Pierre-Yves Gaillard
To each i, j belonging to some set of integers, attach the integer a(i,j). Are there integers x(i) such that x(j)-x(i) is congruent to a(i,j) mod (i,j)? A necessary condition is that a(i,j)+a(j,k) be congruent to a(i,k) mod (i,j,k). This condition is sufficient.
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Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some $Γ_1(N)$ that are eigenforms outside $Np$ and have coefficients in the ring of integers of some number field $K$, we consider the problem of deciding whether $f_1$ and $f_2$ have the same eigenvalues mod ${\mathfrak p}^m$ (where ${\mathfrak p}$ is a fixed prime of $K$ over $p$) for Hecke operators $T_{\ell}$ at all primes $\ell\nmid Np$. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different …
Some congruence relations satisfied by the theta series associated with the Leech lattice are given.
Under relatively mild and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modu…
We describe some new general constructions of $p$-adic $L$-functions attached to certain arithmetically defined complex $L$-functions coming from motives over $\bold Q$ with coefficiens in a number field $T$, with $[T:\bold Q]<\infty$. These constructions are equivalent to proving some generalized Kummer congruences for critical special values of these complex $L$-functions.
We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of s…
Let $E$ be a level 1, vector valued Eisenstein series of half-integral weight, normalized so that the coefficients are all in $\mathbb{Z}$. We show that there is a level one vector valued cusp form $f$ with the same weight as $E$ and with coefficients in $\mathbb{Z}$, which is congruent to $E$ modulo the constant term of $E$.
We use the Jacquet-Langlands correspondence to generalize well-known congruence results of Mazur on Fourier coefficients and L-values of elliptic modular forms for prime level in weight 2 both to nonsquare level and to Hilbert modular forms.
We consider a system of homogeneous quadratic forms with congruence conditions in $n\geq 3$ variables and prove the existence of two linearly independent integral solutions of bounded height. We also show the existence of small height integral zeros of this system avoiding a given set of hyperplanes.
A proof of Wolstenholme's theorem and congruence properties via an Egorychev-type integral
本文通过埃戈里切夫型积分和指数变量变换，详细证明了沃尔斯滕霍尔姆定理，并明确展示了与调和和与伯努利数的联系，推导了模p^4的经典细化，并精确提取了B_{p-3}项。
Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(\mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Γ_{0}(4N)$ for $N=1,2,4$. A proof is based on linear relations among Fourier coefficients of modular forms of half integral weight. As applications we obtain congruences of Borcherds exponents, congruences of quotient of Eisentein series and congruences of values of $L$-functions at a certain point are also studied. Furthermore, the congruences of the Fourier coeff…
Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $Γ_0(N)$ which are supported on finitely many square classes modulo $\ell$, in some cases proving that these forms are congruent to the image of a single variable theta series under some number of iterations of the Ramanujan $Θ$-operator. Here, we study the analogous problem for modular forms of half-integral weight on $\operatorname{SL}_{2}(\mathbb{Z})$. Let $η$ be the Dedekind eta function. For a wide range of weights, we prove that every…
On the quantitative variation of congruence ideals and integral periods of modular forms
本文证明了 Pollack–Weston 关于高权模形式 Ribet 式降级同余的定量猜想，方法依托 $R=\mathbb{T}$ 定理与伴随 $L$ 值比较，并指出其算术应用。
Congruences of first syntomic cohomology groups
研究一阶syntomic上同调群的同余性质，通过比较其模p^n约简与附带Breuil-Kisin模和Nygaard滤子的模p^{2n}约简的同构性，揭示局部Bloch-Kato Selmer群的细化结构。
Quadratic Congruences for half-integral weight cusp forms with the eta multiplier
本文研究半整数权重cuspidal形式的二次同余性质，通过模ular Galois表示理论，证明任意字符下的同余成立并提出关键新结果。
In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke operators, we derive relations of Fourier coefficients of each basis element and obtain congruences of the Fourier coefficients, which extend known congruences for traces of singular moduli.
On the integrality of modular functions over $\mathbb{Z}[j]$ and Kronecker-type congruences
本文研究了模函数在$\mathbb{Z}[j]$上的整性，并证明了在特定条件下，由模函数定义的函数满足克罗内克型同余关系。
The Shimura lift and congruences for modular forms with the eta multiplier
本文证明了带Dedekind eta乘子与Dirichlet特征扭曲的半整权尖点形式的Shimura型对应，确定提升的权、级及Atkin-Lehner特征值，并由此证明一大族半整权模形式空间存在无穷多个素数给出任意幂模的二次同余。
In this paper, we are interested in the interplay between integral ternary quadratic forms and class numbers. This is partially motivated by a question of Petersson.
We address a question posed by Ono, prove a general result for powers of an arbitrary prime, and provide an explanation for the appearance of higher congruence moduli for certain small primes. One of our results coincides with a recent result of Edixhoven, and we hope that the comparison of the methods, which are entirely different, may reveal a connection between the p-adic geometry and the arithmetic of half-integral weight Hecke operators.
In this article, we consider the estimation of exponential sums along the points of the reduction mod $p^{m}$ of a $p$-adic analytic submanifold of $ \mathbb{Z}_{p}^{n}$. More precisely, we extend Igusa's stationary phase method to this type of exponential sums. We also study the number of solutions of a polynomial congruence along the points of the reduction mod $% p^{m}$ of a $p$-adic analytic submanifold of $\mathbb{Z}_{p}^{n}$. In addition, we attach a Poincare series to these numbers, and establish its rationality. In this way, we obtain geometric bounds for the number of solutions of the…
Deligne proved that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g>1 must be integral or half integral. We give a different proof for this. It uses Mennicke's result that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from a paper of Bass-Milnor-Serre.
Congruence conditions for the mod $λ$ values of the Fourier coefficients of classical eigenforms
本文依据模 $\lambda$ Galois 表示的射影像，分类经典尖点特征形式 Fourier 系数满足 $a_p(f)\equiv x\bmod\lambda$ 且与素数 $p$ 同余相关的情形，并用椭圆曲线权 2 新形式举例。
In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.
Let $[x]$ be the integral part of $x$, $n>1$ be a positive integer and $χ_n$ denote the trivial Dirichlet character modulo $n$. In this paper, we use an identity established by Z. H. Sun to get congruences of $T_{m,k}(n)=\sum_{x=1}^{[n/m]}\frac{χ_n(x)}{x^k}\left(\bmod n^{r+1}\right)$ for $r\in \{1,2\}$, any positive integer $m $ with $n \equiv \pm 1 \left(\bmod m \right)$ in terms of Bernoulli polynomials. As its an application, we also obtain some new congruences involving binomial coefficients modulo $n^4$ in terms of generalized Bernoulli numbers.
Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.
On Power Sums and the Kummer Congruences
本文通过研究多重集在乘法作用下的稳定化子，揭示了元素层面的底层结构，为经典Kummer同余提供了新证明，并提出了基于局部信息组合的新视角。
Counterexamples to Dujella's conjecture on integral points on the elliptic curve attached to a Diophantine triple
本文针对丢番图三元组对应椭圆曲线上整数点的杜杰拉猜想，构造无穷多组反例，证明存在x≠-1的整数点使得三个因子均非平方数，否定了该猜想的弱化问题。
Let $f$ be a newform of weight $2$ on $Γ_0(N)$ with Fourier $q$-expansion $f(q)=q+\sum_{n\geq 2} a_n q^n$, where $Γ_0(N)$ denotes the group of invertible matrices with integer coefficients, upper triangular mod $N$. Let $p$ be a prime dividing $N$ once, $p\parallel N$, a Steinberg prime. Then, it is well known that $a_p\in\{1,-1\}$. We denote by $K_f$ the field of coefficients of $f$. Let $λ$ be a finite place in $K_f$ not dividing $2p$ and assume that the mod $λ$ Galois representation attached to $f$ is irreducible. In this paper we will give necessary and sufficient conditions for the existe…
Congruence Jumping in Polynomial Divisibility Systems: Arithmetic Structure and Solution Sets
该论文研究整数系数多项式整除系统 $y\mid A(x), x\mid B(y)$ 的非零整数解，提出同余跳跃方法，建立互反多项式规则并分类二次情形，构造非单位循环四周期，给出解集有限化判据。
Primes represented by shifted quadratic forms: on primitivity and congruence classes
本文扩展了Iwaniec的筛法，证明了移位正定整二元二次型在附加本原性与同余类条件下表示素数的下界，并提供了经计算机验证的局部条件列表以纠正先前错误，旨在为研究Apollonian circle packings中的素数分量提供工具。
A Determinant Congruence Conjectured by Sun
本文证明了孙氏关于二元二次型行列式的一个加强形式猜想，通过分析范德蒙德因子和模p矩阵的秩，得到了复合数和素数情况下的同余结论。
For each positive integer $n$, let $g_Δ(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In this paper, we analyze $g_Δ(n)$ by studying representations of integral quadratic forms by sums of squares with certain congruence condition. We prove that the growth of $g_Δ(n)$ is at most an exponential of $\sqrt{n}$, which is the same as the best known upper bound on the $g$-invariants of the original quadratic Waring's problem. We also determine …
Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational polynomials (taking only integral values) and the function giving the least common multiple of $1,2,\ldots,k$. The tool used to obtain these characterizations is "lifting": if $π\colon X\to Y$ is a surjective morphism, and $f$ a function on $Y$ a lifting of $f$ is a function $F$ on $X$ such that $π\circ F=f\circπ$. In this paper we relate the finite and infinite noti…
Let Gamma be a congruence subgroup of the Picard modular group of an imaginary number field k, and let D be the associated symmetric space. We describe a method to compute the integral cohomology of the locally symmetric space Gamma\D. The method is implemented for the case k=Q(i) and k=Q(sqrt(-3)), and the cohomology is computed for various Gamma.
We give a survey of Denef's rationality theorem on $p$-adic integrals, its uniform in $p$ versions, the relevant model theory, and a number of applications to counting subgroups of finitely generated nilpotent groups and conjugacy classes in congruence quotients of Chevalley groups over rings of integers of local fields. We then state results on analytic properties of Euler products of such $p$-adic integrals over all $p$, and an application to counting conjugacy classes in congruence quotients of certain algebraic groups over the rationals. We then briefly discuss zeta functions arising from …
Let $m$ be a square-free positive integer, $m\equiv 2,3 \; (\bmod \; 4)$. We show that the number field $K=Q(i,\sqrt[4]{m})$ is non-monogene, that is it does not admit any power integral bases of type $\{1,α,\ldots,α^7\}$. In this infinite parametric family of Galois octic fields we construct an integral basis and show non-monogenity using only congruence considerations. Our method yields a new approach to consider monogenity or to prove non-monogenity in algebraic number fields. It is well applicable in parametric families of number fields. We calculate the index of elements as polynomials de…
Reciprocal Dynamics in Polynomial Congruence Jumping: Integral Orbits and Valuation Cones
该研究针对固定次数n,m的多项式对，定义同余跳跃中的互反离散动力系统，通过SL₂(ℤ)矩阵演化常数项赋值，分类不同s=nm regime下的积分轨道，给出局部准则与周期轨道结果。
Recently Ritter and Weiss introduced an equivariant "main conjecture" than generalizes and refines the Main Conjecture of Iwasawa theory. In this paper, we show that, for the prime 2 and a dihedral extension of order 8 over Q, this conjecture is equivalent to a congruence condition on the coefficients of a power series with 2-adic integral coefficients constructed using the 2-adic L-series associated to the extension. We then verify that this congruence condition holds for the first coefficients in a large number of examples.
Efficient Computation and Congruences for Colored Partition Functions
本文针对整数 $\alpha$ 证明了有色分拆计数中指数和的乘性与约化公式，结合截断精度界提出高效精确算法，并验证了新的 Ramanujan 型同余式。
Let $λ$ be an integer, and $f(z)=\sum_{n\gg-\infty} a(n)q^n$ be a weakly holomorphic modular form of weight $λ+\frac 12$ on $Γ_0(4)$ with integral coefficients. Let $\ell\geq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $\ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{\ell^m})(z) = \sum_{n=0}^\infty b(n)q^n$ has the form \[ (f|U_{\ell^m})(z) \equiv b(0) + \sum_{i=1}^{t}\sum_{n=1}^{\infty} b(d_i n^2) q^{d_i n^2} \pmod{\ell}, \] where $d_1, \ldots, d_t$ are square-free positive integers, and the operator $U_\ell$ on formal pow…
Deep congruences + the Brauer-Nesbitt theorem
本文证明了$\mathbb{Z}_p[X]$中多项式模$p$同余与根幂和函数$p$幂同余的等价性，通过组合方法引入分划$p$等价关系，还给出代数证明，推论得到单线性算子Brauer-Nesbitt定理的细化版本。
Let g be a cubic polynomial with integer coefficients and n>9 variables, and assume that the congruence g=0 modulo p^k is soluble for all prime powers p^k. We show that the equation g=0 has infinitely many integer solutions when the cubic part of g defines a projective hypersurface with singular locus of dimension <n-10. The proof is based on the Hardy-Littlewood circle method.
In a recent study of sign-balanced, labelled posets Stanley [13], introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' the conjugate of pi. In [1] Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0, 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partiti…
In a recent study of sign-balanced, labelled posets Stanley, introduced a new integral partition statistic srank(pi) = O(pi) - O(pi'), where O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. Andrews proved the following refinement of Ramanujan's partition congruence mod 5: p[0](5n +4) = p[2](5n + 4) = 0 (mod 5), p(n) = p[0](n) + p[2](n), where p[i](n) (i = 0; 2) denotes the number of partitions of n with srank = i (mod 4) and p(n) is the number of unrestricted partitions of n. Andrews asked for a partition statistic that would divide the partitions enume…
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