Algebraic number theory

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the ring $ \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] $ of all numbers of the form $ a + \sqrt {{ - 1b}} $ with $ a,\;b \in \mathbb{Z} $ and showed that $ \mathbb{Z}\left[ {\sqrt {{ - 1}} } \right] $ is a ring with unique factorization in prime elements (see §2.1). He introduced these numbers for the development of his theory of biquadratic residues. Another motivation for the study of the arithmetic of algebraic numbers comes from the theory of Tiophantine equations. For example, the quadratic form $ f({x_1},\;{x_2}) = x_1^2 - Dx_2^2 $ with $ D \in \mathbb{Z} $, $ \sqrt {{D\; \notin \;\mathbb{Z}}} $ can be written in the form $ \left( {{x_1} - \sqrt {{D{x_2}}} } \right)\left( {{x_1} + \sqrt {{D{x_2}}} } \right) $. Hence the question about the representation of integers by f(a 1, a 2) with $ {a_1},\;{a_2} \in \mathbb{Z} $ can be reformulated as a question of factorization of algebraic numbers of the form $ {a_1} + \sqrt {{D{a_2}}} $. These numbers form a module in the field $ \mathbb{Q}(\sqrt {D} ) $.

Basic Number Theory

For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the direction \"from automorphic to Galois\". Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

Keywords: modular forms ; Langlands conjectures ; density theorems ; Landau-Siegel zeros Reference TAN-ARTICLE-2004-001doi:10.1155/S1073792804132455 Record created on 2008-11-14, modified on 2017-05-12

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On the complex moments of symmetric power $L$-functions at $s = 1$

FELIX KLEIN'S famous enunciation—“Given a manifold and a group of transformations of the same, to develop the theory of invariants relating to that group”—not only describes the spirit of geometry but also reveals that the concept of a group dominated at least one important branch of mathematics. In the fundamental structure of other branches, however, groups are to be found, and the general concepts developed during the latter half of the last century both by Klein and his student friend, Sophus Lie, were responsible for the far-reaching influence which the relevant theory has had on modern mathematical thought. It is not always easy to discover the first important application of a newly developed theory but it is probable that the formulation of the mechanics of the atom made by Heisenberg in 1926 by the application of the theory of matrices—the invention of Cayley in 1858—is a fair illustration of the excursion of that theory into the practical domain. The Theory of Group Representations By Prof. Francis D. Murnaghan. Pp. xi + 369. (Baltimore, Md.: Johns Hopkins Press; London: Oxford University Press, 1938.) 22s. 6d. net.

The Theory of Group Representations

The main objects of study in this article are two classes of Rankin–Selberg L-functions, namely L(s,ƒ×g) and L(s, sym^2(g)× sym^2(g)), where ƒ, g are newforms, holomorphic or of Maass type, on the upper half plane, and sym^2(g) denotes the symmetric square lift of g to GL(3). We prove that in general, i.e., when these L-functions are not divisible by L-functions of quadratic characters (such divisibility happening rarely), they do not admit any LandauSiegel zeros. Such zeros, which are real and close to s=1, are highly mysterious and are not expected to occur. There are corollaries of our result, one of them being a strong lower bound for special value at s=1, which is of interest both geometrically and analytically. One also gets this way a good bound on the norm of sym^2(g).

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On the Exceptional Zeros of Rankin–Selberg L-Functions

In this note we study the symmetric powers of strongly modular icosahedral representations $\rho$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $\rho$, there exists a cuspidal automorphic representation $\Pi = \Pi_{\infty} \otimes \Pi_{f}$ of $GL_{6} (\mathbb{A}_{F})$ such that $L (s, {\rm sym}^{5} (\rho)) = L (s, \Pi_{f})$. One sees that ${\rm sym}^{5} (\rho)$ is twist equivalent to $\rho' \otimes {\rm sym}^{2} (\rho)$ for another modular icosahedral representation $\rho'$, and our theorem is a special case of a cuspidality criterion formulated and proved in this paper, which may be of independent interest, for the Kim--Shahidi automorphic tensor product $\pi \boxtimes {\rm sym}^{2} (\pi')$, where $\pi$ and $\pi'$ are cuspidal automorphic representations of $GL (2) / F$. We also give a complete structure theory of modular icosahedral representations. As a result, we prove that $L (s, {\rm sym}^{m} (\rho) \otimes \chi)$ does not admit any Landau--Siegel zero when it is not divisible by $L$--functions of quadratic characters. In general, there is no such divisibility and and there are no Landau--Siegel zeros for such $L$--functions.

On the symmetric powers of cusp forms on GL(2) of icosahedral type

On Local and Global Conjugacy

The main purpose of this article is to establish an effective version of the Grunwald-Wang theorem, which asserts that given a family of local characters χ
 v
 of K
 v
 * of exponent m, where v ¼ S for a finite set S of primes of K, there exists a global character χ of the idele class group C
 K
 of exponent m (unless some special case occurs, when it is 2m) whose local component at v is χ
 v
 . The effectiveness problem for this theorem is to bound the norm N(χ) of the conductor of χ in terms of K, m, S and N(χ
 v
 ) (v ∈ S). The Kummer case (when K contains μ
 m
 ) is easy since it is almost an application of the Chinese remainder theorem. In this paper, we solve this problem completely in general case, and give three versions of bound, one is with GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix p and l, one gets a good least upper bound for N such that p is not an l-th power mod N. One also gets the least upper bound for N such that l
 r
 | φ(N) and p is not an l-th power mod N. Some part of this article is adopted (with some revision) from the unpublished thesis by Wang (2001).

Grunwald-Wang theorem, an effective version

In this note we discuss and classify LFMO-spcial representations, for which under certain functoriality, it gives the instance of failure of multiplicity one.

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Papers

On the Exceptional Zeros of Rankin–Selberg L-Functions

On the symmetric powers of cusp forms on GL(2) of icosahedral type

On Local and Global Conjugacy

Grunwald-Wang theorem, an effective version

On Local and Global Conjugacy