Search

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn, n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216. [ABSTRACT FROM AUTHOR]

Let q$q$ be a prime power and F=Fq(T)$F=\mathbb {F}_q(T)$ be the rational function field over Fq$\mathbb {F}_q$, the field with q$q$ elements. Let ϕ$\phi$ be a Drinfeld module over F$F$ and p$\mathfrak {p}$ be a nonzero prime ideal of A:=Fq[T]$A:=\mathbb {F}_q[T]$. Over the constant Zp$\mathbb {Z}_p$‐extension of F$F$, we introduce the fine Selmer group associated to the p$\mathfrak {p}$‐primary torsion of ϕ$\phi$. We show that it is a cofinitely generated module over Ap$A_{\mathfrak {p}}$. This proves an analogue of Iwasawa's μ=0$\mu =0$ conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article. [ABSTRACT FROM AUTHOR]

In the present paper, we establish an analytic continuation of Drinfeld logarithms by using the techniques introduced in Furusho (Tunis J Math 4(3):559–586, 2022). This result can be seen as an analogue of the analytic continuation of the elliptic integrals of the first kind for Drinfeld modules. [ABSTRACT FROM AUTHOR]

Let F q be the finite field with q ≥ 5 elements, A : = F q [ T ] and F : = F q (T) . Assume that q is odd and take | · | to be the absolute value at ∞ that is normalized by | T | = q . Given a pair w = (g 1 , g 2) ∈ A 2 with g 2 ≠ 0 , consider the associated Drinfeld module ϕ w : A → A { τ } of rank 2 defined by ϕ T w = T + g 1 τ + g 2 τ 2 . Fix integers c 1 , c 2 ≥ 1 and define | w | : = max { | g 1 | 1 c 1 , | g 2 | 1 c 2 } . I show that when ordered by height, there is a positive density of pairs w = (g 1 , g 2) , such that the T-adic Galois representation attached to ϕ w is surjective. [ABSTRACT FROM AUTHOR]

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the -primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa's result on the p -primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties. [ABSTRACT FROM AUTHOR]

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]

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over ℚ with 1 2 -torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of 1 2 -torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrize such pairs of elliptic curves. A key ingredient in the proof is to construct simple (algebraic) conditions for the 2 , 3 or 4 -torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the j -invariants of the pair of elliptic curves. [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 give asymptotics for the number of isomorphism classes of Drinfeld F q [ T ] -modules over F q (T) of a given height, which satisfy prescribed sets of local conditions. This is done by relating our problem to a problem about counting points on weighted projective stacks. Our results for counting points of bounded height on weighted projective stacks over global function fields may be of independent interest. [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]

We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension, GK-dimension for short, through the study of Nichols algebras over D ∞ , the infinite dihedral group.We find all the irreducible Yetter-Drinfeld modules V over D ∞ , and determine which Nichols algebras B (V) of V are finite GK-dimensional. [ABSTRACT FROM AUTHOR]

Let F be a finite field and G = GL (6 , F). In this paper, we explicitly describe the structure of the twisted Jacquet module π N , ψ A where A is a rank 2 matrix and π is an irreducible cuspidal representation of G. [ABSTRACT FROM AUTHOR]

The classification of irreducible conformal B (p) -modules of finite rank was given in [5] for 0 ≠ p ∈ C and in [6] for p = 0. In this paper, we use a uniform way to give a short proof of this classification. [ABSTRACT FROM AUTHOR]