Étienne Fouvry, Peter Koymans, Carlo Pagano
We show that for $100\%$ of the odd, squarefree integers $n > 0$ the $4$-rank of $\text{Cl}(\mathbb{Q}(i, \sqrt{n}))$ is equal to $ω_3(n) - 1$, where $ω_3$ is the number of prime divisors of $n$ that are $3$ modulo $4$.
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We study the $4$-rank of the ideal class group of $K_n := \mathbb{Q}(\sqrt{-n}, \sqrt{n})$. Our main result is that for a positive proportion of the squarefree integers $n$ we have that the $4$-rank of $\text{Cl}(K_n)$ equals $ω_3(n) - 1$, where $ω_3(n)$ is the number of prime divisors of $n$ that are $3$ modulo $4$.
The SOS Rank of Biquadratic Forms
本文证明了3×2正半定双二次型可表示为四个双线性型平方之和，并提出m×2双二次型可表示为m+1个双线性型平方的猜想。
By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular.
We consider the signature rank of the units in real multiquadratic fields. When the three quadratic subfields of a real biquadratic field $K$ either (a) all have signature rank 2 (that is, fundamental units of norm $-1$), or (b) all have signature rank 1 (that is, have totally positive fundamental units), we provide explicit examples to show there exist infinitely many $K$ having each of the possible unit signature ranks (namely signature rank 3 or 4 in case (a) and signature rank 1,2, or 3 in case (b)). We make some additional remarks for higher rank real multiquadratic fields, in particular …
Greenberg's conjecture and Iwasawa module of Real biquadratic fields II
本文研究实双二次域的$2$-类群秩稳定性，验证了部分数域的格林伯格猜想，并确定了具有平凡$2$-Iwasawa模的实双二次域列表。
Given a pair of modular forms with complex multiplication by distinct imaginary quadratic fields, the four dimensional Galois representation associated to their Rankin--Selberg convolution is induced from a character over an imaginary biquadratic field $F$. Using the Euler system of Lei, Loeffler and Zerbes we bound a Selmer group associated to this character, over the unique $\mathbb{Z}_{p}^{3}$-extension of $F$.
Greenberg's conjecture and Iwasawa module of Real biquadratic fields I
本文研究实双二次域的Greenberg猜想，提出并探讨使Iwasawa模的秩与第一层类群的秩相等的问题，并构造了多族使Iwasawa模平凡或同构于特定循环群及直积的实双二次域，同时给出实三二次域类群2-秩的相关结果。
Minimal rank of universal lattices and number of indecomposable elements in real multiquadratic fields
本文通过给出实多重二次域中泛二次格最小秩及不可分解元素数量的上界，证明了相应域的密度为零，并研究了实双二次域中不可分解元素的结构与具体计算。
Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective
本文从亏元理论角度研究实数双二次域中的欧几里得理想类，证明在特定条件下存在欧几里得理想，并展示相关双二次域集合的密度为零。
Let $p_1\equiv p_2\equiv -q\equiv1 \pmod4$ be different primes such that $\displaystyle\left(\frac{2}{p_1}\right)= \displaystyle\left(\frac{2}{p_2}\right)=\displaystyle\left(\frac{p_1}{q}\right)=\displaystyle\left(\frac{p_2}{q}\right)=-1$. Put $d=p_1p_2q$ and $i=\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\mathbb{Q}(\sqrt{d},i)$ has an elementary abelian $2$-class group, $\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\mathbb{K}_j$ and $\mathbb{L}_j$ within ${k}_2^{(1)}$, that is t…
We introduce Möbius functions of higher rank, a new class of arithmetic functions so that the classical Möbius function is of rank 2. With this idea, we evaluate Dirichlet series on the sum of the reciprocal square of all $r$-free numbers. For the proof, the Riemann zeta function and cyclotomic polynomials play a key role.
Let $p_1 \equiv p_2 \equiv5\pmod8$ be different primes. Put $i=\sqrt{-1}$ and $d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\sqrt{d}, \sqrt{-1})$ has an elementary abelian 2-class group of rank $3$. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group $\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the 2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its seven unramified bicyclic biquadratic extensions $L_i$.
In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isom…
Sails for universal quadratic forms
本文建立广义连分数中的帆与全实数域算术的新联系，并用于双二次域，刻画通用二次型秩和不可分解整数数量随判别式的幂次或对数增长。
We introduce a class of group endomorphisms -- those of finite combinatorial rank -- exhibiting slow orbit growth. An associated Dirichlet series is used to obtain an exact orbit counting formula, and in the connected case this series is shown to have a closed rational form. Analytic properties of the Dirichlet series are related to orbit-growth asymptotics: depending on the location of the abscissa of convergence and the degree of the pole there, various orbit-growth asymptotics are found, all of which are polynomially bounded.
We construct an infinite family of imaginary bicyclic biquadratic number fields $k$ with the 2-ranks of their 2-class groups are $\geq3$, whose strongly ambiguous classes of $k/Q(i)$ capitulate in the absolute genus field $k^{(*)}$, which is strictly included in the relative genus field $(k/Q(i))^*$ and we study the capitulation of the $2$-ideal classes of $k$ in its quadratic extensions included in $k^{(*)}$.
We prove a Voronoi formula for coefficients of a large class of $L$-functions including Maass cusp forms, Rankin-Selberg convolutions, and certain isobaric sums. Our proof is based on the functional equations of $L$-functions twisted by Dirichlet characters and does not directly depend on automorphy. Hence it has wider application than previous proofs. The key ingredient is the construction of a double Dirichlet series.
Let $\mathcal E: y^2 = x^3 + A(T)x + B(T)$ be a nontrivial one-parameter family of elliptic curves over $\mathbb{Q}(T)$, with $A(T), B(T) \in \mathbb Z(T)$, and consider the $k$\textsuperscript{th} moments $A_{k,\mathcal{E}}(p) := \sum_{t (p)} a_{\mathcal{E}_t}(p)^k$ of the Dirichlet coefficients $a_{\mathcal{E}_t}(p) := p + 1 - |\mathcal{E}_t (\mathbb{F}_p)|$. Rosen and Silverman proved a conjecture of Nagao relating the first moment $A_{1,\mathcal{E}}(p)$ to the rank of the family over $\mathbb{Q}(T)$, and Michel proved that if $j(T)$ is not constant then the second moment is equal to $A_{2,…
We generalize Dirichlet's $S$-unit theorem from the usual group of $S$-units of a number field $K$ to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over $S$. Specifically, we demonstrate that the group of algebraic $S$-units modulo torsion is a $\bQ$-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, and that the elements of this vector space which are linearly independent over $\mathbb{Q}$ retain their linear independence over $\mathbb{R}$.
A Weyl group multiple Dirichlet series is a Dirichlet series in several complex variables attached to a root system Phi. The number of variables equals the rank r of the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group W of Phi. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n-th order Gauss sums attached to the root system of type A_2. The basic technique is to construct a rational function in r variables invariant under a certain action of W, and use this to build a ``local factor'' of…
Weyl group multiple Dirichlet series are Dirichlet series in $r$ complex variables, with analytic continuation to $\mathbb{C}^r$ and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank $r$. Such series may be defined for any global field $K$, but in the case when $K$ is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when $K=\mathbb{F}_q(T)$, and describe the denominators and support of the numerators.
Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, which has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. Two constructions of such series are available, one based on summing products of n-th order Gauss sums, the second based on averaging a certain group action over the Weyl group. In this paper we study these constructions and the relationship betwee…
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…
Exact Rank of the Largest Good Lattice Point Matrix
该数学研究证明，对n>2，由小于n且与n互素的整数生成的最大良格点矩阵Uₙ的秩为φ(n)/2+1，通过配对恒等式、奇狄利克雷特征等完成上下界证明，关联狄利克雷定理。
It is shown that the algebra of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that the algebra of bounded Dirichlet series has infinite topological stable rank and infinite Krull dimension.
We fix an elliptic curve $E/\mathbb{F}_q(t)$ and consider the family $\{E\otimesχ_D\}$ of $E$ twisted by quadratic Dirichlet characters. The one-level density of their $L$-functions is shown to follow orthogonal symmetry for test functions with Fourier transform supported inside $(-1,1)$. As an application, we obtain an upper bound of 3/2 on the average analytic rank. By splitting the family according to the sign of the functional equation, we obtain that at least $12.5\%$ of the family have rank zero, and at least $37.5\%$ have rank one. The Katz and Sarnak philosophy predicts that those perc…
We investigate local-global compatibility for cuspidal automorphic representations $π$ for GL(2) over CM fields that are regular algebraic of weight $0$. We prove that for a Dirichlet density one set of primes $l$ and any $ι: \overline{\mathbf{Q}}_l \cong \mathbf{C}$, the $l$-adic Galois representation attached to $π$ and $ι$ has nontrivial monodromy at any $v \nmid l$ in $F$ at which $π$ is special.
Explicit Prime Densities for Lucas Sequence Rank Divisibility
本文研究了Lucas序列中素数p的出现秩ρ_U(p)的Dirichlet密度，推导出闭式公式，完善了Sanna(2022)的工作，覆盖所有Lucas序列和所有d≥1的情况。
We study Euler systems for $\mathbb{G}_m$ over a number field $k$. Motivated by a distribution-theoretic idea of Coleman, we formulate a conjecture regarding the existence of such systems that is elementary to state and yet strictly finer than Kato's equivariant Tamagawa number conjecture for Dirichlet $L$-series at $s=0$. To investigate the conjecture, we develop an abstract theory of `Euler limits' and, in particular, prove the existence of canonical `restriction' and `localisation' sequences in this theory. By using this approach we obtain a variety of new results, ranging from a proof, mod…
Let Phi be a reduced root system of rank r. A Weyl group multiple Dirichlet series for Phi is a Dirichlet series in r complex variables s_1,...,s_r, initially converging for Re(s_i) sufficiently large, that has meromorphic continuation to C^r and satisfies functional equations under the transformations of C^r corresponding to the Weyl group of Phi. A heuristic definition of such series was given in [2], and they have been investigated in certain special cases in [2-6, 11-14]. In this paper we generalize results in [13] to construct Weyl group multiple Dirichlet series by a uniform method, and …
Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^η$ with $0 < η< 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimesχ)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $χ$ is a primitive Dirichlet character modulo $M$. These bounds are attained through an unamplified second moment method using a modified version of the delta method due to R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modification of the delta method.
We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. In type C, they conjecturally arise from the Fourier-Whittaker coefficients of minimal parabolic Eisenstein series on an n-fold metaplectic cover of SO(2r+1). For any odd n, we construct an infinite family of Dirichlet series conjecturally satisfying the above analytic properties. The coefficients of …
Asymptotic independence of class-group 4-ranks in correlated pairs of imaginary quadratic fields
本文研究相关虚二次域对的类群4秩渐近独立性，证明其联合分布收敛于Cohen--Lenstra--Gerth分布乘积，结合理想扩张分解给出双二次族类群分布猜想，并用Smith盒方法与矩反演技术证明相关结果。
A Bombieri-Vinogradov theorem for higher rank groups
该研究针对数域上非自二次扭的GL_n自对偶尖点自守表示，建立了素理想处Dirichlet系数的无条件Bombieri-Vinogradov型定理，改进了依赖未证假设的分布水平，并给出低秩情形的数论应用。
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