关于具有复合模数的代数扭，二

Sums with the Mobius function twisted by characters with powerful moduli

In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums of the Mobius function twisted by characters of the same type, complementing and improving some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of $L$-functions with the same class of moduli.

Galois representations and composite moduli

It is known that for any elliptic curve $E/\mathbb{Q}$ and any integer $m$ co-prime to $30,$ the induced Galois representation $ρ_{E,m}: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \text{GL}_{2}(\mathbb{Z}/m\mathbb{Z})$ is surjective if and only if $ρ_{E,\ell}$ is surjective for any prime $\ell|m.$ In this article, we shall discuss some generalizations, applications, and variants of this phenomenon.

Kloosterman sums with primes to composite moduli

We obtain a new estimate for Kloosterman sum with primes $p\leqslant X$ to composite modulo $q$, that is, for the exponential sum of the type \[ \sum\limits_{p\leqslant X,\;p\,\nmid q}\exp{\biggl(\frac{2πi}{q}\bigl(a\overline{p}+bp\bigr)\,\biggr)},\quad (ab,q)=1,\quad p\overline{p}\equiv 1\pmod{q}, \] which is non-trivial in the case when $q^{\,3/4+\varepsilon}\leqslant X\ll q^{\,3/2}$. We also apply this estimate to the proof of solvability of some congruences with inverse prime residues $\pmod{q}$.

A Generalization of Euler's Criterion to Composite Moduli

A necessary and sufficient condition is provided for the solvability of a binomial congruence with a composite modulus, circumventing its prime factorization. This is a generalization of Euler's Criterion through that of Euler's Theorem, and the concepts of order and primitive roots. Idempotent numbers play a central role in this effort.

Character sums of composite moduli and hybrid subconvexity

Let $M=M_1 M_2 M_3$ be the product of three distinct primes and let $χ=χ_1 χ_2 χ_3$ be a Dirichlet character of modulus $M$ such that each $χ_i$ is a primitive character modulo $M_i$ for $i=1,2,3$. In this paper, we provide a $δ$-symbol method for obtaining non-trivial cancellation in smooth character sums of the form $\sum_{n=1}^\infty χ(n) W(n/N)$, with $N$ roughly of size $\sqrt M$ and $W$ a smooth compactly supported weight function on $(0, \infty)$. As a corollary, we establish hybrid subconvexity bounds for the associated Dirichlet $L$-function.

Niebur-Poincaré Series and Traces of Singular Moduli

We compute the Fourier coefficients of analogues of Kohnen and Zagier's modular forms $f_{k,D}$ of weight $2$ and negative discriminant. These functions can also be written as twisted traces of certain weight $2$ Poincaré series with evaluations of Niebur-Poincaré series as Fourier coefficients. This allows us to study twisted traces of singular moduli in an integral weight setting. In particular, we recover explicit series expressions for twisted traces of singular moduli and extend algebraicity results by Bengoechea to the weight $2$ case. We also compute regularized inner products of these …

A bound for twists of $\rm GL_3\times GL_2$ $L$-functions with composite modulus

Let $π$ be a Hecke-Maass cusp form for $\rm SL_3(\mathbf{Z})$ and let $g$ be a holomorphic or Maass cusp form for $\rm SL_2(\mathbf{Z})$. Let $χ$ be a primitive Dirichlet character of modulus $M=M_1M_2$ with $M_i$ prime, $i=1,2$. Suppose that $M^{1/2+2η}<M_1<M^{1-2η}$ with $0<η<1/8$. Then we have $$ L\left(\frac{1}{2},π\otimes g \otimes χ\right)\ll_{π,g,\varepsilon} M^{3/2-η+\varepsilon}. $$

GL(2) 的扭转模

Twisting moduli for GL(2)

本文通过构造 Γ_0(N) 的特殊矩阵生成集，证明全纯模形式在 N=18、20、24 时无需扭转函数方程，并给出偶权与奇权 Maass 形式对应的 N 范围。

The Markoff Group of Transformations in Prime and Composite Moduli

The Markoff group of transformations is a group $Γ$ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation $x^{2}+y^{2}+z^{2}=xyz$. The fundamental strong approximation conjecture for the Markoff equation states that for every prime $p$, the group $Γ$ acts transitively on the set $X^{*}\left(p\right)$ of non-zero solutions to the same equation over $\mathbb{Z}/p\mathbb{Z}$. Recently, Bourgain, Gamburd and Sarnak proved this conjecture for all primes outside a small exceptional set. In the current paper, we study a group of …