points a , b ∈ K which are F q -linearly independent, we show that there exist positive constants N 0 and c 0 such that for each integer ℓ ⩾ N 0 and for each generator τ of F q ℓ / F q , we have that for all except N 0 values λ ∈ F q ‾ , the corresponding specializations a (τ) and b (τ) cannot have orders of degrees less than c 0 log ⁡ log ⁡ ℓ as torsion points for the Drinfeld module Φ (τ , λ) : F q [ T ] ⟶ End F q ‾ (G a) (where G a is the additive group scheme), given by Φ T (τ , λ) (x) = τ x + λ x q + x q r . [ABSTRACT FROM AUTHOR]

This article suggests a new hypergeometric function for function fields and a new operator such that the Drinfeld logarithm is stable under it, and studies an equation satisfied by the hypergeometric function. As applications, our function is related to a supersingular polynomial of Drinfeld modules, a period of Drinfeld modules, and the Kochubei's polylogarithm, which are function-field analogues of well-known facts for the classical setting. Moreover, we generalize results for the Carlitz modules proven by Thakur to those for the Drinfeld ones. [ABSTRACT FROM AUTHOR]

We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over F q acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed τ -degree between Drinfeld F q [ X ] -modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper. [ABSTRACT FROM AUTHOR]

Let A = F q [ T ] be the polynomial ring over F q , and F be the field of fractions of A. Let ϕ be a Drinfeld A -module of rank r ≥ 2 over F. For all but finitely many primes p ◁ A , one can reduce ϕ modulo p to obtain a Drinfeld A -module ϕ ⊗ F p of rank r over F p = A / p. The endomorphism ring E p = End F p (ϕ ⊗ F p) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K , and let π p ∈ E p be the Frobenius endomorphism of ϕ ⊗ F p. Then we have the inclusion of orders A [ π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg (ϕ) = A , then for arbitrary non-zero ideals n , m of A there are infinitely many p such that n divides the index χ (E p / A [ π p ]) and m divides the index χ (O p / E p). We show that the index χ (E p / A [ π p ]) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r = 2 case we describe an algorithm for computing the orders A [ π p ] ⊂ E p ⊂ O p , and give some computational data. [ABSTRACT FROM AUTHOR]

Let p be a rational prime and q a power of p. Let n be a non-constant monic polynomial in F q [ t ] which has a prime factor of degree prime to q − 1. In this paper, we define a Drinfeld modular curve Y 1 Δ (n) over A [ 1 / n ] and study the structure around cusps of its compactification X 1 Δ (n) , in a parallel way to Katz-Mazur's work on classical modular curves. Using them, we also define a Hodge bundle over X 1 Δ (n) such that Drinfeld modular forms of level Γ 1 (n) , weight k and some type are identified with global sections of its k -th tensor power. [ABSTRACT FROM AUTHOR]

We give an effective algorithm to determine the endomorphism ring of a Drinfeld module, both over its field of definition and over a separable or algebraic closure thereof. Using previous results we deduce an effective description of the image of the adelic Galois representation associated to the Drinfeld module, up to commensurability. We also give an effective algorithm to decide whether two Drinfeld modules are isogenous, again both over their field of definition and over a separable or algebraic closure thereof. Questions of efficiency are left completely out of consideration. [ABSTRACT FROM AUTHOR]

We study tensor powers of rank 1 sign-normalized Drinfeld A -modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of A -motives, we find explicit formulas for the A -action of these modules. Then, by developing the theory of vector-valued Anderson generating functions, we give formulas for the period lattice of the associated exponential function. [ABSTRACT FROM AUTHOR]

We generalize some results of Greither and Popescu to a geometric Galois cover X → Y which appears naturally for example in extensions generated by p n -torsion points of a rank 1 normalized Drinfeld module (i.e. in subextensions of Carlitz–Hayes cyclotomic extensions of global fields of positive characteristic). We obtain a description of the Fitting ideal of class groups (or of their dual) via a formula involving Stickelberger elements and providing a link (similar to the one in [1]) with Goss ζ -function. [ABSTRACT FROM AUTHOR]

Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden. [ABSTRACT FROM AUTHOR]

We present a novel randomized algorithm to factor polynomials over a finite field F q of odd characteristic using rank 2 Drinfeld modules with complex multiplication. The main idea is to compute a lift of the Hasse invariant (modulo the polynomial f ∈ F q [ x ] to be factored) with respect to a random Drinfeld module ϕ with complex multiplication. Factors of f supported on prime ideals with supersingular reduction at ϕ have vanishing Hasse invariant and can be separated from the rest. Incorporating a Drinfeld module analogue of Deligne's congruence, we devise an algorithm to compute the Hasse invariant lift, which turns out to be the crux of our algorithm. The resulting expected runtime of n 3 / 2 + ε (log ⁡ q) 1 + o (1) + n 1 + ε (log ⁡ q) 2 + o (1) to factor polynomials of degree n over F q matches the fastest previously known algorithm, the Kedlaya-Umans implementation of the Kaltofen-Shoup algorithm. [ABSTRACT FROM AUTHOR]

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight 3 2 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves. [ABSTRACT FROM AUTHOR]

In 2000, based on his procedure for constructing explicit towers of modular curves, Elkies deduced explicit equations of rank-2 Drinfeld modular curves which coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth. In 2015, Bassa, Beelen, Garcia, and Stichtenoth constructed a celebrated (recursive and good) tower (BBGS-tower for short) of curves and outlined a modular interpretation of the defining equations. Soon after that, Gekeler studied in depth the modular curves coming from sparse Drinfeld modules. In this paper, to establish a link between these existing results, we propose and prove a generalized Elkies' Theorem which tells in detail how to directly describe a modular interpretation of the equations of rank- m Drinfeld modular curves with m ⩾ 2. [ABSTRACT FROM AUTHOR]

In 2010, V. Futorny and S. Ovsienko gave a realization of U (gl n) as a subalgebra of the ring of invariants of a certain noncommutative ring with respect to the action of S 1 × S 2 × ⋯ × S n , where S j is the symmetric group on j variables. An interesting question is what a similar algebra would be in the invariant ring with respect to a product of alternating groups. In this paper we define such an algebra, denoted A (gl n) , and show that it is a Galois ring. For n = 2 , we show that it is a generalized Weyl algebra, and for n = 3 provide generators and a list of verified relations. We also discuss some techniques to construct Galois orders from Galois rings. Additionally, we study categories of finite-dimensional modules and generic Gelfand-Tsetlin modules over A (gl n). Finally, we discuss connections between the Gelfand-Kirillov Conjecture, A (gl n) , and the positive solution to Noether's problem for the alternating group. [ABSTRACT FROM AUTHOR]

We construct a normal projective rigid analytic compactification of an arbitrary Drinfeld modular variety whose boundary is stratified by modular varieties of smaller dimensions. This generalizes work of Kapranov. Using an algebraic modular compactification that generalizes Pink and Schieder's, we show that the analytic compactification is naturally isomorphic to the analytification of Pink's normal algebraic compactification. We interpret analytic Drinfeld modular forms as the global sections of natural ample invertible sheaves on the analytic compactification and deduce finiteness results for spaces of such modular forms. [ABSTRACT FROM AUTHOR]

• Topological properties of the prime spectrum of a semimodule are studied. • A multiplication semimodule is finitely generated iff the prime spectrum is compact. • For a multiplication semimodule, every basic open set in the prime spectrum is compact. The objective of this paper is to study topological properties of the prime spectrum of a unitary semimodule over a semiring with zero and nonzero identity. We define a top semimodule over a semiring, show that the prime spectrum is a T 0 -space, and prove that each irreducible closed subset of the prime spectrum has a generic point and that the prime spectrum is a compact space if the top semimodule is finitely generated. For a multiplication semimodule over a commutative semiring, we reveal the structure of the radical of a subsemimodule and prove that the multiplication semimodule is finitely generated iff the prime spectrum is a compact space, that in the prime spectrum, every basic open set is compact and the intersection of finitely many basic open sets is also compact, and that the prime spectrum is a spectral space if the multiplication semimodule is finitely generated. [ABSTRACT FROM AUTHOR]

We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and A -motives. We also explain the necessary background on abelian varieties, Drinfeld modules and A -motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication. [ABSTRACT FROM AUTHOR]

We complete the proof of a Siegel type statement for finitely generated Φ-submodules of G a under the action of a Drinfeld module Φ. [ABSTRACT FROM AUTHOR]

We consider the deformed versions of the classical Howe dual pairs (O (r , C) , s l (2 , C)) and (O (r , C) , s p o (2 | 2 , C)) in the context of a rational Cherednik algebra H c = H c (W , h) associated to a finite Coxeter group W at the parameters c and t = 1. For the first pair, we compute the centraliser of the well-known copy of s ≅ s l (2 , C) inside H c. For the second pair, we show that the classical copy of g ≅ s p o (2 | 2 , C) inside the Weyl-Clifford algebra W ⊗ C deforms to a Lie superalgebra inside H c ⊗ C and compute its centraliser algebra. For a generic parameter c such that the standard H c -module is unitary, we compute the joint ((H c) s , s) - and ((H c ⊗ C) g , g) -decompositions of the relevant modules. [ABSTRACT FROM AUTHOR]

In this article we develop the multilinear theory of Drinfeld displays and use it to construct tensor products, symmetric and exterior powers of z -divisible local Anderson modules, which are the function fields analogs of p -divisible groups. [ABSTRACT FROM AUTHOR]

Let A = F q [ T ] be the polynomial ring over F q , and F be the field of fractions of A. Let ϕ be a Drinfeld A -module of rank r ≥ 2 over F. For all but finitely many primes p ◁ A , one can reduce ϕ modulo p to obtain a Drinfeld A -module ϕ ⊗ F p of rank r over F p = A / p. The endomorphism ring E p = End F p (ϕ ⊗ F p) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K , and let π p ∈ E p be the Frobenius endomorphism of ϕ ⊗ F p. Then we have the inclusion of orders A [ π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg (ϕ) = A , then for arbitrary non-zero ideals n , m of A there are infinitely many p such that n divides the index χ (E p / A [ π p ]) and m divides the index χ (O p / E p). We show that the index χ (E p / A [ π p ]) is related to a reciprocity law for the extensions of F arising from the division points of ϕ. In the rank r = 2 case we describe an algorithm for computing the orders A [ π p ] ⊂ E p ⊂ O p , and give some computational data. [ABSTRACT FROM AUTHOR]

Abstract Introduced by Anglès, Pellarin, and Tavares Ribeiro, Drinfeld modules over Tate algebras are closely connected to Anderson log-algebraicity identities, Pellarin L -series, and Taelman class modules. In the present paper we define the de Rham map for Drinfeld modules over Tate algebras, and we prove that it is an isomorphism under natural hypotheses. As part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of Drinfeld modules over Tate algebras. [ABSTRACT FROM AUTHOR]

Abstract We study compactifications of Drinfeld half-spaces over a finite field. In particular, we construct a purely inseparable endomorphism of Drinfeld's half-space Ω (V) over a finite field k that does not extend to an endomorphism of the projective space P (V). This should be compared with theorem of Rémy, Thuillier and Werner that every k -automorphism of Ω (V) extends to a k -automorphism of P (V). Our construction uses an inseparable analogue of the Cremona transformation. We also study foliations on Drinfeld's half-spaces. This leads to various examples of interesting varieties in positive characteristic. In particular, we show a new example of a non-liftable projective Calabi–Yau threefold in characteristic 2 and we show examples of rational surfaces with klt singularities, whose cotangent bundle contains an ample line bundle. [ABSTRACT FROM AUTHOR]

Abstract We investigate color Lie rings over finite group algebras and their universal enveloping algebras. We exhibit these universal enveloping algebras as PBW deformations of skew group algebras: Every color Lie ring over a finite group algebra with a particular Yetter–Drinfeld structure has a universal enveloping algebra that is a quantum Drinfeld orbifold algebra. Conversely, every quantum Drinfeld orbifold algebra of a particular type arising from the action of an abelian group is the universal enveloping algebra of some color Lie ring over the group algebra. One consequence is that these quantum Drinfeld orbifold algebras are braided Hopf algebras. [ABSTRACT FROM AUTHOR]

Abstract We prove that the notion of Drinfeld center defines a functor from the category of indecomposable multi-tensor categories with morphisms given by bimodules to that of braided tensor categories with morphisms given by monoidal bimodules. Moreover, we apply some ideas from the physics of topological orders to prove that the center functor restricted to indecom- posable multi-fusion categories (with additional conditions on the target category) is fully faithful. As byproducts, we provide new proofs to some important known results in fusion categories. In physics, this fully faithful functor gives the precise mathematical description of the boundary-bulk relation for 2+1D anomaly-free topological orders with gapped boundaries. [ABSTRACT FROM AUTHOR]

Abstract We propose and rigorously analyze two randomized algorithms to factor univariate polynomials over finite fields using rank 2 Drinfeld modules. The first algorithm estimates the degree of an irreducible factor of a polynomial from Euler–Poincaré characteristics of random Drinfeld modules. Knowledge of a factor degree allows one to rapidly extract all factors of that degree. As a consequence, the problem of factoring polynomials over finite fields in time nearly linear in the degree is reduced to finding Euler–Poincaré characteristics of random Drinfeld modules with high probability. The second algorithm is a random Drinfeld module analogue of Berlekamp's algorithm. During the course of its analysis, we prove a new bound on degree distributions in factorization patterns of polynomials over finite fields in certain short intervals. [ABSTRACT FROM AUTHOR]

Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all Drinfeld orbifold algebras for symmetric groups acting by the natural permutation representation. This provides, for nonabelian groups, infinite families of examples of Drinfeld orbifold algebras that are not graded Hecke algebras. We include explicit descriptions of the maps recording commutator relations and show there is a one-parameter family of such maps supported only on the identity and a three-parameter family of maps supported only on 3-cycles and 5-cycles. Each commutator map must satisfy properties arising from a Poincaré–Birkhoff–Witt condition on the algebra, and our analysis of the properties illustrates reduction techniques using orbits of group element factorizations and intersections of fixed point spaces. [ABSTRACT FROM AUTHOR]

We present an approach of calculating the group of braided autoequivalences of the category of representations of the Drinfeld double of a finite dimensional Hopf algebra H and thus the Brauer–Picard group of H - mod . We consider two natural subgroups and a subset as candidates for generators. In this article H is the group algebra of a finite group G . As our main result we prove that any element of the Brauer–Picard group, fulfilling an additional cohomological condition, decomposes into an ordered product of our candidates. For elementary abelian groups G our decomposition reduces to the Bruhat decomposition of the Brauer–Picard group, which is in this case a Lie group over a finite field. Our results are motivated by and have applications to symmetries and defects in 3 d -TQFT and group extensions of fusion categories. [ABSTRACT FROM AUTHOR]

Let L be a finite extension of the rational function field in one variable over a finite field F q and E be a Drinfeld module defined over L. Given finitely many elements in E (L) , this paper aims to prove that linear relations among these points can be characterized by solutions of an explicitly constructed system of homogeneous linear equations over F q [ t ]. As a consequence, we show that there is an explicit upper bound for the size of the generators of linear relations among these points. This result can be regarded as an analogue of a theorem of Masser for finitely many K -rational points on an elliptic curve defined over a number field K. [ABSTRACT FROM AUTHOR]

In this paper it is shown that any irreducible representation of a Drinfeld double D ( A ) of a semisimple Hopf algebra A can be obtained as an induced representation from a certain Hopf subalgebra of D ( A ) . This generalizes a well known result concerning the irreducible representations of Drinfeld doubles of finite groups [11] . Using this description we also give a formula for the fusion rules of semisimple Drinfeld doubles. This shows that the Grothendieck rings of these Drinfeld doubles have a ring structure similar to the Grothendieck rings of Drinfeld doubles of finite groups. [ABSTRACT FROM AUTHOR]

We give a product expansion for the Drinfeld discriminant function in arbitrary rank r , which generalises the formula obtained by Gekeler for the rank 2 Drinfeld discriminant function. This enables one to compute the u -expansion of this function much more efficiently. The formula in this article uses an r − 1 -dimensional parameter and as such provides a nice counterpoint to the formula previously obtained by Hamahata, which is written in terms of several 1-dimensional parameters. [ABSTRACT FROM AUTHOR]

In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line X over finite fields. These polynomials are then used to define the generic Ringel–Hall algebra of X as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein. [ABSTRACT FROM AUTHOR]

We determine the representation of the Galois group for the cyclotomic function fields in characteristic p > 0 induced by the natural action on the space of holomorphic differentials via construction of an explicit canonical basis of differentials. This includes those cases which present wild ramification and finite automorphism groups with non-cyclic p -part, which have remained elusive. We also obtain information on the gap sequences of ramified primes and extend these results to rank one Drinfel'd modules. [ABSTRACT FROM AUTHOR]

This paper expands upon our results for the arithmetic of Drinfeld modular forms of level T [12] by providing an interpretation from a representation theoretic point of view. We identify GL ( 2 , F q ) -modules that arise naturally from the theory of Drinfeld modular forms of level T with classical GL ( 2 , F q ) -modules. Using the arithmetic of the so-called modified Eisenstein series all isomorphisms are stated explicitly. In particular, we examine the close connection between Drinfeld modular forms of level T and the theory of symmetric powers of the tautological representation of GL ( 2 , F q ) described in previous work [13] . [ABSTRACT FROM AUTHOR]

In this article we give a Drinfeld modular interpretation for various towers of function fields meeting Zink's bound. [ABSTRACT FROM AUTHOR]

We describe Lagrangian algebras in twisted Drinfeld centres for finite groups. Using the full centre construction, we establish a 1-1 correspondence between Lagrangian algebras and module categories over pointed fusion categories. [ABSTRACT FROM AUTHOR]

The Drinfeld double D of the bosonization of a finite-dimensional Nichols algebra B ( V ) over a finite non-abelian group G is called a quantum group at a non-abelian group . We introduce Verma modules over such a quantum group D and prove that a Verma module has simple head and simple socle. This provides two bijective correspondences between the set of simple modules over D and the set of simple modules over the Drinfeld double D ( G ) . As an example, we describe the lattice of submodules of the Verma modules over the quantum group at the symmetric group S 3 attached to the 12-dimensional Fomin–Kirillov algebra, computing all the simple modules and calculating their dimensions. [ABSTRACT FROM AUTHOR]