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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 paper we prove the \pm-main conjecture formulated by Kobayashi for elliptic curves with supersingular reduction at p such that a_p = 0. We also prove as a corollary the p-part of the BSD formula at supersingular primes when the analytic rank is 0.

We improve upon the local bound in the depth aspect for sup-norms of newforms on D^×, where D is an indefinite quaternion division algebra over Q. Our sup-norm bound implies a depth-aspect subconvexity bound for L(1/2,f×θ_χ), where f is a (varying) newform on D^× of level p^n, and θ_χ is an (essentially fixed) automorphic form on GL_2 obtained as the theta lift of a Hecke character χ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via p-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

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.

A vast class of exponential functions are shown to be deterministic. This class includes functions whose exponents are polynomial-like or ``piece-wise'' close to polynomials after differentiation. Many of these functions are proved to be disjoint from the M\"obius function.

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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 construct the Frobenius structure on a rigid connection Be_{\hat G} on G_m for a split reductive group \hat G introduced by Frenkel-Gross. These data form a \hat G-valued overconvergent F-isocrystal Be^†_{\hat G} on G_{m,F_p}, which is the p-adic companion of the Kloosterman \hat G-local system Kl_{\hat G} constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be^†_{\hat G} when \hat G is almost simple (which recovers the calculation of monodromy group of Kl_{\hat G} due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth-Ngô-Yun on the functoriality between different Kloosterman \hat G-local systems. We show that the Frobenius Newton polygons of Kl_{\hat G} are generically ordinary for every \hat G and are everywhere ordinary on |G_{m,F_p}| when \hat G is classical or G_2.