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A variant of Brauers induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.
It is shown that a system of $r$ quadratic forms over a ${mathfrak p}$-adic field has a non-trivial common zero as soon as the number of variables exceeds $4r$, providing that the residue class field has cardinality at least $(2r)^r$.
Generalizing the completed cohomology groups introduced by Matthew Emerton, we define certain spaces of ordinary $p$-adic automorphic forms along a parabolic subgroup and show that they interpret all classical ordinary automorphic forms.
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Cole
Serre obtained the p-adic limit of the integral Fourier coefficient of modular forms on $SL_2(mathbb{Z})$ for $p=2,3,5,7$. In this paper, we extend the result of Serre to weakly holomorphic modular forms of half integral weight on $Gamma_{0}(4N)$ for
We show that Hidas families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-ad