Conjectures in arithmetic algebraic geometry: a survey

Until recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.

FROM FERMAT TO GAUSS. Fermat, Euler and Quadratic Reciprocity. Lagrange, Legendre and Quadratic Forms. Gauss, Composition and Genera. Cubic and Biquadratic Reciprocity. CLASS FIELD THEORY. The Hilbert Class Field and p = x 2 + ny 2 . The Hilbert Class Field and Genus Theory. Orders in Imaginary Quadratic Fields. Class Fields Theory and the Cebotarev Density Theorem. Ring Class Field and p = x 2 + ny 2 . COMPLEX MULTIPLICATION. Elliptic Functions and Complex Multiplication. Modular Functions and Ring Class Fields. Modular Functions and Singular j--Invariants. The Class Equation. Ellpitic Curves. References. Index.

Olga Taussky’s interest in class field theory began when she was a student with Ph. Furtwangler at the University of Vienna (she received her doctorate under his supervision in 1930). At this time, E. Artin, using his general reciprocity law, reformulated the “Principal Ideal Theorem” as a purely grouptheoretic question regarding the triviality of the “Verlagerung” homomorphism from a finite group to its commutator subgroup. Since Furtwangler had been developing group theoretic methods in his own number theoretic investigations, Artin communicated this new idea to him. This was considered to be one of the central problems in the subject at the time, and so there was a good deal of excitement when Furtwangler eventually succeeded in proving the result using this new approach. Olga Taussky had been looking for a thesis topic and Furtwangler suggested several questions related to the finer structure of the Principal Ideal Theorem. The optimistic atmosphere, and the prospect of contributing at the frontiers of class field theory, sparked in Olga a deep interest in this problem which lasted her entire career. She returned to the questions of “capitulation”, a term coined by one of her co-authors Arnold Scholz, several times in her life, always with the sense that these were questions of deep arithmetic significance. I believe that it was the experience of her early contact with class field theory which dominated her research interests in number theory and which provided her a fertile source of inspiration for most of her career. I will describe some of her work in this area below. I would like to take this opportunity to thank the referee for many valuable suggestions. For a number field F , (i.e., an extension of finite degree over the rational field Q) its Hilbert class field H = H(F ) is the maximal Galois extension of F which is everywhere unramified and whose Galois group Gal(H/F ) is abelian. It is a consequence of Artin’s reciprocity law that Gal(H/F ) is isomorphic to the ideal class group C(F ) of F. The Principal Ideal Theorem is the statement that every ideal of F becomes a principal ideal when considered

Let K be a number field. For which primes p does there exist an elliptic curve $E / K$ admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.

We denote by A_{3.n}, the number of the form 3^{2^n}+2, where n is an integer \geq0; in other words, for every integer n\geq0,A_{3.n}=3^{2^n}+2 [it is easy to check that A_{3.0}=5,A_{3.1}=11,A_{3.2}=83, A_{3.3}=6563 and A_{3.4}=43046723; moreover, for every j\in{0,1,2,3},A_{3.j} is prime and A_{3.4} is composite ( since A_{3.4}\equiv0mod[19] )]. In this paper, we show [via elementary arithmetic congruences] the following result (T.). For every positive integer n such that n\equiv4mod[6], we have A_{3.n}-2\equiv17mod[19]; and for every integer n\geq2, we have A_{3.n}-2\equiv1mod[5]. Result (T.) immediately implies that there are infinitely many composite numbers of the form A_{3.n}. That being so, using result (T.) coupled with a special case of a Theorem of Dirichlet on arithmetic progressions, we explain why it is natural to conjecture that there are infinitely many prime numbers of the form A_{3.n}. A Fermat number is a number of the form F_n=2^{2^n}+1, where n is an integer \geq0. A Fermat composite (see Dickson (1952) or Hardy, & Wright, (1979) or Annouk, (2012)) is a non prime Fermat number and a Fermat prime is a prime Fermat number. Fermat composites and Fermat primes are characterized via divisibility in Annouk, (2012) and in Annouk, (2014). It is known (see [4] ) that for every j\in{0,1,2,3,4},F_j is a Fermat prime \left(F_0=2^{2^0}+1=3\right. and 3 is prime, F_1=2^{2^1}+1=5 and 5 is prime, F_3=2^{2^3}+1=257 and 257 is prime, and F_4=2^{2^4}+1=65537 and 65537 is prime), and it is also known (see Hardy, & Wright (1979) or Hoffman (2000)or Křížek, et al. (2002) or Keller, (2020) or Euler, (1738)or Scharlau, & Opolka, (1985) that F_5 and F_6 are Fermat composites \left(F_5=2^{2^5}+1=641\times6700417\right., and since 2013, it is known that F_{2747497}=2^{2^{2747497}}+1 is Fermat composite number). Fermat numbers have importance and their application to other sciences such as cryptography, neural networks, electronic computer, polygons with straightedge, filtering, autocorrelation, and related areas with conventional computing are clearly known. Factorization of Fermat numbers (specially for big one) is a very hard problem in number theory and cryptography too. The biggest known Fermat prime is F_4=2^{2^4}+1=65537, and this number is used in cryptography because of the fact that this prime helps the cryptography less vulnerable to the public exponent attack lowly (as mentioned "Coppersmith's short pad attack" in the literature). Practical and efficient methods are still in need for factorization of such numbers, even there are many algorithms to factorize some of such composite numbers. That being so, in this paper, we show [via elementary arithmetic congruences] the following result (E.). For every integer n>0 such that n\equiv1mod[2], we have F_n-1\equiv4mod[7]; and for every integer n\geq2, we have F_n-1\equiv1mod[j], where j\in{3,5}. Result (E.) immediately implies that there are infinitely many composite numbers of the form 2+F_n. Result (E.) also implies that the only prime of the form 4+F_n is 7 and the only primes of the form 8+F_n are twin primes 11 and 13. That being said, using result (E.) and a special case of a Theorem of Dirichlet, we conjecture that there are infinitely many primes of the form 2+F_n.

This thesis presents some contributions in additive combinatorics and arithmetic Ramsey theory. More specifically, it deals with the interaction between combinatorics, number theory and additive combinatorics. This area saw a great improvement with the Szemerédi Regularity Lemma and some of the results that followed. The Regularity Lemma and its consequences have become a widely used tool in graph theory, combinatorics and number theory. Furthermore, its language and point of view has deeply changed the face of additive number theory, a fact universally acknowledged by the Abel award given to Szemerédi in 2012. One of the main reasons for the prize has been Szemerédi's theorem, a result regarding the existence of arbitrarily long arithmetic progressions in dense sets of the integers, the proof of which uses the Regularity Lemma in a key step. One of the earlier consequences of the Regularity Lemma was the Removal Lemma for graphs that was used by Ruzsa and Szemerédi to show Roth theorem, regarding the existence of 3-term arithmetic progressions in dense sets of the integers, in a combinatorial way. The Removal Lemma states that in any graph K with few copies of a subgraph, say a triangle, we can remove few edges from K so that the result contains no copy of the subgraph. This has become a key tool in the applications of the so-called Regularity Method, which has extensive literature in combinatorics, graph theory, number theory and computer science. In 2005 Green introduced a regularity lemma for Abelian groups as well as an algebraic removal lemma. The removal lemma for groups states that, for a given finite Abelian group G, if there are o(|G|^3) solution to x+y+z+t=0 with the variables taking values in S, a subset of G, then we can remove o(|G|) elements from S to make the set S solution-free. The main contributions of this work corresponds to extensions of the removal lemma for groups to either more general contexts, like non-necessary Abelian finite groups, or to linear systems of equations for finite Abelian groups. The main goal is to give a comprehensive and more general framework for many results in additive number theory like Szemerédi Theorem. In particular, we show that the removal lemma for groups by Green can be extended to non-necessary Abelian finite groups. Moreover, we prove a removal lemma for linear systems on finite fields: for every e>0 there exists a d>0 such that if A is a (k x m) linear system of equations with coefficients in a finite field F and the number of solutions to Ax=b, where each variable takes values from a subset Si in F is less than d times |F| raised to m-k, then by removing less than e|F| elements in each Si we can make the resulting sets solution-free, thus solving a conjecture by Green to that respect. Even more, if A is an integer linear system, G is a finite Abelian group, and the determinantal of A and |G| are coprime, then a similar statement holds. Let us mention that the last result allows us to characterize those linear systems where any set S with size proportional to G has a nontrivial solution in S, provided |G| is large enough. This extends the validity of Szmerédi's theorem to finite Abelian groups. These extensions of the removal lemma have been used in arithmetic Ramsey theory to obtain counting results for the number of monochromatic solutions of linear systems. The main result from a work by Frankl, Graham and Rödl in '88 states that the number of monochromatic solutions of regular systems in integer intervals is in fact a positive proportion of the total number of solutions. We give analogous results for solutions in Abelian groups with bounded exponent, for which the main tool in the torsion-free case cannot be applied. Density versions of these counting results are also obtained, in this case with a full characterization.

Landau's theorem asserts that the asymptotic density of sums of two squares in the interval $1\leq n\leq x$ is $K/{\sqrt{\log x}}$, where $K$ is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals $|n-x|\leq x^{\epsilon}$ for a fixed $\epsilon$ and $x\to\infty$. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic $f_0\in\Bbb{F}_q[T]$ of degree $n$ and take $\epsilon$ with $1>\epsilon\geq {2\over n}$. Then the asymptotic density of polynomials $f$ in the "interval" $\deg(f-f_0)\leq\epsilon n$ that are of the form $f=A^2+TB^2$, $A,B\in\Bbb{F}_q[T]$ is ${1\over 4^n}\big(\matrix{2n\atop n}\big)$ as $q\to\infty$. This density agrees with the asymptotic density of such monic $f$'s of degree $n$ as $q\to\infty$, as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of $f(-T^2)$, where $f$ is a polynomial of degree $n$ with a few variable coefficients: The Galois group is the hyperoctahedral group of order $2^nn!$.

In this paper, we show that some new variants of the Very Smooth Hash (VSH) hash function are susceptible to similar types of preimage attacks as the original VSH. We also generalise the previous mathematical results, which have been used in the preimage attacks. VSH is a hash function based on the multiexponentiation of prime numbers modulo some large product of two primes. The security proof of VSH is based on some computational problems in number theory, which are related to the problem of factoring large integers. However, the preimage resistance of VSH has been studied and found somewhat lacking especially in password protection. There have been many different variants of VSH proposed by the original authors and others. Especially the discrete logarithm version of VSH has been proposed in order to make the hash values shorter. Further proposals have used the discrete logarithm in finite fields and elliptic curves to gain even more advantage to the hash length. Our results demonstrate that even for these new variants, the same ideas for preimage attacks can be applied as for the original VSH and they result in effective preimage attacks.

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisentr\"ager-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.

In this survey, we discuss some basic problems in representation theory of finite groups, including some long-standing conjectures of Alperin, Brauer, and others. A possible approach to some of these problems is to use the classification of finite simple groups to reduce the problem in consideration to some, more specific, questions about simple groups. We will describe recent progress on reduction theorems in this direction. We will also outline applications of these results to various problems in group theory, number theory, and algebraic geometry.

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

Through basic research in mathematics, the NTT Institute for Fundamental Mathematics aims to enrich the "fountain of knowledge" that nourishes science and technology.In this article, we first provide an overview of the research being carried out at the Institute then introduce the Institute's core research areas: number theory, especially arithmetic dynamics; algebraic and arithmetic geometry; and representation theory and automorphic forms.

We generalise the classical Bombieri–Vinogradov theorem for short intervals to a non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are “twisted” by a splitting condition in a Galois extension of number fields. Using this result in conjunction with the recent work of Maynard, we prove that rational primes with a given splitting condition in a Galois extensionL/$\mathbb{Q}$exhibit bounded gaps in short intervals. We explore several arithmetic applications related to questions of Serre regarding the non-vanishing Fourier coefficients of cuspidal modular forms. One such application is that for a given modularL-functionL(s, f), the fundamental discriminantsdfor which thed-quadratic twist ofL(s, f) has a non-vanishing central critical value exhibit bounded gaps in short intervals.

This paper contains a summary table of the author’s computation of the Gaussian primes of the form a + i a + i . For the values x = 1000 , 10000 , 100000 , 180000 x = 1000, 10000, 100000, 180000 , and 500000 (500000) 14000000, the following values are tabulated: G ( x ) G(x) , the numbers of Gaussisn primes a + i a + i with a ≦ 14000000 ; π 1 ( x ) a \leqq 14000000;{\pi _1}(x) , the number of primes ≦ x \leqq x congruent to 1 m o d 4 1 mod 4 ; π 3 ( x ) {\pi _3}(x) , the number of primes ≦ x \leqq x congruent to 3 m o d 4 3 mod 4 ; and G ( x ) / π 3 ( x ) G(x)/{\pi _3}(x) .