A remarkable identity in class numbers of cubic rings

On the analytic properties of a cubic Dirichlet series associated to a cubic metaplectic form

In this paper we study the analytic properties of a certain cubic Dirichlet series associated to a metaplectic form $f$ over the cubic cover of $GL_2.$ Such a sum generalizes the work of Shimura in studying a similar quadratic Dirichlet series for a half-weight modular form $f.$ Shimura connects the analytic properties of his Dirichlet series to the L-function of a holomorphic modular form via a converse theorem. This connection, and its higher cover generalizations, has been given the name: Shimura's correspondence. Even assuming Shimura's correspondence for the cubic cover of $GL_2,$ the ana…

Nicomachus恒等式的平移与推广

Translations and extensions of the Nicomachus identity

本文通过引入平移参数、变量参数、数列乘积和附加数来寻找Nicomachean恒等式，在由平移参数得到的立方方程研究中发现了定二次型和不定二次型的解，并研究了几何上向满足该恒等式的数列附加两个数的性质。

Automatic Solving of Cubic Diophantine Equations Inspired by Ramanujan

In Ramanujan's Lost Notebook there is an amazing identity that furnishes infinitely many "almost counterexamples" to the cubic Fermat's Last Theorem, with no indication whatsoever how he discovered it. In 1995, Michael Hirschhorn explained, in a brilliant way, how Ramanujan may have done it, based on a certain polynomial identity for a sum of four cubes. Much earlier, Eri Jabotinsky, in an article published in 1946 (in a mathematics journal for teenagers) explained how Ramanujan may have discovered these polynomial identities needed for Hirschhorn's approach. Here we combine these two brillian…

Franklin's argument proves an identity of Zagier

Recently Zagier proved a remarkable q-series identity. We show that this identity can also be proved by modifying Franklin's classical proof of Euler's pentagonal number theorem.

关于\(Γ_0(2)^+\)上的类数为一的三次弗里克伴随式及精确的楚德诺夫斯基桥

Class-Number-One Cubic Fricke Companions on $Γ_0(2)^+$ and an Exact Chudnovsky Bridge

证明了\(\Gamma_0(2)^+\)上RS级数的结构分类及桥接结果，确定特定判别式下不可约三次弗里克尖点伴随式情况，给出\(z\)与\(X\)的显式桥接等

On the Ohno-Nakagawa Theorem

In this paper we give a new proof of the Ohno-Nakagawa Theorem using the techniques of $L$-series. By applying Eisenstein's parametrization of binary cubic forms on the one hand, and a class field theory interpretation of Datskovsky \& Wright's Theorem on the other, we reduce the Ohno-Nakagawa Theorem to an identity involving the $L$-series and the truncated $L$-series of quadratic orders. We prove this identity by establishing a general relation among these two types $L$-series.

Identities for field extensions generalizing the Ohno-Nakagawa relations

In previous work, Ohno conjectured, and Nakagawa proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of `extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms. In the present paper we generalize their result by proving a similar identity relating certain degree l fields with Galois groups D_l and F_l respectively, for any odd prime l, and in particular we give ano…

Beyond endoscopy for the Symmetric Cube L-function

This paper is a first attempt at getting information on a symmetric power representation of a $GL_2$ automorphic form via a trace formula that is beyond endoscopic techniques. In particular, we study the symmetric third power representation for all forms over $\Q$ adjoined the cube roots of unity. A key tool needed in the study is an identity relating cubic exponential sums to Kloosterman sums. This very same identity is crucial to the fundamental lemma of a trace formula comparison in work of Mao and Rallis.

On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. These forms are related by Watson's transformations. To prove this identity we employ the Siegel--Weil and the Smith--Minkowski product formulas.

Bernoulli Number Identities from Quantum Field Theory and Topological String Theory

We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli number convolution identity. We then generalize each of these three identities into new families of convolution identities depending on a continuous parameter. We rederive a cubic generalization of Miki'…

十变量布尔三次型的分类

Classification of Boolean Cubic Forms in Ten Variables

通过秩分层枚举，对十变量布尔三次型在GL(10,2)作用下的轨道进行分类，得到3691560个非零轨道，并提供了快速完全不变量。