We announce a very general statement involving the rational quartic residue symbol $(m/p)_4$ and, more generally, Legendre symbols of the type ${a+b\sqrt{m}/p$. We show how our main theorem can be used to produce many older results such as Scholz's, Lehmer's or Burde's reciprocity laws and many others. It is very likely that all existing reciprocity laws of this type can be obtained from our result. This is a corrected and improved version of version 2.
Parshin's higher Witt pairing on an arithmetic surface can be combined with the higher tame pairing to form a symbol taking values in the absolute abelian Galois group of the function field. We prove reciprocity laws for this symbol using techniques of Morrow for the Witt symbol and Romo for the higher tame symbol.
Dedekind symbols are generalizations of the classical Dedekind sums (symbols). There is a natural isomorphism between the space of Dedekind symbols with Laurent polynomial reciprocity laws and the space of modular forms. We will define a new elliptic analogue of the Apostol-Dedekind sums. Then we will show that the newly defined sums generate all odd Dedekind symbols with Laurent polynomial reciprocity laws. Our construction is based on Machide's result on his elliptic Dedekind-Rademacher sums. As an application of our results, we discover Eisenstein series identities which generalize certain …
I use polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 . to prove six identities involving the $q$-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.
In this article we estimate the number of integers up to $X$ which can be represented by a positive-definite, binary integral quadratic form of discriminant which is small relative to $X$. This follows from understanding the vector of signs when computing the Legendre symbol of small integers $n$ at multiple primes.
For $0\leq α<1$ and prime number $p$ let $L(α,p)$ be the sum of the first $[αp]$ values of Legendre symbol modulo $p$. We study positivity of $L(α,p)$ and prove that for $|α-\frac13|<2\cdot 10^{-6}$ and for rational $α\leq \frac12$ with denominators in the set $\{1,2,3,4,5,6,8,12\}$ the inequality $L(α,p)\geq 0$ holds for majority of primes.
For a reciprocity functor $\mathcal{M}$ we consider the local symbol complex $\mathcal{M}\otimes^{M}\mathbb{G}_{m}(η_{C})\to\oplus_{P\in C}\mathcal{M}(k)\to\mathcal{M}(k)$, where $C$ is a smooth complete curve over an algebraically closed field $k$ with generic point $η_{C}$ and $\otimes^{M}$ is the product of Mackey functors. We prove that if $\mathcal{M}$ satisfies certain conditions, then the homology of the above complex is isomorphic to the $K$-group of reciprocity functors $T(\mathcal{M},\underline{CH}_{0}(C)^{0})(Spec k)$.
The evaluations of determinants with Legendre symbol entries have close relation with character sums over finite fields. Recently, Sun posed some conjectures on this topic. In this paper, we prove some conjectures of Sun and also study some variants. For example, we show the following result: Let $p=a^2+4b^2$ be a prime with $a,b$ integers and $a\equiv1\pmod4$. Then for the determinant $$S(1,p):={\rm det}\bigg[\left(\frac{i^2+j^2}{p}\right)\bigg]_{1\le i,j\le \frac{p-1}{2}},$$ the number $S(1,p)/a$ is an integral square, which confirms a conjecture posed by Cohen, Sun and Vsemirnov.
In this paper we study multiple Dedekind symbols and the associated multiple reciprocity functions. There is a bijection between the two sets of them after a normalization. By this bijection we define products of multiple reciprocity functions, and study the relationship to the shuffle property. We construct and calculate shuffled multiple Dedekind symbols and shuffled multiple reciprocity functions from modular forms by regularized iterated integrals. Also we give a decomposition for them.
In this article we present the history of auxiliary primes used in proofs of reciprocity laws from the quadratic to Artin's reciprocity law. We also show that the gap in Legendre's proof can be closed with a simple application of Gauss's genus theory.
In this paper we mainly focus on some determinants with Legendre symbol entries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\frac dp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0<k<\frac p4:\ (\frac kp)=-1\}|}&\text{if}\ p\equiv1\pmod4, \\(-1)^{\lfloor(p+1)/8\rfloor}&\text{if}\ p\equiv3\pmod4,\end{cases}$$ where $$S(d,p)=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}$$ and $$W_p=\det\left[\left(\frac{i^2-((p-1)/2)!j}p\right)\right]_{0\le i,j\le(p-1)/2}.$$ We …
The aim of this work is to offer a general theory of reciprocity laws for symbols on arbitrary vector spaces, and to show that classical explicit reciprocity laws are particular cases of this theory (sum of valuations on a complete curve, Residue Theorem, Weil Reciprocity Law and the Reciprocity Law for the Hilbert Norm Residue Symbol). Moreover, several reciprocity laws introduced over the past few years by D. V. Osipov, A. N. Parshin, I. Horozov, I. Horozov - M. Kerr and the author -together with D. Hernández Serrano-, can also be deduced from this general expression.
Let $m$ be any positive integer and let $δ_1,δ_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=δ_1\ \quad\text{and}\ \quad\left(\frac{p_{n+j}}{p_{n+i}}\right)=δ_2$$ for all $0\le i<j\le m$, where $p_k$ denotes the $k$-th prime, and $(\frac {\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any di…
We study multilinear symbols on fields taking values in reciprocity sheaves. We prove that any such symbol satisfying natural axioms automatically has Steinberg-type relations, which is a manifestation of the geometry of modulus pairs lying behind.
On a determinant involving linear combinations of Legendre symbols
School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University（西安利物浦大学数学与物理学院）；School of Mathematics, Nanjing University（南京大学数学系）；School of Physical and Mathematical Sciences, Nanjing Tech University（南京工业大学物理与数学科学学院）
We define a two-dimensional Contou-Carrère symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrère symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We relate also the two-dimensional Contou-Carrère symbol with the two-dimensional class field theory.
The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special---our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol--based construction.
Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind symbols. We prove that these newly introduced operators are compatible with Hecke operators on the space of modular forms. As an application, we present formulae to give Fourier coefficients of Hecke ei…
Let $p>3$ be a prime. We investigate Legendre symbol of $\displaystyle \prod_{0<i<j<p/2 \atop p\nmid f(i,j) } f(i,j) \ $, where $i,j\in \Bbb Z, f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=ai^2+bij+cj^2$ and $p\nmid c(a+b+c)$ , we prove that to evaluate the product is equivalent to determine $ \displaystyle \sum_{y=1}^{p-1} \bigg(\frac{y(y+1)(y+k)}{p}\bigg) \pmod{16}$ , where $4c(a+b+c)k \equiv (4ac-b^2)\pmod{p}.$ Parallel results are given for $\displaystyle \prod_{i,j=1 \atop p\nmid f(i,j) }^{(p-1)/2} \bigg(\frac{ f(i,j) }{p}\bigg).$ Then we show that $ \displayst…
Suppose that $p$ is an odd prime and $\genfrac{(}{)}{}{}{\cdot}{p}$ denotes the Legendre symbol modulo $p$. If $p$ is has the form $p= n^2+1$ then one easily verifies that $\genfrac{(}{)}{}{}{a}{p} = \genfrac{(}{)}{}{}{-a}{p}$ for all $a\in \mathbb Z/p\mathbb Z$. We identify various symmetry properties of sequential matrices over $\mathbb Z/(n^2+1)\mathbb Z$ regardless of whether $n^2+1$ is prime. We deduce from these results a collection of symmetries involving Jacobi symbol modulo $n^2+1$ which generalize our above observation on the Legendre symbol.
Usually the boundary map in K-theory localization only gives the tame symbol at $K_{2}$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at p=2. Our idea is to pinch singularities near the ramification locus. This fattens up K-theory and makes the wild symbol vi…
Large sieve inequality for sums of Legendre symbols over short intervals
Université Jean Monnet, Centrale Lyon, INSA Lyon, Université Claude Bernard Lyon 1, CNRS, ICJ UMR5208（让·莫内大学、里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、法国国家科学研究中心、ICJ UMR5208）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematics, University of Bristol（布里斯托大学数学系）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Since the development of higher local class field theory, several explicit reciprocity laws have been constructed. In particular, there are formulas describing the higher-dimensional Hilbert symbol given, among others, by M. Kurihara, A. Zinoviev and S. Vostokov. K. Kato also has explicit formulas for the higher-dimensional Kummer pairing associated to certain (one-dimensional) $p$-divisible groups. In this paper we construct an explicit reciprocity law describing the Kummer pairing associated to any (one-dimensional) formal group. The formulas are a generalization to higher-dimensional local …
We consider an algebraic surface. For an irreducible curve on this surface and for a point on this curve one can associate an artinian ring, which is a sum of two-dimensional local fields. An example of two-dimensional local field is iterated Laurent series. We consider an integer value symbol on a two-dimensional local field. This symbol appears for the description of non-ramified coverings of two-dimensional local fields and for the description of intersection index of divisors on surfaces by adeles. We interpret this symbol as the commutator of lifting of elements in the central extension o…
We explicitly study Kato's residue homomorphisms in Milnor $K$-theory, which are closely related to Contou-Carrère symbols. As applications we establish several reciprocity laws for certain locally defined maps on $K$-groups that are associated to arithmetic surfaces.
The notion of determinant groupoid is a natural outgrowth of the theory of the Sato Grassmannian and thus well-known in mathematical physics. We briefly sketch here a version of the theory of determinant groupoids over an artinian local ring, taking pains to put the theory in a simple concrete form suited to number-theoretical applications. We then use the theory to give a simple proof of a reciprocity law for the Contou-Carrère symbol. Finally, we explain how from the latter to recover various classical explicit reciprocity laws on nonsingular complete curves over an algebraically closed fiel…
We generalize the theory of Contou-Carrère symbols to higher dimensions. To an $(n+1)$-tuple $f_0,\dots,f_n \in A((t_1))\cdots((t_n))^{\times}$, where $A$ denotes a commutative algebra over a field $k$, we associate an element $(f_0,\dots,f_n) \in A^{\times}$, compatible with the higher tame symbol for $k = A$, and earlier constructions for $n = 1$, by Contou-Carrère, and $n = 2$ by Osipov--Zhu. Our definition is based on the notion of \emph{higher commutators} for central extensions of groups by spectra, thereby extending the approach of Arbarello--de Concini--Kac and Anderson--Pablos Romo. F…
In a recent work, the present author generalized a fundamental result of Gauss related to quadratic reciprocity, and also showed that the above result of Gauss is equivalent to a special case of a well-known result of Sylvester related to the Frobenius coin problem. In this note, we use this equivalence to show that the above generalization of the result of Gauss naturally leads to an interesting generalization of the result of Sylvester. To be precise, for given positive coprime integers $a$ and $b$, and for a family of values of $k$ in the interval $0 \leq k < (a-1)(b-1)$, we find the number…
One aspect of Chebyshev's bias is the phenomenon that a prime number, $ q $, modulo another prime number, $ p$, experimentally seems to be slightly more likely to be a nonquadratic residue than a quadratic residue. We thought it would be interesting to model this residue bias as a "random" walk using Legendre symbol values as steps. Such a model would allow us to easily visualize the bias. In addition, we would be able to extend our model to other number fields. In this report, we first outline underlying theory and some motivations for our research. In the second section, we present our findi…
We consider a functional analytic variant of the notion of Tate space, namely the category of those topological vector spaces which have a direct sum decomposition where one summand is nuclear Frechet space and the other is the dual of a nuclear Frechet. We show that, both in the complex and in the p-adic setting, one can use this formalism to define symbols for analytic functions which satisfy Weil-type reciprocity laws.
We prove Fermat's Last Theorem over ${\mathbb Q}(\sqrt{5})$ and ${\mathbb Q}(\sqrt{17})$ for prime exponents $p \ge 5$ in certain congruence classes modulo $48$ by using a combination of the modular method and Brauer-Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of ${\mathbb Q}(\sqrt{5})$ is a generalization to a real quadratic base field of the one used by Chen-Siksek. For the case of ${\mathbb Q}(\sqrt{17})$, this is insufficient, and we generalize a reciprocity constraint of Bennett-Chen-Dahmen-Yazdani using Hilbe…
We will formulate and prove a certain reciprocity law relating certain residues of the differential symbol dlog^2 from the K_2 of a Mumford curve to the rigid analytic regulator constructed by the author in a previous paper. We will use this result to deduce some consequences on the kernel and image of the rigid analytic regulator analogous to some old conjectures of Beilinson and Bloch on the complex analytic regulator. We also relate our construction to the symbol defined by Contou-Carrere and to Kato's residue homomorphism, and we show that Weil's reciprocity law directly implies the recipr…
We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using it in the context of two-dimensional local fields and two-dimensional adelic theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.
We show some fundamental results concerning $3$-dimensional foliated dynamical systems (FDS$^3$ for short) introduced by Deninger. Firstly, we give a decomposition theorem for an FDS$^3$, which yields a classification of FDS$^3$'s. Secondly, for each type of the classification, we construct concrete examples of FDS$^3$'s. Finally, by using the integration theory for smooth Deligne cohomology, we introduce geometric analogues of local symbols and show a Hilbert type reciprocity law for an FDS$^3$. Our results answer the question posed by Deninger.
Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we solve some conjectures of Z.W. Sun concerning $\sum_{k=0}^{p-1}\binom{2k}k^3/m^k\pmod{p^2}$, $\sum_{k=0}^{p-1}\binom{2k}k\b{4k}{2k}/m^k\pmod p$ and $\sum_{k=0}^{p-1}\binom{2k}k^2\b{4k}{2k}/m^k\pmod {p^2}.$ In particular, we show that $\sum_{k=0}^{\frac{p-1}{2}}\binom{2k}k^3\equiv 0\pmod {p^2}$ for $p\equiv 3,5,6\pmod 7$. Let $P_n(x)$ be the Legendre polynomials. In the paper we also show that $ P_{[\frac {p}{4}]}(t)\equiv -\big(\frac{-6}{p}\big)\sum_{x=0}^{p-1} \big(\frac{x^3-3/2(3t+5)x-9t-7}{p}\big)\pmod p$ and d…
We introduce the Shifted Legendre Symbol Problem and some variants along with efficient quantum algorithms to solve them. The problems and their algorithms are different from previous work on quantum computation in that they do not appear to fit into the framework of the Hidden Subgroup Problem. The classical complexity of the problem is unknown despite the various results on the irregularity of Legendre Sequences.
