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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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Benedict Gross's paper on companion forms

In the page 458 of his paper(A tameness criterion for Galois representations associated to modular forms), Gross wrote the following "A detailed analysis of $U_p(Af)+V_p(<p>f)$ shows that it vanishes …