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Soit G un groupe réductif défini sur un corps algébriquement clos de caractéristique positive. Nous montrons que le foncteur contraction par Frobenius de la catégorie des G-modules est adjoint à droite de celui de tensorisation deux fois par le module de Steinberg du tordu de Frobenius du G-module de départ. Il s’ensuit en particulier que le foncteur de contraction par Frobenius préserve le caractère injectif et l’existence de bonne filtration mais pas la semi-simplicité. (For a reductive group G defined over an algebraically closed field of positive characteristic, we show that the Frobenius contraction functor of $Gs-modules is right adjoint to the Frobenius twist of the modules tensored with the Steinberg module twice. It follows that the Frobenius contraction functor preserves injectivity, good filtrations, but not semi-simplicity.)

We prove several formulas related to Hodge theory and the KodairaSpencerKuranishi deformation theory of Khler manifolds. As applications, we present a construction of globally convergent power series of integrable Beltrami differentials on CalabiYau manifolds and also a construction of global canonical family of holomorphic ( n , 0 ) -forms on the deformation spaces of CalabiYau manifolds. Similar constructions are also applied to the deformation spaces of compact Khler manifolds.

In 1998, Jeffrey Hoffstein, Jill Pipher, and Joseph H. Silverman introduced the famous NTRU cryptosystem, and called it "A ring-based public key cryptosystem". Actually, it turns out to be a lattice based cryptosystem that is resistant to Shor's algorithm. There are several modifications to the original NTRU and two of them are selected as round 2 candidates of NIST post quantum public key scheme standardization. In this paper, we present a simple attack on the original NTRU scheme. The idea comes from Ding et al.'s key mismatch attack. Essentially, an adversary can find information on the private key of a KEM by not encrypting a message as intended but in a manner which will cause a failure in decryption if the private key is in a certain form. In the present, NTRU has the encrypter generating a random polynomial with "small" coefficients, but we will have the coefficients be "large". After this, some further work will create an equivalent key.