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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 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]

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]

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]

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]

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]

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]

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of Kirillov and Ostrik (Adv Math 171(2):183–227, 2002) in the semisimple setup. Examples of non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided. In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification by Davydov (J Algebra 323(5):1321–1348, 2010) to arbitrary characteristic. [ABSTRACT FROM AUTHOR]

We obtain an equivariant class formula for z-deformation of Anderson modules. Under mild conditions, it allows us to get an equivariant class formula for Anderson modules. [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]

It is well known from the results of Ferrero and Kida [2,7] that the Z 2 -torsion part of the unramified abelian Iwasawa module X ∞ of any imaginary quadratic number field is trivial or cyclic of order 2. We will determine an infinite family of CM number fields, in which the Z 2 -torsion of the Iwasawa module X ∞ is of arbitrary large rank, giving also the exact value of the rank of X ∞ . [ABSTRACT FROM AUTHOR]

In this paper, for two weak Hopf monoids H and B with invertible antipode, we define a functor between the category of left-left H ⊗ B-Yetter--Drinfeld modules and the category of H-B-Long dimodules. We also show that, if moreover H is quasitriangular and B is coquasitriangular, this functor is a retraction of the well-known injective functor between left-left H-B-Long dimodules and left-left H ⊗ B-Yetter--Drinfeld modules. [ABSTRACT FROM AUTHOR]

Let (R,⋅,∘) be a braided Hopf brace, i.e. a Hopf brace in the category of Yetter–Drinfeld modules over a Hopf brace (H, ⋅, ∘). In this note, we give the sufficient and necessary conditions which make Radford’s biproduct (R♯H,⋅) and (R♯H∘,∘) become a Hopf brace. [ABSTRACT FROM AUTHOR]

For a quasi-triangular Hopf algebra (H, R), there is a notion of transmuted braided group HR of H introduced by Majid. The transmuted braided group HR is a Hopf algebra in the braided category H M . The R-adjoint-stable algebra associated with any simple left HR-comodule is defined by the authors, and is used to characterize the structure of all irreducible Yetter-Drinfeld modules in H H Y D . In this note, we prove for a semisimple factorizable Hopf algebra (H, R) that any simple subcoalgebra of HR is H-stable and the R-adjoint-stable algebra for any simple left HR-comodule is anti-isomorphic to H. As an application, we characterize all irreducible Yetter-Drinfeld modules. [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]

Let q be an odd prime power, denote by \mathbb {F}_q the finite field with q elements, and set A ≔\mathbb {F}_q[T], F ≔\mathbb {F}_q(T). Let \psi : A \to F\{\tau \} be a Drinfeld A-module over F, of rank r \geq 2, with End_{\overline {F}}(\psi) = A. For a non-zero ideal \mathfrak {n} of A, denote its unique monic generator by n, denote the degree of n as a polynomial in T by \deg n, and denote the \mathfrak {n}-division field of \psi by F(\psi [\mathfrak {n}]). A reciprocity law for \psi asserts that, if \gcd (charF, r) = 1 or if \mathfrak {n} is prime, then a non-zero prime ideal \mathfrak {p} \nmid \mathfrak {n} of A splits completely in F(\psi [\mathfrak {n}]) if and only if the Frobenius trace a_{1, \mathfrak {p}}(\psi) of \psi at \mathfrak {p} and the first component b_{1, \mathfrak {p}}(\psi) of the Frobenius index of \psi at \mathfrak {p} satisfy the congruences a_{1, \mathfrak {p}}(\psi) \equiv -r \pmod n and b_{1, \mathfrak {p}}(\psi) \equiv 0 \pmod n. We find the Dirichlet density of the set of non-zero prime ideals \mathfrak {p} for which the latter congruence never holds, that is, for which b_{1, \mathfrak {p}}(\psi) = 1. Using similar methods, we prove an asymptotic formula for the function of x defined by the average \frac {1}{\#\{\mathfrak {p}: \ \deg p = x \}} \sum _{\mathfrak {p}: \ \deg p = x} \tau _A(b_{1, \mathfrak {p}}(\psi)), where \mathfrak {p} = A p denotes an arbitrary non-zero prime ideal of A whose monic generator p \in A has degree x and where \tau _A(b_{1, \mathfrak {p}}(\psi)) denotes the number of monic divisors of b_{1, \mathfrak {p}}(\psi). [ABSTRACT FROM AUTHOR]

Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t-modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t-modules which lack the height property. In particular, let A be the polynomial ring in one variable \theta over the finite field \mathbb {F}_q with the quotient field K. Let (E, \phi) be an iterated extension of the tensor powers of the Carlitz module defined over K. We show that the \mathbb {F}_q[t]-module E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free \mathbb {F}_q[t]-module of rank \aleph _0. [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]

Let (H , ⋅ , ∘) be a Hopf brace. We first show that the category of left compatible modules over (H , ⋅ , ∘) is a tensor category. Next, we show that its Drinfeld centre is equivalent to the category H H ′ Y D of left Yetter–Drinfeld modules over (H , ⋅ , ∘) as a braided tensor category. Finally, we show that Radford's biproduct (R ♯ H , ⋅) and (R ♯ H ∘ , ∘) constitute a Hopf brace, where R is a Hopf algebra in H H ′ Y D . This presents some new method to construct Hopf braces. [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]

We prove a function field analogue of the Stark conjecture. [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]

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]

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]

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 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]

Let H be a Hopf algebra with bijective antipode. In this paper we prove that, if A is a Hopf brace with some projection on H, then there exists a compatible braided Hopf brace R such that A is isomorphic to R ♯ H as Hopf braces, where R ♯ H is some Radford's biproduct Hopf algebra. This should be viewed as the brace version of well-known Radford's theorem about Hopf algebras with a projection. This provides a new method to construct Hopf braces by some braided Hopf algebras. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a special function on the Drinfeld period domain \Omega ^{r} for r\geq 2 which gives the false Eisenstein series of Gekeler when r=2. We also study its functional equation and relation with quasi-periodic functions of a Drinfeld module as well as transcendence of its values at CM points. [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]

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]

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]

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 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]

This paper proposes the algebraic soft decoding (ASD) for one-point elliptic codes, where the interpolation problem is solved from the perspective of module basis reduction. In ASD, the interpolation polynomial $\mathcal {Q}(x, y, z)$ is the minimum candidate of a Gröbner basis. Based on a multiplicity matrix, an interpolation ideal can be defined. With the decoding output list size, an equivalent interpolation module can be led to. By further defining the set of interpolation points, a sequence of modules from the elliptic curve coordinate ring can be obtained. Based on the Lagrange interpolation functions over elliptic function field, a basis of the interpolation module can be constructed. The desired Gröbner basis that contains $\mathcal {Q}$ can be determined by reducing the module basis. Re-encoding transform (ReT) is further introduced to reduce the basis reduction complexity. It is also shown that the interpolation can be facilitated by assessing the degree of the Lagrange interpolation polynomials. The decoding complexity is analyzed, which is verified by numerical results. That shows the advantage of this interpolation technique over the conventional Kötter’s interpolation. The ASD performance of elliptic codes is also presented. [ABSTRACT FROM AUTHOR]

Pink has given a qualitative answer to the Mumford-Tate conjecture for Drinfeld modules in the 90s. He showed that the image of the \mathfrak {p}-adic Galois representation is \mathfrak {p}-adically open in the motivic Galois group for any prime \mathfrak {p}. In contrast to this result, we provide a family of uniformizable Anderson t-modules for which the Galois representations of their t-adic Tate-modules are "far from" having t-adically open image in their motivic Galois groups. Nevertheless, the image is still Zariski-dense in the motivic Galois group which is in accordance to the Mumford-Tate conjecture. For the proof, we explicitly determine the motivic Galois group as well as the Galois representation for these t-modules. [ABSTRACT FROM AUTHOR]

We generalize our work on Carlitz prime power torsion extension to torsion extensions of Drinfeld modules of arbitrary rank. As in the Carlitz case, we give a description of these extensions in terms of evaluations of Anderson generating functions and their hyperderivatives at roots of unity. We also give a direct proof that the image of the Galois representation attached to the -adic Tate module lies in the -adic points of the motivic Galois group. This is a generalization of the corresponding result of Chang and Papanikolas for the t -adic case. [ABSTRACT FROM AUTHOR]

We describe a systematic investigation into the existence of congruences between the mod p torsion modules of elliptic curves defined over Q, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod p congruences between twisted elliptic curves over number fields, giving global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes p≥7 to the elliptic curves in the LMFDB database, which currently includes all elliptic curves of conductor less than 500000. We also show that while such congruences exist for each p≤17, there are none for p≥19 in the database, in line with a strong form of the Frey-Mazur conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal of q [ T ] , the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over q (T) contains a rational -torsion point if the reduction of ϕ at almost all primes of q [ T ] contains a rational -torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2 , but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive. [ABSTRACT FROM AUTHOR]

The goal of this paper is to define an analogue of the Weil-pairing for Drinfeld modules using explicit formulas and to deduce its main properties from these formulas. Our result generalizes the formula given for rank 2 Drinfeld modules by van der Heiden and works as a more explicit, elementary proof of the Weil-pairing's existence given by van der Heiden. [ABSTRACT FROM AUTHOR]

Let X be a smooth, projective curve defined over a finite field Fr, and ∞ a (closed) point of X. Let κ be the function field of X and A the elements of κ regular everywhere except possibly at ∞. Let ϕ:A→K{τ} be a Drinfeld module defined over K, where ι:A→K is an injective map that identifies K as a finite extension of κ. Suppose that the rank of ϕ is n⩾2. For a place ℘ of good reduction for ϕ, reducing the coefficients of ϕ modulo ℘ equips the residue field F℘ with an A‐module structure. We establish that the number of places ℘ of K of good reduction for ϕ, of degree x, and such that ϕ(F℘) is a cyclic A‐module, has a natural density. Furthermore, this density is positive if and only if there are infinitely many primes ℘ such that ϕ(F℘) is cyclic as an A‐module. We do not make further restrictions on either A or the ring of endomorphisms EndKsep(ϕ). [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]

In the present paper, we prove that any finite nontrivial irreducible module over a rank two Lie conformal algebra ℋ is of rank one. We also describe the actions of ℋ on its finite irreducible modules explicitly. Moreover, we show that all finite nontrivial irreducible modules of finite Lie conformal algebras whose semisimple quotient is the Virasoro Lie conformal algebra are of rank one. [ABSTRACT FROM AUTHOR]

Zeta values in Tate algebras were introduced by Pellarin in 2012. They are generalizations of Carlitz zeta values and play an increasingly important role in function field arithmetic. In this paper we prove a conjecture of Pellarin on identities for these zeta values. The proof is based on arithmetic properties of Carlitz zeta values and an explicit formula for Bernoulli-type polynomials attached to zeta values in Tate algebras. [ABSTRACT FROM AUTHOR]