arXiv math/04082212004-08math.NT

Converse theorems assuming a partial Euler product

David W. Farmer, Kevin Wilson

备注：: 12 pages, LaTeX. Final version. To appear in The Ramanujan Journal

英文摘要：

Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of a particular form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $Γ(1)$. Weil extended this result to $Γ_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. Conrey and Farmer extended Hecke's result for certain small $N$, assuming that the local factors in the Euler product of $F(s)$ were of a special form. We make the same assumption on the Euler product and describe an approach to the converse theorem using certain additional assumptions. Some of the assumptions may be related to second order modular forms.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Twists, Euler products and a converse theorem for $L$-functions of degree 2 in the Selberg class

We prove a general result relating the shape of the Euler product of an $L$-function to the analytic properties of certain linear twists of the $L$-function itself. Then, by a sharp form of the transformation formula for linear twists, we check the required analytic properties in the case of $L$-functions of degree 2 and conductor 1 in the Selberg class. Finally we prove a converse theorem, showing that $ζ(s)^2$ is the only member of the Selberg class satisfying the above conditions and, moreover, having a pole at $s=1$.

2012-07-10math.NT

A conjectural extension of Hecke's converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.

2017-09-27math.NT

关于假设非分歧扭曲的 Hilbert 模形式逆定理

On converse theorems for Hilbert modular forms assuming unramified twists

本文证明了次数大于1的全实域上 Hilbert 模形式的两个逆定理，分别通过非分歧 Hecke 特征扭曲的 L-级数函数方程，以及结合 Euler 乘积恢复出预期水平的模形式，推广了 GL2 逆定理并完全去除了对分歧扭曲的假设。

2025-11-03math.NT

A converse theorem for $Γ_0(13)$

We prove that a Dirichlet series with a functional equation and Euler product of a particular form can only arise from a holomorphic cusp form on the Hecke congruence group $Γ_0(13)$. The proof does not assume a functional equation for the twists of the Dirichlet series. The main new ingredient is a generalization of the familiar Weil's lemma that played a prominent role in previous converse theorems.

2006-01-23math.NT

A converse theorem for degree 2 elements of the Selberg class with restricted gamma factor

We prove a converse theorem for a family of L functions of degree 2 with gamma factor coming from a holomorphic cuspform. We show these L functions coincide with either those coming from a newform or a product of L functions arising from Dirichlet characters. We require some analytic data on the Euler factors, but don't require anything on the shape. We also suppose that the twisted L functions satisfy expected functional equations. We allow the non-trivial twists to have arbitrary poles.

2021-10-07math.NT

An extension of Hecke's converse theorem

Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$ has a functional equation of the appropriate form, then $F(s)=L_f(s)$ for some holomorphic newform $f(z)$ on $Γ(1)$. Weil extended this result to $Γ_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet characters. We show that, at least for small $N$, the assumption on twists can be replaced by an assumption on the local factors of the Euler product of $F(s)$.

1995-02-22math.CVmath.NT

更多相关论文收起更多论文

半整数权中的一个逆定理

A converse theorem in half-integral weight

本文利用半整数权 Petersson 迹公式，将 Booker、Farmer 和 Lee 的逆定理推广至半整数权模形式，并以带加法扭转 $L$-级数的函数方程为判定条件。

2024-09-09math.NT

三次阿贝尔域的Euler-Kronecker常数的极值

Extreme values of Euler-Kronecker constants of abelian cubic fields

Waseda University（早稻田大学）

本文在广义黎曼猜想下，研究三次阿贝尔域和二次域的Euler-Kronecker常数的极值，给出了相应结果。

2026-09-25math.NT

On the local structure of the set of values of Euler's $φ$ function

Assuming the validity of Dickson's conjecture, we show that the set $\mathcal{V}$ of values of the Euler's totient function $φ$ contains arbitrarily large arithmetic progressions with common difference 4. This leads to the question of proving unconditionally that this set $\mathcal{V}$ has a positive upper Banach density.

2021-03-01math.NT

Heegner点的例外零点与Gross-Zagier和Kolyvagin定理的p-逆定理

Exceptional zeros for Heegner points and $p$-converse to the theorem of Gross-Zagier and Kolyvagin

本文证明了在素数p>3处具有乘性约化的椭圆曲线的Gross-Zagier和Kolyvagin定理的p-逆定理，关键工具是Bertolini-Darmon-Prasanna的p-进公式的推广和Heegner点的例外零点公式。

2024-09-02math.NT

Ruelle type L-functions versus determinants of Laplacians for torsion free abelian groups

We study Ruelle's type zeta and $L$-functions for a torsion free abelian group $\G$ of rank $\n\ge 2$ defined via an Euler product. It is shown that the imaginary axis is a natural boundary of this zeta function when $\n=2,4$ and 8, and in particular, such a zeta function has no determinant expression. Thus, conversely, expressions like Euler's product for the determinant of the Laplacians of the torus $\bR^{\n}/\G$ defined via zeta regularizations are investigated. Also, the limit behavior of an arithmetic function arising from the Ruelle type zeta function is observed.

2007-09-25math.NT

中心处的欧拉乘积及其对切比雪夫偏差的应用

Euler Products at the Centre and Applications to Chebyshev's Bias

本文在广义黎曼假设及必要时Ramanujan–Petersson猜想下，证明了GL_n自守L-函数部分欧拉乘积在中心点的渐近公式，证实Kurokawa猜想，并应用于Aoki-Koyama框架中的切比雪夫偏差问题。

2024-05-02math.NT

函数方程刻画狄利克雷特征

Functional Equations Characterize Dirichlet Characters

The Ohio State University（俄亥俄州立大学）

本文证明，在温和假设下，形如∑_{n≥1} f(n)n^{-s}的L级数满足的函数方程迫使系数函数f为原初狄利克雷特征，从而也迫使欧拉乘积存在。

2026-07-02math.NT

局部逆定理与Langlands参数

Local converse theorems and Langlands parameters

本文为可接受对偶群的拟分裂约化群提出并证明了局部逆定理，并探讨了其变体在经典群和G2上的应用。

2025-10-28math.NTmath.RT

椭圆曲线 $L$-函数的欧拉乘积渐进行为

Euler Product Asymptotics for $L$-functions of Elliptic Curves

本文基于椭圆曲线 $L$-函数的欧拉乘积渐进行为，证明了在黎曼假设下，点数的渐近行为与秩的关系，并推导出相关猜想的反向结果。

2026-01-15math.NT

A demonstration of a theorem on the order observed in the sums of divisors

Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved a recurrence relation for the sum of divisors function. In E244 Euler proves the pentagonal number theorem, and then repeats the proof of the recurrence. In Proposition 1, Euler proves that $\prod_{n=1}^\infty (1-a_n)=1-\sum_{n=1}^\infty a_n(1-a_1)...(1-a_{n-1})$. In Proposition 2 he explicitly states the case $a_n=x^n$. Then in Proposition 3, Euler applies…

2009-07-30math.HOmath.NT

Finite Euler products and the Riemann Hypothesis

We show that if the Riemann Hypothesis is true, then in a region containing most of the right-half of the critical strip, the Riemann zeta-function is well approximated by short truncations of its Euler product. Conversely, if the approximation by products is good in this region, the zeta-function has at most finitely many zeros in it. We then construct a parameterized family of non-analytic functions with this same property. With the possible exception of a finite number of zeros off the critical line, every function in the family satisfies a Riemann Hypothesis. Moreover, when the parameter i…

2007-04-26math.CVmath.NT

关于塞尔伯格（Selberg）类狄利克雷级数：小次数情形

On the Selberg class of Dirichlet series: small degrees

本文梳理了具有算术意义的狄利克雷级数研究中，学界围绕函数方程、欧拉乘积与黎曼假设的关联，s=1单极点函数的结构，以及自守形式梅林变换的逆定理这三方面形成的相关猜想。

1992-04-01math.NT