Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.

Additional excerpts

/pdf/self-dual-kahler-metrics-of-neutral-signature-on-complex-3yi9ziljqm.pdf

A Course in Arithmetic

Self-dual Kähler metrics of neutral signature on complex surfaces

We study the distribution of graded degrees of freedom in classically stable oriented closed string vacua and use the Rankin-Selberg transform to link it to the finite one-loop vacuum energy. In particular, we find that the spectrum of physical excitations not only must enjoy asymptotic supersymmetry but actually, at very large mass, bosonic and fermionic states must follow a universal oscillating pattern, whose frequencies are related to the zeros of the Riemann zeta-function. Moreover, the convergence rate of the overall number of the graded degrees of freedom to the value of the vacuum energy is determined by the Riemann hypothesis. We discuss also attempts to obtain constraints in the case of tachyon-free open-string theories.

/pdf/vacuum-stability-string-density-of-states-and-the-riemann-4numcbeqzc.pdf

Vacuum stability, string density of states and the Riemann zeta function

Lectures on K3- and enriques surfaces

We propose a new homomorphic public-key cryptosystem over arbitrary nonidentity finite group based on the difficulty of the membershipproblem for groups of integer matrices. Besides, a homomorphic cryptosystem is designed for the first time over finite commutative rings.

https://hal.archives-ouvertes.fr/hal-03047076/document

Homomorphic public-key cryptosystems over groups and rings

In this paper, we consider the family of hyperelliptic curves over having a fixed genus and a marked rational non-Weierstrass point. We show that when , a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, in Automorphic representations and -functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

A Course in Arithmetic

Citations

Self-dual Kähler metrics of neutral signature on complex surfaces

Vacuum stability, string density of states and the Riemann zeta function

Lectures on K3- and enriques surfaces

Homomorphic public-key cryptosystems over groups and rings

Rational points on hyperelliptic curves having a marked non-Weierstrass point

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