In this paper, we study a multidimensional divisor problem concerning the symmetric j^th power L-function. More precisely, we establish an asymptotic formula for the discrete mean square of the Fourier coefficients of the symmetric j^th power L-function in k-dimensions.
Inspired by the existence of multiple alternative definitions of continued fraction expansions for elements in ℚ_p , we study the p-adic convergence of periodic continued fractions with partial quotients in ℤ[1/p] from a geometric point of view. To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the p-adically convergent loci of these varieties, describing the zero and one-dimensional cases by combining tools from algebraic geometry, arithmetic, and the theory of Pell equations and of linear recurrences.
Let p>3 be a prime. We obtain explicit congruences for ∑ _k=0^p-1w(k)( [ 2k; k ]) ^3/(-8)^k, ∑ _k=0^p-1w(k)( [ 2k; k ]) ^2( [ 3k; k ]) /(-192)^k, ∑ _k=0^p-1w(k)( [ 2k; k ]) ^2( [ 4k; 2k ]) /(-144)^k, ∑ _k=0^p-1w(k)( [ 2k; k ]) ^2( [ 4k; 2k ]) /648^k, ∑ _k=0^(p-1)/2( [ 2k; k ]) ^3/(-8)^k(k+1)^r, ∑ _k=0^p-2( [ 2k; k ]) ^2( [ 3k; k ]) /(-192)^k(k+1)^r, ∑ _k=0^p-2( [ 2k; k ]) ^2( [ 4k; 2k ]) /(-144)^k(k+1)^r, ∑ _k=0^p-2( [ 2k; k ]) ^2( [ 4k; 2k ]) /648^k(k+1)^r, ∑ _k=0^p-1k^r( [ 2k; k ]) ^3/64^k, ∑ _k=0^p-1k^r( [ 2k; k ]) ^2( [ 3k; k ]) /108^k, ∑ _k=0^p-1k^r( [ 2k; k ]) ^2( [ 4k; 2k ]) /256^k, ∑ _k=0^p-1k^r( [ 2k; k ]) ( [ 3k; k ]) ( [ 6k; 3k ]) /1728^k mod p^2 and partial results for ∑ _k=0^(p-1)/2( [ 2k; k ]) ^31/m^k(k+1)^r and ∑ _k=0^p-1( [ 2k; k ]) ^3w(k)/m^k mod p^2 , where w(k)∈{k^2,k^3, 1/2k-1,1/(2k-1)^2} , r∈{1,2,3} and m∈{1,16,-64,256,-512,4096} .
We describe how the partial quotients of a nearest integer continued fraction for a complex irrational can be used to generate a sequence of nested Farey quadrilaterals in the complex plane ℂ , each of which contains the target point. Our approach begins by establishing a connection between the Farey tessellation of hyperbolic space ℍ^3 and the Schmidt arrangement on the boundary of ℍ^3 . We show that Farey octahedra are represented uniquely by Farey quadrilaterals among the Farey circles and dual Farey circles in the Schmidt arrangement. Next, we interpret continued fractions in terms of products of Möbius maps, making use of Beardon’s observation that Möbius maps act as isometries of hyperbolic space. Factors of the initial product of Möbius maps alternate between parabolic Möbius maps that fix infinity and those that fix zero. A closer analysis of elliptic Möbius maps that leave vertices of the fundamental octahedron invariant reveals that factors of certain elliptic Möbius maps appear naturally within the product of parabolic factors. These elliptic factors need to be isolated from the parabolic factors, adjusting the coefficients of the parabolic factors as required in the process. The structural connection between Farey quadrilaterals and the adjusted coefficients yields a new visual interpretation of how complex nearest integer continued fractions generate successive approximations for irrational complex numbers.
Let & ell; and n be positive integers with & ell; prime. The modular curves X1(& ell;n) and X0(& ell;n) are algebraic curves over Q whose non-cuspidal points parameterize elliptic curves with a distinguished point of order & ell;n or a distinguished cyclic subgroup of order & ell;n, respectively. We wish to understand isolated points on these curves, which are roughly those not belonging to an infinite parameterized family of points having the same degree. Our first main result is that there are precisely 15 j-invariants in Q which arise as the image of an isolated point x is an element of X1(& ell;n) under the natural map j:X1(& ell;n)-> X1(1). This completes a prior partial classification of Ejder. We also identify the 19 rational j-invariants which correspond to isolated points on X0(& ell;n).
The Lang-Trotter conjecture on primitive points is the analogue for elliptic curves of Artin’s conjecture on primitive roots. Indeed, if we have an elliptic curve E over ℚ with a rational point P of infinite order, we may count the primes p of good reduction for which (P p) generates E(𝔽_p) . In this work, we formulate and investigate two natural variants of the Lang-Trotter conjecture. For one of them, we require that the group E(𝔽_p) and its subgroup < (P p)> have the same exponent, namely the cyclic subgroup is as large as possible. We conjecture that the set of primes p such that this condition holds admits a natural density, whose value is a rational multiple of the product over all primes ℓ of the natural densities (which we prove to exist and be rational) of those p such that the exponents of E(𝔽_p) and < (P p)> have the same ℓ -adic valuation. Numerical examples support the validity of our conjectures.
After having stated several conjectures regarding potential families of normal numbers, we construct various new families of normal numbers using different concepts, in particular the whole set of partitions of the set {0,1,… ,k} , the smallest prime divisor of n which is larger than log n , and finally a certain regroupment of the whole set of primes known as a disjoint classification of primes.
We obtain best possible results for the number of coprime positive integer solutions of the equation in the title when a is a positive integer, b = pm, 2pm or 4pm, where m is a non-negative integer, p is prime, gcd (a2, b) is squarefree and X2 - (a2 + b) Y2 = -4 has a solution in positive integers. We prove our results by establishing best possible bounds for the number of distinct squares in certain binary recurrence sequences, including those associated with such equations.
In this short note we observe that the gamma factor defined by Gelfand and Kazhdan coincides with the Rankin-Selberg root number defined by Jacquet, Piatetskii-Shapiro and Shalika.
Using a differential operator which sends a scalar-valued hermitian modular form to the tensor product of two vector-valued hermitian modular forms, under a certain condition, we give the pullback formula for vector-valued hermitian modular forms on any CM field. We also give equivalence conditions for differential operators to have the above properties, which is an extension of Ibukiyama’s result for hermitian modular forms.
We derive asymptotic estimates for some average values of the Jordan function evaluated over shifted smooth numbers in arithmetic progressions whose sum of digits is in arithmetic progression. In particular, we generalize an earlier work of Loiperdinger and Shparlinski.
In this article we study the endomorphism algebras of abelian varieties A defined over a given number field K with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of A to be defined over K(A[2]), the field extension generated by its 2-torsion. When K= ℚ and Gal(ℚ(A[2])/ℚ) is cyclic of prime order p = 2 (A) +1 , we prove that there are only finitely many possibilities for the geometric endomorphism algebra End(A) ⊗ℚ . In fact, when (A) ∉{3,5,9,21,33,81} , we show End(A) ⊗ℚ is a proper subfield of the p-th cyclotomic field. In particular, when g=2 , End(A) ⊗ℚ is isomorphic to either ℚ or ℚ(√(5)) .
Let ξ∈ℝ be an irrational number and η∈ℝ . Dirichlet’s theorem on Diophantine approximation and its inhomogeneous version, due to Kronecker, tells us that |ξ -m/n|=𝒪(1/n^2) and |nξ -m-η |=𝒪(1/n) for infinitely many pairs of integers n,m∈ℤ . We derive two refinements telling us that for any A>0 , ρ >0 , η rational, and 0<ε <2 there are infinitely many pairs (n,p)∈ℕ^*×ℤ such that n^1-ε |nξ -p-η | ∼ A, resp. n^2(1-ε )( nξ -p-η) ^2 ∼ρlog n+A. Then we apply these facts to revisit results by F. Luca and J.C. Saunders on the set of cluster points of sequences of the form f^n(sin (α n)) and n^s |sin (α n)|^n^r and their analogues for the cosine function. A very special case e.g. will tell us that for every α for which α /π is irrational, the cluster set of (1-sin ^6 (α n) )^n^5 is [0, 1], whereas this no longer holds for (1-sin ^6(π√(2) n) )^n^6. Several results depend heavily on the irrationality exponent associated with α /π . In the last section we study the asymptotic behavior of |sin (n_k+m_k)|^ ε _k for various sequences ε _k>0 whenever (sin n_k)^n_k and (sin m_k)^n_k converge to non-zero numbers. It will finally be shown that for 0<|λ |<1 the existence of the limit λ :=(sin n_k)^n_k implies that lim _k |sin (p n_k) |^pn_k exists, too, and equals |λ |^p^3 if p is odd and 0 if p is even.
We define the notion of k-almost consecutive partitions, and study associated combinatorial and modular aspects. We establish quantum Jacobi properties of their corresponding two-variable partition generating functions. Further, we provide related asymptotics, formulas, and combinatorial identities, and make connections to Ramanujan’s third order mock-theta function ψ (q) and Cohen’s σ ^*(q) . We conclude by recording a proof of a related conjecture of Xiong.
We construct an explicit Cartan–Dieudonné decomposition for orthogonal group elements of binary quadratic forms over non-archimedean local fields of characteristic zero, expressing each group element as a product of reflections defined by vectors. A key feature of our construction is its stability: we establish quantitative control on how the reflection matrices vary under small perturbations of the underlying vectors. Using this decomposition, we establish an effective result for the equivalence of binary quadratic forms over number fields. Specifically, let K be a number field and S a finite set of non-archimedean places of K. Given two K-equivalent binary quadratic forms integrally equivalent at every prime in S, we provide an explicit search bound for finding a K-equivalence that are integral at all primes in S.
The study of special values of adjoint L-functions and congruence ideals is gradually becoming a classical theme in number theory, driven by the Bloch-Kato conjecture and generalisations of Wiles-Lenstra's numerical criterion. In this paper, we relate L ( 1 , π , Ad ∘ ) to the congruence ideals for cohomological cuspidal automorphic representations π of GL n over any number field. We then use this result to deduce relationships between the congruences of automorphic forms and adjoint L-functions. For CM fields, using the existence of Galois representations, we apply the result to obtain a lower bound on the cardinality of certain Selmer groups in terms of L ( 1 , π , Ad ∘ ) . This can be viewed as partial progress on the Bloch-Kato conjecture. The main technical ingredients are a careful study of the cohomology associated with the locally symmetric space of GL n , its relation to automorphic representations, and the establishment of some algebraic properties of the congruence ideals. We anticipate that the methods developed here will find further applications in related problems, particularly in the study of congruence modules and their relation to the arithmetic of automorphic forms.
