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We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with r-modifications for an even integer r. We prove an identity between (1) The r-th central derivative of the quadratic base change L-function associated to an everywhere unramified cuspidal automorphic representation π of PGL2; (2) The self-intersection number of the π-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of L-functions.

In this paper we derived a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalized Venkatesh and Woodbury’s work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors.

We give a new construction of (φ,Ĝ )-modules using the theory of prisms developed by Bhatt and Scholze. As an application, we give a new proof about the equivalence between the category of prismatic F-crystals in finite locally free Δ-modules over (K)Δ and the category of lattices in crystalline representations of GK, where K is a complete discretely valued field of mixed characteristic with perfect residue field. We also generalize this result to semi-stable representations using the absolute logarithmic prismatic site defined by Koshikawa.

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Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three nonzeros. Recently, the authors of [37]constructed a class of cyclic codes with arbitrary number of nonzeros, and computed the weight distribution for several cases. In this paper, we determine the weight distribution for a new family of such codes. This is achieved by introducing certain new methods, such as the theory of Jacobi sums over finite fields and subtle treatment of some complicated combinatorial identities.

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In this article, we define a “ recursive local” algorithm in order to construct two reccurent numerical sequences of positive prime numbers (𝑈2𝑛) and (𝑉2𝑛), ((𝑈2𝑛) function of (𝑉2𝑛)), such that for any integer n≥ 2, their sum is 2n. To build these , we use a third sequence of prime numbers (𝑊2𝑛) defined for any integer n≥ 3 by : 𝑊2𝑛 = Sup(p∈IP : p ≤ 2n-3), where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers 2n between 4 and 4.1018. . In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of 2n= 101000 . Thus, thanks to this algorithm of “ascent and descent”, we can validate the strong Euler-Goldbach conjecture.

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We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products q_n d(q_n θ,Z) where the qn's are the denominators of the convergents associated with the real number θ by the ordinary continued fraction algorithm. Beside these two main results, we show that when d≥2, for almost all vectors θ∈R^d, lim inf_{n→∞} q_{n+k} d(q_n θ,Z^d)=0 for all positive integers k, where (q_n)_{n∈N} is the sequence of best approximation denominators of θ.

We present an integral representation for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. Our construction is uniform over all classical groups, and is applicable to all cuspidal representations; it does not require genericity. The main new ideas of the construction are the use of generalized Speh representations as inducing data for the Eisenstein series and the introduction of a new (global and local) model, which generalizes the Whittaker model. Here we consider linear groups, but our construction also extends to arbitrary degree metaplectic coverings; this will be the topic of an upcoming work.

In this paper, based on an idea of Tian we establish a new sufficient condition for a positive integer n to be a congruent number in terms of the Legendre symbols for the prime factors of n. Our criterion generalizes previous results of Heegner, Birch–Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving the criterion is to give formulae for the analytic Tate–Shafarevich number $\mathcal{L}(n)$ in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the generalized Gross–Zagier formula of Yuan–Zhang–Zhang.

Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about periods.

We study the geometry of the p‑adic analogues of the complex analytic period spaces first introduced by Griffiths. More precisely, we prove the Fargues–Rapoport conjecture for p‑adic period domains: for a reductive group G over a p‑adic field and a minuscule cocharacter μ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,μ) is fully Hodge–Newton decomposable.

The aim of this paper is to present a function field analogue of the classical Kronecker limit formula. We first introduce a “non-holomorphic” Eisenstein series on the Drinfeld half plane, and connect its “second term” with Gekeler’s discriminant function. One application is to express the Taguchi height of rank 2 Drinfeld modules with complex multiplication in terms of the logarithmic derivative of the corresponding zeta functions. Moreover, from the integral form of the Rankin-type L-function associated to two “Drinfeld-type” newforms, we then derive a formula for a non-central special derivative of the L-function in question. Adapting the classical approach, we also obtain a Kronecker-type solution for Pell’s equation over function fields.

We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules dened over the polynomial ring Fq[\theta]. This generalizes results of Anderson for the rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.

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The aim of the present paper is to suggest that statistical physics provides the correct language to understand the practical behavior of the LLL algorithm, most of which are left unexplained to this day. To this end, we propose sandpile models that imitate LLL with compelling accuracy, and prove for these models some of the most desired statements regarding LLL. We also formulate a few conjectures that formally capture our heuristics and would serve as milestones for further development of the theory.

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This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a$λ$-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a$λ$-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the$λ$-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kähler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to étale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.

In this paper, we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively. As applications, we prove a non-archimedean analogue of the Calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems.

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