Physics as a generalized number theory program involves three threads: various p-adic physics and their fusion together with real number based physics to a larger structure, the attempt to understand basic physics in terms of classical number fields (in particular, identifying associativity condition as the basic dynamical principle), and infinite primes whose construction is formally analogous...

We study the algebraicity of Stark-Heegner points on a modular elliptic curve E. These objects are p-adic points on E given by the values of certain p-adic integrals, but they are conjecturally defined over ring class fields of a real quadratic field K. The present article gives some evidence for this algebraicity conjecture by showing that linear combinations of Stark-Heegner points weighted b...

After formulating Conjecture A for p-adic L-functions defined over ordinary Hilbert modular Hida deformations on a totally real field F of degree d, we construct two p-adic L-functions of d+1-variable depending on the parity of weight as a partial result on Conjecture A. We will also state Conjecture B which is a corollary of Conjecture A but is important by itself. Main issues of the construct...

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half ...

We give an example of a projective smooth surface X over a p-adic field K such that for any prime l different from p, the l-primary torsion subgroup of CH0(X), the Chow group of 0-cycles on X, is infinite. A key step in the proof is disproving a variant of the Block-Kato conjecture which characterizes the image of an l-adic regulator map from a higher Chow group to a continuous étale cohomology...

We study the asymptotical behavior of the p-adic singular Fourier integrals Jπα,m;φ(t) = 〈 fπα;m(x)χp(xt), φ(x) 〉 = F [ fπα;mφ ] (t), |t|p → ∞, t ∈ Qp, where fπα;m ∈ D ′(Qp) is a quasi associated homogeneous distribution (generalized function) of degree πα(x) = |x| α−1 p π1(x) and orderm, πα(x), π1(x), and χp(x) are a multiplicative, a normed multiplicative, and an additive characters of the fi...

Let p be a fixed prime number. Throughout this paper, p, p , and p will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of p , respectively. Let be the set of natural numbers, and let ∪ {0}. Let νp be the normalized exponential valuation of p with |p|p p−νp p 1/p. Let q be regarded as either a complex number q ∈ or a p-adic n...

In (Arch. Math. 57 (1991), pp. 446–455), R. Farré proved a positivstellensatz for real-series closed fields. Here we consider p-valued fields 〈K, vp〉 with a non-trivial valuation v which satisfies a compatibility condition between vp and v. We use this notion to establish the p-adic analogue of real-series closed fields; these fields are called henselian residually p-adically closed fields. Fir...

In quantum-mechanical experiments, as well as in all measurements, numerical results belong to the field of rational numbers Q. In principle, the corresponding theoretical models could be made using only Q, but it would missed usual effectiveness and beauty of mathematical analysis. So, instead of Q one traditionally applies the field of real numbers R in classical mechanics and the field of co...

We generalize the theory of p-adic geometric zeta functions of Y. Ihara and K. Hashimoto to the higher rank case. We give the proof of rationality of the zeta function and the connection of the divisor to group cohomology, i.e. the p-adic analogue of the Patterson conjecture. Introduction. In [13] and [14] Y. Ihara defined geometric zeta functions for the group PSL2. This was the p-adic counter...