Symmetric primes revisited

The Rabin cryptosystem revisited

The Rabin public-key cryptosystem is revisited with a focus on the problem of identifying the encrypted message unambiguously for any pair of primes. In particular, a deterministic scheme using quartic reciprocity is described that works for primes congruent 5 modulo 8, a case that was still open. Both theoretical and practical solutions are presented. The Rabin signature is also reconsidered and a deterministic padding mechanism is proposed.

再探小子群上的指数和

Exponential sums over small subgroups, revisited

本文以阐述方式讲解 Bourgain、Glibichuk 与 Konyagin 的定理证明，给出素有限域极小型乘法子群上指数和的非平凡界。

A structure theorem for sets of small popular doubling, revisited

We prove that every set $A\subset\mathbb{Z}/p\mathbb{Z}$ with $\mathbb{E}_x\min(1_A*1_A(x),t)\le(2+δ)t\mathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $δ,t$ are small real numbers. This shows that the Vosper theorem is stable in the case of a single set.

Iwasawa theory for symmetric powers of CM modular forms at nonordinary primes, II

Continuing the study of the Iwasawa theory of symmetric powers of CM modular forms at supersingular primes begun by the first author and Antonio Lei, we prove a Main Conjecture equating the "admissible" $p$-adic $L$-functions to characteristic ideals of "finite-slope" Selmer modules constructed by the second author. As a key ingredient, we improve Rubin's result on the Main Conjecture of Iwasawa theory for imaginary quadratic fields to an equality at inert primes.

Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain finite fields

We obtain new uniform bounds for the symmetric tensor rank of multiplication in finite extensions of any finite field Fp or Fp2 where p denotes a prime number greater or equal than 5. In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on sufficiently dense families of modular curves defined over Fp2 attaining the Drinfeld-Vladuts bound and on the descent of these families to the definition field Fp. These families are obtained thanks to prime number density theorems of type Hoheisel, in particular a result due to Dudek (2016).

Modular periodicity of exponential sums of symmetric Boolean functions and some of its consequences

This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Boolean functions. In particular, the periodicity modulo $p$ ($p$ odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo $p$, bounds for periods and relations between them are obtained for these exponential sums. The concept of avoiding primes is also introduced. This concept and the bounds presented in this work are used to show that some classes of symmetric Boolean functions are not balanced. In particular, every elementary symmetric Boo…

Algebraic tori revisited

Let $K/k$ be a finite Galois extension and $π= \fn{Gal}(K/k)$. An algebraic torus $T$ defined over $k$ is called a $π$-torus if $T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n$ for some integer $n$. The set of all algebraic $π$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(π)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let $π$ be a finite group. Then $T(π)\simeq C(Ω_{\bm{Z}π})$ where $Ω_{\bm{Z}π}$ is a maximal $\bm{Z}$-order in $\bm{Q}π$ containing $\bm{Z}π$ and $C(Ω_{\bm{Z}π})$ is the local…

Some works of Furtwängler and Vandiver revisited and Fermat's last theorem

From some works of P. Furtwängler and H.S. Vandiver, we put the basis of a new cyclotomic approach to Fermat's last theorem for p>3 and to a stronger version called SFLT, by introducing governing fields of the form Q(exp(2 i pi/q-1)) for prime numbers q. We prove for instance that if there exist infinitely many primes q, q not congruent to 1 mod p, q^(p-1) not congruent to 1 mod p^2, such that for Q dividing q in Q(exp(2 i pi /q-1)), we have Q^(1-c) = A^p . (alpha), with alpha congruent to 1 mod p^2 (where c is the complex conjugation), then Fermat's last theorem holds for p. More generally, t…

Iwasawa Theory for the Symmetric Square of a CM Modular Form at Inert Primes

Let f be a CM modular form and p an odd prime which is inert in the CM field. We construct two p-adic L-functions for the symmetric square of f, one of which has the same interpolating properties as the one constructed by Delbourgo-Dabrowski, whereas the second one has a similar interpolating properties but corresponds to a different eigenvalue of the Frobenius. The symmetry between these two p-adic L-functions allows us to define the plus and minus p-adic L-functions à la Pollack. We also define the plus and minus p-Selmer groups analogous to Kobayashi's Selmer groups. We explain how to relat…

New bounds for Szemeredi's theorem, Ia: Progressions of length 4 in finite field geometries revisited

Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and published in Proc. London Math. Society. Unfortunately the proof had a gap, and we issue an erratum for that paper here. Our new argument is different and significantly shorter. In fact we prove a stronger result, which can be viewed as a quantatitive version of some previous results of Bergelson-Host-Kra and the authors.

Ramanujan and Labos primes, their generalizations and classifications of primes

Considering Ramanujan primes and the symmetric to them so-called Labos primes, we study their parallel properties, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems.

Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes

Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p-adic L-functions for the symmetric powers of f, thus verifying general conjectures of Dabrowski and Panchishkin in this special case. We also construct their "mixed" plus and minus counterparts and prove an analogue of Pollack's decomposition of the admissible p-adic L-functions into mixed plus and minus p-adic L-functions. On the arithmetic side, we define corresponding mixed plus and minus Selmer groups. We unite the arithmetic with the analytic by firs…

对模形式对称立方转移的两个性质研究

Two properties of symmetric cube transfers of modular forms

本文研究对称立方转移的自守表示的导数和局部自守表示类型，通过ε因子的变化来分类可能的转移类型，重点分析p=3的情况。

On the symmetric determinantal representations of the Fermat curves of prime degree

We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation between symmetric matrices with entries in linear forms and non-effective theta characteristics on smooth plane curves. We also use some results of Gross-Rohrlich on the rational torsion points on the Jacobian varieties of the Fermat curves of prime degree.

Almost all entries in the character table of the symmetric group are multiples of any given prime

We show that almost every entry in the character table of $S_N$ is divisible by any fixed prime as $N\to\infty$. This proves a conjecture of Miller.

对称群特征值被素数幂整除的性质

Divisibility of character values of the symmetric group by prime powers

本文证明了Miller猜想，表明当n趋于无穷时对称群S_n特征表的几乎所有元素都能被任意给定素数幂整除，推广了此前关于素数整除性的结果。