Diagonal forms of prime degree p in a P-adic ring
نویسندگان
چکیده
منابع مشابه
Zeros of p-adic forms
A variant of Brauer’s 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.
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with coefficients aij in O. Write the degree as k = pm with p m. A solution x = (x1, . . . , xN) ∈ K is called non-trivial if at least one xj is non-zero. It is a special case of a conjecture of Emil Artin that (∗) has a non-trivial solution whenever N > Rk. This conjecture has been verified by Davenport and Lewis for a single diagonal equation over Qp and for a pair of equations of odd degree ...
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This paper is concerned with non-trivial solvability in p-adic integers of systems of additive forms. Assuming that the congruence equation ax + by + cz ≡ d (mod p) has a solution with xyz 6≡ 0 (mod p) we have proved that any system of R additive forms of degree k with at least 2 · 3R−1 · k + 1 variables, has always non-trivial p-adic solutions, provided p k. The assumption of the solubility of...
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Let k > 1 be an integer and let p be a prime. We show that if pa 6 k < 2pa or k = paq + 1 (with q < p/2) for some a = 1, 2, 3, . . . , then the set { (n k ) : n = 0, 1, 2, . . . } is dense in the ring Zp of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.
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We are interested in understanding and describing the p-adic properties of Jacobi forms. As opposed to the case of modular forms, not much work has been done in this area. The literature includes [3, 4, 7]. In the first section, we follow Serre’s ideas from his theory of p-adic modular forms. We study Jacobi forms whose Fourier expansions have integral coefficients and look at congruences betwe...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1972
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-20-2-171-173