Displaying similar documents to “Self-intersection of the relative dualizing sheaf on modular curves X 1 ( N )

Generalized divisor problem for new forms of higher level

Krishnarjun Krishnamoorthy (2022)

Czechoslovak Mathematical Journal

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Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ 0 ( N ) with normalized eigenvalues λ f ( n ) and let X > 1 be a real number. We consider the sum 𝒮 k ( X ) : = n < X n = n 1 , n 2 , ... , n k λ f ( n 1 ) λ f ( n 2 ) ... λ f ( n k ) and show that 𝒮 k ( X ) f , ϵ X 1 - 3 / ( 2 ( k + 3 ) ) + ϵ for every k 1 and ϵ > 0 . The same problem was considered for the case N = 1 , that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k 5 . Since the result is valid...

An arithmetic Riemann-Roch theorem for pointed stable curves

Gérard Freixas Montplet (2009)

Annales scientifiques de l'École Normale Supérieure

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Let ( 𝒪 , Σ , F ) be an arithmetic ring of Krull dimension at most 1, 𝒮 = Spec 𝒪 and ( π : 𝒳 𝒮 ; σ 1 , ... , σ n ) an n -pointed stable curve of genus g . Write 𝒰 = 𝒳 j σ j ( 𝒮 ) . The invertible sheaf ω 𝒳 / 𝒮 ( σ 1 + + σ n ) inherits a hermitian structure · hyp from the dual of the hyperbolic metric on the Riemann surface 𝒰 . In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ω 𝒳 / 𝒮 ( σ 1 + ... + σ n ) hyp . The theorem is applied to modular curves X ( Γ ) , Γ = Γ 0 ( p ) or Γ 1 ( p ) , p 11 prime, with sections given by the cusps. We show Z ' ( Y ( Γ ) , 1 ) e a π b Γ 2 ( 1 / 2 ) c L ( 0 , Γ ) , with p 11 m o d 12 when Γ = Γ 0 ( p ) . Here Z ( Y ( Γ ) , s ) is the Selberg...

Group algebras whose groups of normalized units have exponent 4

Victor Bovdi, Mohammed Salim (2018)

Czechoslovak Mathematical Journal

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We give a full description of locally finite 2 -groups G such that the normalized group of units of the group algebra F G over a field F of characteristic 2 has exponent 4 .

-invariants and Darmon cycles attached to modular forms

Victor Rotger, Marco Adamo Seveso (2012)

Journal of the European Mathematical Society

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Let f be a modular eigenform of even weight k 2 and new at a prime p dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D f F M and an -invariant f F M . The first goal of this paper is building a suitable p -adic integration theory that allows us to construct a new monodromy module D f and -invariant f , in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two -invariants...

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

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Let C be a hyperelliptic curve of genus g 1 over a number field K with good reduction outside a finite set of places S of K . We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g , S and K . In particular, we obtain that for any given number field K , finite set of places S of K and integer g 1 one can in principle determine the set of K -isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside...

Asymptotic values of modular multiplicities for GL 2

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo p of irreducible algebraic representations V of the group Res K / p GL 2 for K a finite extension of p . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of V and the central character of its reduction modulo p . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

On higher moments of Hecke eigenvalues attached to cusp forms

Guodong Hua (2022)

Czechoslovak Mathematical Journal

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Let f , g and h be three distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 and k 3 for the full modular group Γ = SL ( 2 , ) , respectively, and let λ f ( n ) , λ g ( n ) and λ h ( n ) denote the n th normalized Fourier coefficients of f , g and h , respectively. We consider the cancellations of sums related to arithmetic functions λ g ( n ) , λ h ( n ) twisted by λ f ( n ) and establish the following results: n x λ f ( n ) λ g ( n ) i λ h ( n ) j f , g , h , ε x 1 - 1 / 2 i + j + ε for any ε > 0 , where 1 i 2 , j 5 are any fixed positive integers.

The local lifting problem for actions of finite groups on curves

Ted Chinburg, Robert Guralnick, David Harbater (2011)

Annales scientifiques de l'École Normale Supérieure

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Let k be an algebraically closed field of characteristic p &gt; 0 . We study obstructions to lifting to characteristic 0 the faithful continuous action φ of a finite group G on k [ [ t ] ] . To each such  φ a theorem of Katz and Gabber associates an action of G on a smooth projective curve Y over k . We say that the KGB obstruction of φ vanishes if G acts on a smooth projective curve X in characteristic  0 in such a way that X / H and Y / H have the same genus for all subgroups H G . We determine for which G the KGB...

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let C be a smooth curve of genus g . For each positive integer r the birational r -gonality s r ( C ) of C is the minimal integer t such that there is L Pic t ( C ) with h 0 ( C , L ) = r + 1 . Fix an integer r 3 . In this paper we prove the existence of an integer g r such that for every integer g g r there is a smooth curve C of genus g with s r + 1 ( C ) / ( r + 1 ) > s r ( C ) / r , i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails.

Variations on a question concerning the degrees of divisors of x n - 1

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial x n - 1 : “How often does x n - 1 have a divisor of every degree between 1 and n ?” In a previous paper, we considered the situation when x n - 1 is factored in [ x ] . In this paper, we replace [ x ] with 𝔽 p [ x ] , where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p .

Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

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Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k 2 and of the same level N , both eigenfunctions of the Hecke operators, and both normalized so that a 1 ( f ) = a 1 ( E ) = 1 . The main result we prove is that when E and f are congruent mod a prime 𝔭 (which we take in this paper to be a prime of ¯ lying over a rational prime p &gt; 2 ), the algebraic parts of the special values L ( E , χ , j ) and L ( f , χ , j ) satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

Thompson’s conjecture for the alternating group of degree 2 p and 2 p + 1

Azam Babai, Ali Mahmoudifar (2017)

Czechoslovak Mathematical Journal

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For a finite group G denote by N ( G ) the set of conjugacy class sizes of G . In 1980s, J. G. Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N ( G ) = N ( L ) , then G L . We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z ( G ) = 1 and N ( G ) = N ( A i ) is necessarily isomorphic to A i , where i { 2 p , 2 p + 1 } .