Jeffrey Stopple
An algorithm is given to efficiently compute $L$-functions with large conductor in a restricted range of the critical strip. Examples are included for about 21000 dihedral Galois representations with conductor near $10^7$. The data shows good agreement with a symplectic random matrix model. Also included are an elliptic curve $L$-function and one for a quadratic Dirichlet character.
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We extend the algorithm of Darmon-Green and Darmon-Pollack for computing p-adic Darmon points on elliptic curves to the case of composite conductor. We also extend the algorithm of Darmon-Logan for computing ATR Darmon points to treat curves of nontrivial conductor. Both cases involve an algorithmic decomposition into elementary matrices in congruence subgroups Γ(N) for ideals N in certain rings of S-integers. We use these extensions to provide additional evidence in support of the conjectures on the rationality of Darmon points.
Three-Torsion Subgroups and Wild Conductors of Genus 3 Hyperelliptic Curves
本文提出用复逼近、同伦延拓和格基约化计算亏格3超椭圆曲线雅可比簇三挠子群的实用方法，并将其应用于计算导子在2处局部指数的野部分。
An algorithm to compute Dirichlet $L$-functions for many quadratic characters is derived. The algorithm is optimal (up to logarithmic factors) provided that the conductors of the characters under consideration span a dyadic window.
We extend previous lists by numerically computing approximations to many L-functions of degree $d=3$, conductor $N=1$, and small spectral parameters. We sketch how previous arguments extend to say that for very small spectral parameters there are no such L-functions. Using the case $(d,N) = (3,1)$ as a guide, we explain how the set of all L-functions with any fixed invariants $(d,N)$ can be viewed as a landscape of points in a $(d-1)$-dimensional Euclidean space. We use Plancherel measure to identify the expected density of points for large spectral parameters for general $(d,N)$. The points f…
We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and get much improved lower bounds on the smallest conductor. For small Galois types we use complete tables of number fields to determine the actual smallest conductor.
We obtain a Weyl law with power savings for the universal families of cuspidal automorphic representations, ordered by analytic conductor, of $\mathrm{GL}_2$ over $\mathbb{Q}$, as well as for Hecke characters over any number field. The method proceeds by establishing the requisite analytic properties of the underlying conductor zeta function.
Non-vanishing of Dirichlet $L$-functions with smooth conductors
针对大的无平方因子光滑导子，研究证明至少35.9%的本原狄利克雷L函数中心值非零，推进了狄利克雷L函数非零性的定量结果。
We prove a Weyl-exponent subconvex bound for any Dirichlet $L$-function of cube-free conductor. We also show a bound of the same strength for certain $L$-functions of self-dual $\mathrm{GL}_2$ automorphic forms that arise as twists of forms of smaller conductor.
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
In this paper we study the forbidden values of the conductor $q$ of the $L$-functions of degree 2 in the extended Selberg class by a novel technique, linking the problem to certain continued fractions and to their weight $w_q$. Our basic result states that if an $L$ function with conductor $q$ exists, then the weight $w_q$ is unique in a suitable sense. From this we deduce several results, both of theoretical and computational nature.
We relate non-critical special values $p$-adic $L$-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to $p$-adic Coleman function called the $p$-adic Eisenstein-Kronecker series, when the conductors of the algebraic Hecke characters are divisible by $p$.
The Mixed Second Moment of Quadratic Dirichlet $L$-functions with Prime Conductors
本文在函数域语境下，针对首一不可约多项式平均的情况，建立了含完备L函数二阶导数的混合二阶矩的渐近公式。
In this paper we extend the hybrid Euler-Hadamard product model for quadratic Dirichlet $L$-functions associated to irreducible polynomials over function fields. We also establish an asymptotic formula for the first twisted moment in this family of L-functions and then we provide further evidence for the conjectural asymptotic formulas for its moments.
We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of $L$-functions associated to quadratic twists of a fixed elliptic curve over $\mathbb{F}_q(t)$ by monic irreducible polynomials, which allows us to show that there exists a monic irreducible polynomial such that the analytic rank of the corresponding twisted elliptic curve is…
On the Minimality of the Conductor for Elliptic Curve $L$-Functions
本文证明在解析秩界中，任何满足二次函数方程和解析延拓的算术不变量必须不小于导子，从而说明导子是此类框架下的最小不变量。
We survey a number of different methods for computing $L(χ,1-k)$ for a Dirichlet character $χ$, with particular emphasis on quadratic characters. The main conclusion is that when $k$ is not too large (for instance $k\le100$) the best method comes from the use of Eisenstein series of half-integral weight, while when $k$ is large the best method is the use of the complete functional equation, unless the conductor of $χ$ is really large, in which case the previous method again prevails.
We give a full description of the functions $F$ of degree 2 and conductor 1 in the general framework of the extended Selberg class. This is performed by means of a new numerical invariant $χ_F$, which is easily computed from the data of the functional equation. We show that the value of $χ_F$ gives a precise description of the nature of $F$, thus providing a sharp form of the classical converse theorems of Hecke and Maass. In particular, our result confirms, in the special case under consideration, the conjecture that the functions in the Selberg class are automorphic $L$-functions.
$L$-functions of degree $2$ and conductor $1$: underlying ideas and a generalisation
本文简述了扩展Selberg类中次数为2且导子为1的$L$-函数分类定理，提出了涉及两个Dirichlet级数的函数方程的新推广，并纠正了原证明的疏漏。
Mean square of inverses of Dirichlet $L$-functions involving conductors
本文研究Dirichlet $L$-函数的负平方矩，得到涉及导子的负二阶矩渐近公式，并应用于改进分圆数域中恢复短生成元算法的成功概率下界。
Furusawa has given an integral representation for the degree 8 L-function of GSp(4) x GL(2) and has carried out the unramified calculation. The local p-adic zeta integrals were calculated in our earlier work under the assumption that the GSp(4) representation πis unramified and the GL(2) representation τhas conductor p. In the present work we generalize to the case where the GL(2) representation has arbitrarily high conductor. The result is that the zeta integral represents the local Euler factor L(s,π\times τ) in all cases. As a global application we obtain a special value result for a GSp(4)…
In the analytic study of trace functions of $\ell$-adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such a transform is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating exampl…
The Second Moment of Rankin-Selberg $L$-Functions in Conductor-Dropping Regimes
研究Rankin-Selberg L-函数在降阶 regime 中二次矩的渐进行为，采用holomorphic Hecke cusp forms等方法，得到渐近公式。
The value-distribution of Artin $L$-functions associated with cubic fields in conductor aspect
本文研究了导子方面非Galois三次域族Artin $L$-函数的值分布，证明其均值可由可显式构造的密度函数积分表示，并借助类数公式应用于三次域类数分布的研究。
Conductors of twisted Weil--Deligne representations
本文研究扭曲下特定 Weil--Deligne 表示关联的 L 函数导子行为，并为每个全局域 K 证明了一对阿贝尔簇关联的 Rankin--Selberg L 函数导子的精确上界。
In a previous article (a joint work with J. Manoharmayum) the modularity of a large class of rigid Calabi-Yau threefolds was established. To make that result more explicit, we recall (and re-prove) a result of Serre giving a bound for the conductor of an "integral" 2-dimensional compatible family of Galois representations and apply this result to give an algorithm that determines the level of a modular rigid Calabi-Yau threefold. We apply the algorithm to three examples.
On partial T-adic exponential sums and partial exponential sums with p-power conductor
本文推广了Liu和Wan的T进指数和理论到部分情形，证明了相关L函数的T进亚纯性，从而给出p幂导子特征部分L函数有理性的p进证明，并给出Newton-over-Hodge估计。
We prove that more than nine percent of the central values $L(\frac{1}{2},χ_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $χ_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},χ_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.
Extreme central values of quadratic Dirichlet $L$-functions with prime conductors
研究素数p≡1 mod 8时L(1/2,χ_p)的下界结果，采用分析方法证明极值下限。
Counting abelian extensions by Artin-Schreier conductor
本文通过定义 Artin-Schreier 导子，对特征 p 整体函数域的有限阿贝尔 p-群 étale 扩张进行计数，并证明其生成函数为有理函数，从而得到精确计数结果。
Computing Euler products and coefficients of classical modular forms for twisted L-functions
提出一种快速计算经典模形式系数及欧拉乘积的算法，并在大导子三重积L-函数上验证其效率。
We give an explicit description of the stable reduction of superelliptic curves of the form $y^n=f(x)$ at primes $\p$ whose residue characteristic is prime to the exponent $n$. We then use this description to compute the local $L$-factor of the curve and the exponent of conductor at $\p$.
Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor
研究D4四次函数域的Artin L-函数低频零点，证明按判别式排序时至少77%的L-函数在中心点非消失，扩展了Durlanik在有理数域上的结果。
The conductor operator acts on a function through multiplying it with the logarithm of the norm of the variable both in position and in momentum space and adding the outcomes. It makes sense at each completion of an arbitrary number field and arose in previous papers by the author where it was shown to be intimately connected with the Explicit Formula of Analytic Number Theory. I complete here its spectral analysis: the conclusion is that the local contribution to the Explicit Formula expressed as an integral over the critical line is completely equivalent to, indeed precisely realizes this sp…
It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve $L$-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-…
We describe a procedure for determining the existence, or non-existence, of an algebraic variety of a given conductor via an analytic calculation involving L-functions. The procedure assumes that the Hasse-Weil L-function of the variety satisfies its conjectured functional equation, but there is no assumption of an associated automorphic object or Galois representation. We demonstrate the method by finding the Hasse-Weil L-functions of all hyperelliptic curves of conductor less than 500.
In a previous paper we proved that if an $L$-function $F$ from the Selberg class has degree $2$, its conductor $q_F$ is a prime number and $F$ is weakly twist-regular at all primes $p\neq q_F$, then $F$ has a polynomial Euler product. In this paper we extend this result to $L$-functions of degree 2 with square-free conductor $q_F$, which are weakly twist-regular at all primes $p\nmid q_F$
In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.
Katz showed that the L-functions of all Dirichlet characters of F_q(t), with conductor a fixed power of a degree one prime, are equidistributed in the limit as q goes to infinity. We generalize this statement to the L-functions of twists of an arbitrary Galois representation by Dirichlet characters, including independence of the L-functions of twists of different representations by the same Dirichlet character. A similar generalization, without the independence statement, for characters with squarefree conductor, was proven by Hall, Keating, and Roditty-Gershon.
We study the minimal resultant divisor of self-maps of the projective line over a number field or a function field and its relation to the conductor. The guiding focus is the exploration of a dynamical analog to Theorem 1.1, which bounds the degree of the minimal discriminant of an elliptic surface in terms of the conductor. We study minimality and semi-stability, considering what conditions imply minimality (Theorem 4.4) and whether semi-stable models and presentations are minimal, proving results in the degree two case (Theorems 4.6, 4.7). We prove the singular reduction of a semi-stable pre…
This article studies the variation of the Swan conductor of a lisse étale sheaf of $\mathbb{F}_{\ell}$-modules $\mathcal{F}$ on the rigid unit disc $D$ over a complete discrete valuation field $K$ with algebraically closed residue field of characteristic $p\neq \ell$. We associate to $\mathcal{F}$ a function ${\rm sw}_{\rm AS}(\mathcal{F}, \cdot): \mathbb{Q}_{\geq 0}\to \mathbb{Q}$, defined with the Abbes-Saito logarithmic ramification filtration, which measures, at each $t\in \mathbb{Q}_{\geq 0}$, the ramification of the restriction of $\mathcal{F}$ to the subdisc of radius $t$ along the spec…
Averages of Arithmetic Functions over Conductors of Function Fields
研究函数域上算术函数在导数群中的平均值，并推进正特征情况下Boston-Markin猜想的类比研究。
We propose a random matrix model for families of elliptic curve L-functions of finite conductor. A repulsion of the critical zeros of these L-functions away from the center of the critical strip was observed numerically by S. J. Miller in 2006; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the one-level density of eigenvalues of orthogonal matrices after appropriate rescaling).Our purpose here is to provide a random matrix model for Miller's surprising discovery. We consider the fami…
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