Let K ∈ L1(R) and let f ∈ L∞(R) be two functions on R.The convolution(Kf)(x) =∫R K(x-y)f(y)dy can be considered as an average of f with weight defined by K.Wiener's Tauberian theorem says that under suitable c...Let K ∈ L1(R) and let f ∈ L∞(R) be two functions on R.The convolution(Kf)(x) =∫R K(x-y)f(y)dy can be considered as an average of f with weight defined by K.Wiener's Tauberian theorem says that under suitable conditions,if lim x→∞(K f)(x) = lim x→∞(K A)(x) for some constant A,then lim x→∞ f(x) = A.We prove the following-adic analogue of this theorem:Suppose K,F,G are perverse-adic sheaves on the affine line A over an algebraically closed field of characteristic p(p=l).Under suitable conditions,if(K F)|η∞≌(K G)|η∞,then F|η∞≌ G|η∞,where η∞ is the spectrum of the local field of A at ∞.展开更多
In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups： proper push-forward, fiat pull-back, bas...In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups： proper push-forward, fiat pull-back, base change, cap-product, etc. In particular, on singular varieties, this kind of l-adic homology behaves much better than the classical l-adie cohomology. As an application, we give a much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields. And we prove that these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.展开更多
We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers ...We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Michel and Sawin(2017) on sum-products of Kloosterman sheaves.Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.展开更多
The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N wh...The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the Langlands functoriality.展开更多
Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) ...Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) with the prime number p = r^2+s^2 and s is even, then C_1h^-≡B_((p-1)/_4)B_(3(p-1)/4) (mod p) for p≡1 (mod 8); and C_2h^-≡E_((p-5)/8)E_((3p-7)/8)(mod p) for p≡5 (mod 8)where B_n and E_n are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(5^(1/2))))^(1/2),then C_3h^-≡h(Q((-v)^(1/2))) h (Q((-5v)^(1/2))) (mod 5). If 3 ramifies in K = Q(θ^(1/2)), then C_4h(K)≡h(K~*) (mod 3) with K~* = Q((-3θ^(1/2))). All the above C_i are explicitly given constants.Some relations between the factors of class numbers h^- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.展开更多
We give a proof of the irrationality of p-adic zeta-values ξp（κ） for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we...We give a proof of the irrationality of p-adic zeta-values ξp（κ） for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function.展开更多
Let K = $ k(\sqrt \theta ) $ be a real cyclic quartic field, k be its quadratic subfield and $ \tilde K = k(\sqrt { - \theta } ) $ be the corresponding imaginary quartic field. Denote the class numbers of K, k and $ \...Let K = $ k(\sqrt \theta ) $ be a real cyclic quartic field, k be its quadratic subfield and $ \tilde K = k(\sqrt { - \theta } ) $ be the corresponding imaginary quartic field. Denote the class numbers of K, k and $ \tilde K $ by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h ? = h K /h K and $ \tilde h^ - = h_{\tilde K} /h_k $ are obtained via studying the p-adic L-functions of the fields.展开更多
基金supported by National Natural Science Foundation of China (Grant No.10525107)
文摘Let K ∈ L1(R) and let f ∈ L∞(R) be two functions on R.The convolution(Kf)(x) =∫R K(x-y)f(y)dy can be considered as an average of f with weight defined by K.Wiener's Tauberian theorem says that under suitable conditions,if lim x→∞(K f)(x) = lim x→∞(K A)(x) for some constant A,then lim x→∞ f(x) = A.We prove the following-adic analogue of this theorem:Suppose K,F,G are perverse-adic sheaves on the affine line A over an algebraically closed field of characteristic p(p=l).Under suitable conditions,if(K F)|η∞≌(K G)|η∞,then F|η∞≌ G|η∞,where η∞ is the spectrum of the local field of A at ∞.
文摘In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups： proper push-forward, fiat pull-back, base change, cap-product, etc. In particular, on singular varieties, this kind of l-adic homology behaves much better than the classical l-adie cohomology. As an application, we give a much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields. And we prove that these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.
基金supported by National Natural Science Foundation of China (Grant Nos. 12025106 and 11971370)。
文摘We prove non-trivial upper bounds for general bilinear forms with trace functions of bountiful sheaves,where the supports of two variables can be arbitrary subsets in F_(p) of suitable sizes.This essentially recovers the Polya-Vinogradov range,and also applies to symmetric powers of Kloosterman sums and Frobenius traces of elliptic curves.In the case of hyper-Kloosterman sums,we can beat the Pólya-Vinogradov barrier by combining additive combinatorics with a deep result of Kowalski,Michel and Sawin(2017) on sum-products of Kloosterman sheaves.Two Sato-Tate distributions of Kloosterman sums and Frobenius traces of elliptic curves in sparse families are also concluded.
基金National Natural Science Foundation of China (Grant No. A010102-11671380)One Hundred Talents Program at Chinese Academy of Sciences, National Basic Research Program of China (Grant No. 2013CB834202)National Science Foundation of USA (Grant No. DMS9729992)。
文摘The issue of local and global conjugacy is closely related to the multiplicity one property in representation theory and the Langlands program. In this article we give first families of connected instances for SO2N where the multiplicity one fails in both aspects of representation theory and automorphic forms with certain assumptions on the Langlands functoriality.
基金Project supported by the National Natural Science Foundation of China.
文摘Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) with the prime number p = r^2+s^2 and s is even, then C_1h^-≡B_((p-1)/_4)B_(3(p-1)/4) (mod p) for p≡1 (mod 8); and C_2h^-≡E_((p-5)/8)E_((3p-7)/8)(mod p) for p≡5 (mod 8)where B_n and E_n are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(5^(1/2))))^(1/2),then C_3h^-≡h(Q((-v)^(1/2))) h (Q((-5v)^(1/2))) (mod 5). If 3 ramifies in K = Q(θ^(1/2)), then C_4h(K)≡h(K~*) (mod 3) with K~* = Q((-3θ^(1/2))). All the above C_i are explicitly given constants.Some relations between the factors of class numbers h^- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.
文摘We give a proof of the irrationality of p-adic zeta-values ξp（κ） for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function.
基金supported by National Natural Science Foundation of China (Grant No. 10771111)
文摘Let K = $ k(\sqrt \theta ) $ be a real cyclic quartic field, k be its quadratic subfield and $ \tilde K = k(\sqrt { - \theta } ) $ be the corresponding imaginary quartic field. Denote the class numbers of K, k and $ \tilde K $ by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h ? = h K /h K and $ \tilde h^ - = h_{\tilde K} /h_k $ are obtained via studying the p-adic L-functions of the fields.
