On the Landau-Siegel Zeros Conjecture

Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.

Discrete mean estimates and the Landau-Siegel zero

Let $χ$ be a real primitive character to the modulus $D$. It is proved that $$ L(1,χ)\gg (\log D)^{-2022} $$ where the implied constant is absolute and effectively computable. In the proof, the lower bound for $L(1,χ)$ is first related to the distribution of zeros of a family of Dirichlet $L$-functions in a certain region, and some results on the gaps between consecutive zeros are derived. Then, by evaluating certain discrete means of the large sieve type, a contradiction can be obtained if $L(1,χ)$ is too small.

素数丢番图逼近与Landau–Siegel零点

Diophantine approximation by primes and Landau--Siegel zeros

本文在假设Siegel零点存在或足够强的条件下，证明了有限型无理数$\alpha$与实数$\beta$的线性组合$\alpha p+\beta$可被素数$p$逼近到$p^{-1/3+\varepsilon}$或$p^{-1/3-1/20}$的精度，给出了素数丢番图逼近的新结果。

三重积L函数的兰道-西格尔零点

Landau-Siegel Zeros of Triple Product L-functions

本文研究三重积L函数的兰道-西格尔零点问题，证明在一般类型情况下无例外零点，并给出非一般类型情况下的零点可能条件。

在兰道-西格尔零的存在下算术进展中的素数

Primes in arithmetic progressions under the presence of Landau-Siegel zeroes

本文在假设兰道-西格尔零存在的情况下，研究了算术进展中素数的分布，并在更宽的模数范围内获得了ψ(x;q,a)的渐近公式。

关于兰道-西格尔零点的数值计算

Numerical computations concerning Landau-Siegel zeros

该研究通过新的算法验证了大量模数下二次狄利克雷 $L$-函数在特定区域内的非零性，突破了之前的研究成果。

Theta block conjecture for Siegel paramodular forms

The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite series of theta blocks of weights 2 and 3. The proof is based on Scheithauer's classification of reflective modular forms of singular weight.

Siegel零点与Hardy-Littlewood猜想

Siegel Zeros and the Hardy-Littlewood Conjecture

本文针对Siegel零点界问题，给出一种比Fei和Jia的条件证明略简化的新论证，重新证明相同结论。

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

We formulate an explicit refinement of Böcherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the relevant local integrals that appear in the refined global Gan-Gross-Prasad conjecture for Bessel periods as proposed by Yifeng Liu. We note several consequences of our conjecture to arithmetic and analytic properties of L-functions and Fourier coefficients of Siegel modular forms.

Chowla猜想与Landau-Siegel零点

The Chowla conjecture and Landau-Siegel zeroes

本文在Landau-Siegel零点存在的条件下，对Chowla猜想中的相关和式建立了非平凡界，改进了先前结果。

Uniform behavior of families of Galois representations on Siegel modular forms and the Endoscopy Conjecture

We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.

A Generalization of Siegel's Theorem and Hall's Conjecture

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upon the number of prime factors dividing a fixed coordinate? If the bound is zero, then Siegel's Theorem guarantees that there are only finitely many such points. We consider, theoretically and computationally, two conjectures: one is a generalization of Siegel's Theorem and the other is a refinement which resonates with Hall's conjecture.

Efficiently generated spaces of classical Siegel modular forms and the Boecherer conjecture

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this generating set we verify the Boecherer conjec- ture for non-rational eigenforms. As one further application we verify another conjectures for weights up to 150 and investigate an analogue of the Victor-Miller basis. Additionally, we describe some arithmetic properties of the basis we found.

Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals

We study the distribution of prime numbers under the unlikely assumption that Siegel zeros exist. In particular we prove for \[ \sum_{n \leq X} Λ(n) Λ(\pm n+h) \] an asymptotic formula which holds uniformly for $h = O(X)$. Such an asymptotic formula has been previously obtained only for fixed $h$ in which case our result quantitatively improves those of Heath-Brown (1983) and Tao and Teräväinen (2021). Since our main theorems work also for large $h$ we can derive new results concerning connections between Siegel zeros and the Goldbach conjecture and between Siegel zeros and primes in almost al…

Siegel's Theorem and the Shafarevich Conjecture

It is known that in the case of hyperelliptic curves the Shafarevich conjecture can be made effective, i.e., for any number field k and any finite set of places S of k, one can effectively compute the set of isomorphism classes of hyperelliptic curves over k with good reduction outside S. We show here that an extension of this result to an effective Shafarevich conjecture for Jacobians of hyperelliptic curves of genus g would imply an effective version of Siegel's theorem for integral points on hyperelliptic curves of genus g.

Applications of Arthur's multiplicity formula to Siegel modular forms

We give two applications of Arthur's multiplicity formula to Siegel modular forms. The one is a lifting theorem for vector valued Siegel modular forms, which contains Miyawaki's conjectures and Ibukiyama's conjectures. The other is the strong multiplicity one theorem for Siegel modular forms of scalar weights and level one.

Siegel's theorem and the abc conjecture

Following N. Elkies ("ABC implies Mordell") we show that the abc conjecture of Masser-Oesterle implies an effective version of Siegel's theorem about integral points on algebraic curves, i.e. an upper bound for the S-integral points where the dependence on S is explicit. The converse statement is also announced in this note. For both results, the main geometric tool is a theorem of G.V. Belyi.

Landau-Siegel Zeroes and Black Hole Entropy

There has been some speculation about relations of D-brane models of black holes to arithmetic. In this note we point out that some of these speculations have implications for a circle of questions related to the generalized Riemann hypothesis on the zeroes of Dirichlet $L$-functions.

关于Landau-Siegel零点及其他相关量的数值估计

Numerical estimates on the Landau-Siegel zero and other related quantities

本文对不超过10^7的奇素数模的Dirichlet L-函数进行数值计算，给出L(1,χ)的显式下界和Landau-Siegel零点位置的显式上界，并附带得到虚二次域类数相关界的信息。

On Euler-Kronecker constants and the generalized Brauer-Siegel conjecture

As a natural generalization of the Euler-Mascheroni constant $γ$, Y. Ihara introduced the Euler-Kronecker constant $γ_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $γ_K$ in a tower of number fields $\mathcal{K}$ implies the generalized Brauer-Siegel conjecture for $\mathcal{K}$ as formulated by Tsfasman and Vlǎduţ. Moreover, we use known bounds on $γ_K$ for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function $ζ_K(s)$ in the critical strip.

On the generalized Brauer-Siegel theorem for asymptotically exact families with solvable Galois closure

In 2002, M. A. Tsfasman and S. G. Vlăduţ formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable normal closure.

The Hardy--Littlewood--Chowla conjecture in the presence of a Siegel zero

Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$, we establish an asymptotic formula for $$\sum_{n\leq x}Λ(n+h_1)\cdots Λ(n+h_k)λ(n+h_{1}')\cdots λ(n+h_{\ell}')$$ for any $0\leq k\leq 2$ and $\ell \geq 0$. Specializing to either $\ell=0$ or $k=0$, we deduce the previously known results on the Hardy--Littlewood (or twin primes) conjecture and the Chowla conjecture under the existence of Siegel zeros, due to H…

几乎 Sn 字段上数字段族的 Brauer-Siegel 定理

Brauer-Siegel theorem for families of number fields over almost Sn fields

本文在几乎 Sn 字段上建立了 Brauer-Siegel 定理的新形式，并推广了 Tsfasman-Vladut 的广义猜想。

Siegel单位的调节子及其应用

Regulators of Siegel units and applications

本文给出了两个Siegel单位调节子的L值公式，并应用于验证导子为14、21、35、48和54的椭圆曲线的Boyd与Zagier猜想。

On the conditional bounds for Siegel zeros

Under a weakened version of Hardy-Littlewood Conjecture on the number of representations in Goldbach problem, J. H. Fei proved bounds for the Siegel zeros. Recently G. Bhowmik and K. Halupczok generalized Fei's result under a weaker conjecture. In the first version of this paper on arXiv, we pointed out a defect in the paper of G. Bhowmik and K.Halupczok, and assumed a new conjecture and used new discussion to overcome this defect to recover their result. Afterwards, in the second version of their paper, G. Bhowmik and K. Halupczok assumed previous conjecture and used our new discussion to get…

On the cuspidality of pullbacks of Siegel Eisenstein series and applications to the Bloch-Kato conjecture

Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…

Repulsive behavior in an exceptional family

The existence of a Landau-Siegel zero leads to the Deuring-Heilbronn phenomenon, here appearing in the 1-level density in a family of quadratic twists of a fixed genus character L-function. We obtain explicit lower order terms describing the vertical distribution of the zeros, and realize the influence of the Landau-Siegel zero as a resonance phenomenon.

关于OpenAI证明Landau-Siegel零点不存在的工作

On the OpenAI proof that Landau-Siegel zeros do not exist

针对OpenAI关于Landau-Siegel零点不存在的工作，本文详细阐述其带显式常数的直接证明，结合Lu等人的计算得到导体相关界，核心方法为双二次域上的插值行列式论证，显式常数与表述为新内容。

关于Siegel模形式的Lang--Trotter猜想

On the Lang--Trotter conjecture for Siegel modular forms

对于亏格2的尖点Siegel模特征形式，证明了与其相关的Galois表示相容系的adelic开像定理，推广了Ribet和Momose关于椭圆模形式的结果，并据此研究了Hecke特征值a_p的分布，得到了固定a∈C时集合{p≤x: a_p=a}大小的上界，类似于椭圆曲线的Lang--Trotter猜想。

Sur l'effectivite du theoreme de Siegel et la conjecture abc

Nous montrons qu'un raffinement du théorème de Siegel sur les points entiers de courbes algébriques impliquerait la conjecture abc de Masser-Oesterlé. Nous formulons une hypothèse "Siegel uniforme" qui est une majoration de la hauteur des points S-entiers de la courbe, en termes du corps de rationalité et de l'ensemble de places S. La validité de l'hypothèse pour une quelconque courbe algébrique de caractéristique d'Euler-Poincaré strictement négative, impliquerait une version de la conjecture abc. Ceci étend aux points S-entiers des résultats précédents de L. Moret-Bailly, et est en quelque s…

Condtional Bounds on Siegel Zeros

We present an overview of bounds on zeros of $L$-functions and obtain some improvements under weak conjectures related to the Goldbach problem.

Siegel Zeros and Sarnak's Conjecture

Assuming the existence of Siegel zeros, we prove that there exists an increasing sequence of positive integers for which Chowla's Conjecture on $k$-point correlations of the Liouville function holds. This extends work of Germán and Kátai, where they studied the case $k=2$ under identical hypotheses. An immediate corollary, which follows from a well-known argument due to Sarnak, is that Sarnak's Conjecture on Möbius disjointness holds. More precisely, assuming the existence of Siegel zeros, there exists a subsequence of the natural numbers for which the Liouville function is asymptotically orth…

Proofs of Ibukiyama's conjectures on Siegel modular forms of half-integral weight and of degree 2

We prove Ibukiyama's conjectures on Siegel modular forms of half-integral weight and of degree 2 by using Arthur's multiplicity formula on the split odd special orthogonal group $\SO_5$ and Gan-Ichino's multiplicity formula on the metaplectic group $\Mp_4$. In the proof, the representation theory of the Jacobi groups also plays an important role.

算术Siegel-Weil公式对于球Hecke代数

Arithmetic Siegel-Weil for the spherical Hecke algebra

本文提出并部分验证了算术Siegel-Weil公式的推广，将酉Rapoport-Zink空间上环的交点重数与局部Whittaker函数的中心导数相联系，并引入Hecke对应的作用，同时推广了表示密度理论。

关于Landau-Siegel零点与奇异模的高度

On Landau-Siegel zeros and heights of singular moduli

本文改进Granville-Stark结果，给出L'/L(1,χ_D)与奇异模高度的渐近公式，并基于一致abc猜想推出Landau-Siegel零点不存在的新界，对光滑判别式有改进。

Heuristics in direction of a p-adic Brauer--Siegel theorem

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence of a constant C\_p>0 such that log(\#T\_K) $\le$ C\_p log(\sqrt(D\_K)) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer--Siegel Theorem, wearing here on the valuation of the residue at s=1 (essentially equal to \#T\_K) of the p-adic …

Generalizations of Siegel's and Picard's Theorems

We prove new theorems which are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. These include results on integral points over varying number fields of bounded degree and results on Kobayashi hyperbolicity. We give a number of new conjectures describing, from our point of view, how we expect Siegel's and Picard's theorems to optimally generalize to higher dimensions. In some special cases we will be able to relate our conjectures to existing conjectures. In this respe…

Derivative of the standard $p$-adic $L$-function associated with a Siegel form

In this paper we construct a two variables $p$-adic $L$-function for the standard representation associated with a Hida family of parallel weight genus $g$ Siegel forms, using a method previously developed by Böcherer--Schmidt in one variable. When a form of weight $g+1$ is Steinberg at $p$, a trivial zero appears and, using the method of Greenberg--Stevens, we calculate the first derivative of this $p$-adic $L$-function and show that it has the form predicted by a conjecture of Greenberg on trivial zeros.

向量值Siegel自守形式的锥猜想

The cone conjecture for vector-valued Siegel automorphic forms

本文证明了特征$p$下秩为$n$的Siegel模簇上自守向量丛的有效锥由$G$-zip叠编码，证明了零阶上同调群在zip锥外消失，并在$n=3$时给出显式消失条件，同时定义了平凡型自守形式。

Siegel模形式的非尖点Hida理论与p进L-函数的平凡零点

Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions

本文通过发展非尖点Siegel模形式的Hida理论，利用Greenberg-Stevens方法构造改进的p进L-函数，研究了P-普通Siegel模形式的标准p进L-函数在半稳定平凡零点处的导数，从而部分证明了Greenberg关于p进L-函数平凡零点的阶数与首项系数的猜想。

Siegel modular forms of genus 2 and level 2: cohomological computations and conjectures

We study the cohomology of certain local systems on moduli spaces of principally polarized abelian surfaces with a level 2 structure. The trace of Frobenius on the alternating sum of the étale cohomology groups of these local systems can be calculated by counting the number of pointed curves of genus 2 with a prescribed number of Weierstrass points over the given finite field. This cohomology is intimately related to vector-valued Siegel modular forms. The corresponding scheme in level 1 was carried out in [FvdG]. Here we extend this to level 2 where new phenomena appear. We determine the cont…

与Siegel尖点形式相关的Lambert级数的渐近展开

An asymptotic expansion for a Lambert series associated to Siegel cusp forms

本文研究Siegel尖点形式相关的Lambert级数，发现其常数项可类似地用Riemann zeta函数的非平凡零点表达，推广了Hafner和Stopple关于Zagier猜想的结果。

Smallness of Faltings heights of CM abelian varieties

We prove that assuming the Colmez conjecture and the ``no Siegel zeros" conjecture, the stable Faltings height of a CM abelian variety over a number field is less than or equal to the logarithm of the root discriminant of the field of definition of the abelian variety times an effective constant depending only on the dimension of the abelian variety. In view of the fact that the Colmez conjecture for abelian CM fields, the averaged Colmez conjecture, and the ``no Siegel zeros" conjecture for CM fields with no complex quadratic subfields are already proved, we prove unconditional analogues of t…

Motivic action for Siegel modular forms

We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of a Hecke eigensystem to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint $L$-function of $f$. We also prove some unconditional results in special case…

Arithmetic degrees of special cycles and derivatives of Siegel Eisenstein series

Let V be a rational quadratic space of signature (m,2). A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with SO(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m+1. We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel-Weil formulas at the archimed…

关于西格尔问题与G-函数的德沃克猜想

On Siegel's problem and Dwork's conjecture for $G$-functions

本文否定回答了G-函数的西格尔问题，构造出无穷多个非超几何代数拉回的几何秩二局部系统，为德沃克猜想提供新反例并回答了Krammer的问题。

仿模西格尔三重fold的几何 Jacquet-Langlands 对应

A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds

本文研究仿模层级西格尔模三重fold的 Picard-Lefschetz 公式，证明权-单值猜想，并借助 Rapoport-Zink 一致化构造 GSp₄ 与定内形式间的几何 Jacquet-Langlands 对应，同时得到整数值结果与降层级定理。

关于Shafarevich和Siegel定理

Sobre los teoremas de Shafarevich y Siegel

本文提供了一个新的Shafarevich定理证明，不依赖迪菲安特逼近或超越理论，为有限性迪菲安特定理提供了新切入点。

素数元组与 Siegel 零点

Prime Tuples and Siegel Zeros

本文在无穷多个近实轴 Siegel 零点假设下，证明素数 m 元组间距可降至 ≪e^{1.9828m}，并改进三至六元组间距上界。

Harder猜想与Hermitian自守形式

Harder's conjecture and Hermitian automorphic forms

研究Hermitian Klingen-Eisenstein提升与尖形式之间的同余关系，证明同余中的Hermitian尖特征形式是Siegel尖特征形式的Hermitian旋提升，从而得到Harder猜想预言的旋量L-多项式同余。

特征 $p$ 下向量值 Siegel 自守形式的消失定理

A vanishing theorem for vector-valued Siegel automorphic forms in characteristic $p$

本文证明了特征 p 下向量值 Siegel 自守形式在权处于显式轨迹之外时为零，从而验证了与 Goldring 提出的 Hodge 型 Shimura 簇一般猜想的一个特例。

On the Ribenboim-Landau Conjecture

We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a positive solution.

Level lowering on Siegel modular threefold of paramodular level

In this article we prove several level lowering results for cuspidal automorphic representations occurring in the cohomology of the Siegel modular threefold with paramodular level structure by adapting a method of Ribet in his proof of the Serre's epsilon conjecture. The proof is purely geometric and relies on the description of the supersingular locus of certain quaternionic unitary Shimura variety and an arithmetic level raising result on this Shimura variety. The heart of the proof is a comparison of the dimension of the space of vanishing cycles on the paramodular Siegel modular threefold …

On vector-valued Siegel modular forms of degree 2 and weight (j,2)

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.