Abstract We classify rational cuspidal curves of degrees 6 and 7 in the complex projective plane, up to symplectic isotopy. The proof uses topological tools, pseudoholomorphic techniques, birational transformations.

let $r$ be a commutative noetherian ring with non-zero identity and $fa$ an ideal of $r$. let $m$ be a finite $r$--module of finite projective dimension and $n$ an arbitrary finite $r$--module. we characterize the membership of the generalized local cohomology modules $lc^{i}_{fa}(m,n)$ in certain serre subcategories of the category of modules from upper bounds. we define and study the properti...

In this note we explore the relationships between the motivic cohomology operations and the (classical) cohomology operations defined on mod-l étale cohomology. More precisely we show that the cohomology operations on motivic cohomology transform to the (classical) cohomology operations on mod-l étale cohomology upon inverting the motivic Bott element.

Brief introduction to cyclotomic theory over Q using adeles. Discussion of the definitions of modular forms and automorphic forms. Introducing the adelic automorphic forms via strong approximation theorem. Discussion of the connected components of Shimura varieties (modular curves). Smooth/admissible representations of locally finite groups. Definition and admissibility of (cuspidal) automorphi...

Let π = ⊗πv and π ′ = ⊗π ′ v be two irreducible, automorphic, cuspidal representations of GLm (AK). Using the logarithmic zero-free region of Rankin-Selberg L-function, Moreno established the analytic strong multi-plicity one theorem if at least one of them is self-contragredient, i.e. π and π ′ will be equal if they have finitely many same local components πv, π ′ v , for which the norm of pla...

Let π = ⊗πv and π ′ = ⊗π ′ v be two irreducible, automorphic, cuspidal representations of GLm (AK). Using the logarithmic zero-free region of Rankin-Selberg L-function, Moreno established the analytic strong multi-plicity one theorem if at least one of them is self-contragredient, i.e. π and π ′ will be equal if they have finitely many same local components πv, π ′ v , for which the norm of pla...

In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order q-recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, r...