Recent zbMATH articles in MSC 11Dhttps://www.zbmath.org/atom/cc/11D2021-04-16T16:22:00+00:00WerkzeugThue Diophantine equations.https://www.zbmath.org/1456.110342021-04-16T16:22:00+00:00"Waldschmidt, Michel"https://www.zbmath.org/authors/?q=ai:waldschmidt.michelThis survey presents the essentials concerning Thue (mainly) and Thue-Mahler equations. In this reviewer' s opinion, it is very much appropriate for graduate students of Mathematics looking for their way in Number Theory and for any mathematician who would like to know about what is about these equations and what kind of Mathematics are involved in their study.
Description of the paper's contents. Section 1: Basic definitions. The special case of Thue equation in which positive definite binary forms are involved, and CM-fields. General Thue equations and their relation to Diophantine Approximation. Thue-Siegel-Roth Theorem and Thue's Theorem (without proof of course). Description of Thue's method by a concrete example.
Section 2. Solving Thue equations by Baker's method (use of linear forms in logarithms of algebraic numbers); a list of book references for further study is given. Thue equation and Siegel's Unit equation. Lower bounds for linear forms in logarithms and Siege's Unit equation.
Section 3. Families of Thue equations and finiteness of solutions to such families. Short description of the author's research project jointly with Claude Levesque, and a related conjecture.
Section 4. A guide for further references.
For the entire collection see [Zbl 1444.11004].
Reviewer: Nikos Tzanakis (Iraklion)The Galois action and cohomology of a relative homology group of Fermat curves.https://www.zbmath.org/1456.112172021-04-16T16:22:00+00:00"Davis, Rachel"https://www.zbmath.org/authors/?q=ai:davis.rachel"Pries, Rachel"https://www.zbmath.org/authors/?q=ai:pries.rachel-j"Stojanoska, Vesna"https://www.zbmath.org/authors/?q=ai:stojanoska.vesna"Wickelgren, Kirsten"https://www.zbmath.org/authors/?q=ai:wickelgren.kirsten-gLet \(p\) be a prime satisfying Vandiver's conjecture, i.e., such that \(p\) does not divide the order of \(h^+\) of the class group of \(\mathbb{Q}(\zeta+\zeta^{-1})\), where \(\zeta\) is a \(p\)-th root of unity. Let \(X\) be the degree \(p\) Fermat curve \(x^p+y^p=z^p\). Let \(U\subset X\) be the affine open given by \(z\neq 0\). Consider the closed subscheme \(Y\subset U\) defined by \(xy=0\). Let \(H_1(U,Y;\mathbb{Z}/p)\) denote the étale homology group with \(\mathbb{Z}/p \) coefficients, of the pair \((U\otimes \bar{K},Y\otimes\bar{K})\). By [\textit{G. W. Anderson}, Duke Math. J. 54, 501--561 (1987; Zbl 1370.11069)], the group \(H_1(U,Y;\mathbb{Z}/p)\) is a free rank-one \(\mathbb{Z}/p[\mu_p\times\mu_p]\)-module with generator \(\beta\). The Galois action of \(\sigma\in G_{\mathbb{Q}(\zeta)}\) is then determined by \(\sigma\beta=B_\sigma\beta\), for some \(B_\sigma\in \mathbb{Z}/p[\mu_p\times\mu_p]\). Anderson theoretically described \(B_\sigma\). In this paper, a closed form formula for \(B_\sigma\) is given. Intermediate results by the same authors [\textit{R. Davis} et al., Assoc. Women Math. Ser. 3, 57--86 (2016; Zbl 1416.11045)] about the isomorphism class of the Galois group of the field extension through the action of \(G_{\mathbb{Q}(\zeta)}\) factors, are strongly used.
The first application of this formula is that the norm of \(B_\sigma\) is \(0\) for almost all \(\sigma\). This is important in computing Galois cohomology as in Section 6 where a method for the efficient computation of the first cohomology group \(H^1(G_{\mathbb{Q}(\eta)}, H_1(U,Y;\mathbb{Z}/p))\) is given. This will eventually play a key role in understanding obstructions for rational points on Fermat curves as Ellenberg's obstruction related to the non-abelian Chabauty method.
A second application of the main formula is a proof of the fact that \(H_1(U;\mathbb{Z}/p)\) is trivialized by the product of \(\lfloor 2p/3\rfloor\) terms of the form \((B_\sigma-1)\).
Reviewer: Elisa Lorenzo García (Rennes)Diophantine equation generated by the maximal subfield of a circular field.https://www.zbmath.org/1456.112032021-04-16T16:22:00+00:00"Galyautdinov, I. G."https://www.zbmath.org/authors/?q=ai:galyautdinov.ildarkhan-galyautdinovich"Lavrentyeva, E. E."https://www.zbmath.org/authors/?q=ai:lavrentyeva.elena-evgenevnaSummary: Using the fundamental basis of the field \(L_9=\mathbb{Q} (2\cos(\pi/9))\), the form \(N_{L_9}(\gamma)=f(x, y, z)\) is found and the Diophantine equation \(f(x,y,z)=a\) is solved. A similar scheme is used to construct the form \(N_{L_7}(\gamma)=g(x,y,z)\). The Diophantine equation \(g (x, y, z)=a\) is solved.Twists of Hooley's \(\Delta\)-function over number fields.https://www.zbmath.org/1456.111842021-04-16T16:22:00+00:00"Sofos, Efthymios"https://www.zbmath.org/authors/?q=ai:sofos.efthymiosSummary: We prove tight estimates for averages of the twisted Hooley \(\Delta\)-function over arbitrary number fields.On twists of smooth plane curves.https://www.zbmath.org/1456.111172021-04-16T16:22:00+00:00"Badr, Eslam"https://www.zbmath.org/authors/?q=ai:badr.eslam-e"Bars, Francesc"https://www.zbmath.org/authors/?q=ai:bars.francesc"Lorenzo García, Elisa"https://www.zbmath.org/authors/?q=ai:lorenzo-garcia.elisaLet \(C\) be a projective, smooth, non-hyperelliptic, curve and genus \(g \geq 3\) defined over a field \(k\). Denote by \(\bar{k }\) a fixed separable closure of \(k\) and by \(\bar{C}\) the curve \(C \times_k \bar{k}\). A twist of \(C\) over \(k\) is a projective, non-singular \(C^{\prime}\) defined over \(k\) with a \(\bar{k}\)-isomorphism \(\varphi : \overline{C^{\prime}} \rightarrow \bar{C}\). The paper under review deals with the following question: Assuming that \(C\) admits a smooth \(\bar{k}\)-plane model, does it have a smooth plane model over \(k\)? And if the answer is yes, does every twist of C over
\(k\) also have smooth plane model over \(k\)? The answer, in general, is negative. The twists possessing such models are characterized and an example of a twist not admitting any non-singular plane model over \(k\) is given. An interesting consequence is that explicit equations for a non-trivial Brauer-Severi surface are obtained. Furthermore, for smooth plane curves defined over \(k\) with a cyclic automorphism group generated by a diagonal matrix, a general theoretical result to
compute all its twists is presented.
Reviewer: Dimitros Poulakis (Thessaloniki)Elliptic curves over finite fields with Fibonacci numbers of points.https://www.zbmath.org/1456.111162021-04-16T16:22:00+00:00"Bilu, Yuri"https://www.zbmath.org/authors/?q=ai:bilu.yuri-f"Gómez, Carlos A."https://www.zbmath.org/authors/?q=ai:gomez.carlos-alexis"Gómez, Jhonny C."https://www.zbmath.org/authors/?q=ai:gomez.jhonny-c"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florianThe paper studies and solves a curious problem: let \(E\)\, be an elliptic curve defined over a finite field \(\mathbb{F}_q\)\, with \(\sharp(E)=q+1-a\)\, and let us denote \(E_m(q,a);\,\, m\ge 1\)\, the number of points of \(E\)\, over \(\mathbb{F}_{q^m}\); let \(\{F_n\}_{n\ge 1}\)\, be the Fibonacci sequence. When the intersection of \(\{E_m(q,a)\}_{m\ge 1}\)\, and \(\{F_n\}_{n\ge 1}\)\, has two elements at least?, i.e. to find the solutions of \(\sharp E_{m_1} = F_{n_1}\),\, \(\sharp E_{m_2} = F_{n_2}\), etc.
Section 1 discusses the problem and formulates the obtained result (Theorem 1.1): the cardinal of the intersection is two for four couples \((q,a)\)\, and three in only a case (for \(q=a=2\)). The rest of the paper is devoted to the proof of that theorem.
Section 2 studies the equation \(E_m(q,a)=F_n\)\, in terms of linear forms in logarithms and discusses how the problem can be reduced to three cases: (i) \(q\)\, small (\(q\le 10.000\)), (ii) \(n_1\)\, small and (iii) \(m_2\)\, small. This provides a list of the possible values for \(q\).
Section 3 gathers some necessary tools regarding linear forms in logarithms and continued fractions. Assuming proved that \(n_2\le 1.000\)\, Section 4 first studies the problem for \(q\le 10.000\)\, and, using the computational package \texttt{Mathematica} finds the five solutions of Theorem 1.1. Then, for \(q> 10.000\),\, also using Mathematica, proves that there is no solution.
The rest of the paper assumes \(n_2> 1.000\)\, and Section 5 to 10 deals with the three cases (i), (ii) and (iii) and, always with the computational help of Mathematica (the paper says that ``the total calculation time for the Mathematica software for this paper was 20 days on 25 parallel desktop computers'') finishes the proof of Theorem 1.1.
Reviewer: Juan Tena Ayuso (Valladolid)Elliptic curves arising from the triangular numbers.https://www.zbmath.org/1456.111032021-04-16T16:22:00+00:00"Juyal, Abhishek"https://www.zbmath.org/authors/?q=ai:juyal.abhishek"Kumar, Shiv Datt"https://www.zbmath.org/authors/?q=ai:datt-kumar.shiv"Moody, Dustin"https://www.zbmath.org/authors/?q=ai:moody.dustinGiven a positive integer \(t\), the \(t\)-th triangular number is \(t(t+1)/2\) the sum of the natural numbers up to \(t\). In the paper under review, the authors considered the elliptic curves of the Legendre's form associated to triangular numbers given by \[E_t: y^2 = x (x-1)(x-\frac{t (t+1)}{2}).\] The main results of the paper are about the rank of Mordell-Weil group
of \(E_t\) over \(\mathbb Q(t)\) and \({\bar{\mathbb Q} (t)}\). In Theorem 1 of the paper, they prove
that the elliptic surface associated to \(E_t\) is rational. The ranks of \(E_t\) over \(\mathbb Q(t)\) and \({\bar{\mathbb Q} (t)}\) are \(0\) and \(1\), respectively. Furthermore, the torsion subgroup of \(E_t(\mathbb Q(t))\) is isomorphic to \({\mathbb Z}_2 \times {\mathbb Z}_2\). The proof of these results are included in Section 3. The authors provide infinite families of positive rank ans a subfamilies of tank two in Section 4. Finally, they tried to find high rank elliptic curve over \(\mathbb Q\) by searching in those families. They find only one elliptic curve of rank \(6\) and several of rank \(5\).
Reviewer: Sajad Salami (Rio de Janeiro)Markoff-Rosenberger triples with Fibonacci components.https://www.zbmath.org/1456.110332021-04-16T16:22:00+00:00"Tengelys, Szabolcs"https://www.zbmath.org/authors/?q=ai:tengelys.szabolcs\textit{F. Luca} and \textit{A. Srinivasan} [Fibonacci Q. 56, No. 2, 126--129 (2018; Zbl 06985849)] showed that the solutions triples with all Fibonacci components of
the Markoff equation, \(x^2+y^2+z^2=3xyz\) are given by \((1, F_{2n-1}, F_{2n+1})\) where \(n\ge 0\) is an integer. The author here solves the same problem for the Markoff-Rosenberger equation, \(ax^2+by^2+cz^2=dxyz\), where \(a,b\) and \(c\) are positive integers that divide \(d\). Rosenberger proved that this equation has non-trivial solutions if and only if \((a, b, c, d)\) is one of the following:
\(\{(1, 1, 1, 1), (1, 1, 1, 3), (1, 1, 2, 4), (1, 2, 3, 6), (1,1, 2, 2), (1, 1, 5, 5)\}.
\)
The current author lists the finite number of solution triples \((F_i, F_j, F_n)\), with all Fibonacci components for the Markoff-Rosenberger equations other than the Markoff equation. He first bounds \(i\) as in the work of Luca and Srinivasan, using the Binet formula. Then he bounds \(n-j\) similarly and examines the Markoff-Rosenberger equation for each case given above, and for each value of \(i\) and \(n-j\). Using modular considerations and by reducing the equation to a quartic genus \(1\) curve, he is able to finish his proof, using Magma for obtaining integral points on this curve.
Reviewer: Anitha Srinivasan (Madrid)On a Diophantine inequality over primes.https://www.zbmath.org/1456.111932021-04-16T16:22:00+00:00"Zhang, Min"https://www.zbmath.org/authors/?q=ai:zhang.min.1"Li, Jinjiang"https://www.zbmath.org/authors/?q=ai:li.jinjiangThe following analogue of the Waring-Goldbach problem was considered by \textit{I. I. Piatetski-Shapiro} [Mat. Sb., Nov. Ser. 30(72), 105--120 (1952; Zbl 0047.28001)]. Let \(c>1\) be non-integer and let \(H(c)\) denote the least \(r\) such that the inequality \(|p^c_1+ p^c_2+\cdots+ p^c_r-N|<\varepsilon\) has a solution in prime numbers \(p_1,p_2,\dots, p_r\) for every \(\varepsilon>0\) and \(N>N_0(c,\varepsilon)\). Piatetski-Shapiro proved that \(H(c)\le (4+ o(1))c\log c\) as \(c\to\infty\) and that \(H(c)\le 5\) when \(1<c<3/2\). (The upper bound \(3/2\) for \(c\) with \(H(c)\le 5\) was improved by many authors.)
\textit{W. Zhai} and \textit{X. Cao} [Adv. Math., Beijing 32, 63--73 (2003)] proved that \(H(c)\le 4\) for \(1<c<81/68\). \textit{Q. Mu} [Adv. Math., Beijing 44, No. 4, 621--637 (2015; Zbl 1349.11082)] showed that \(H(c)\le 4\) for \(1<c<97/81\).
In the paper under review, the authors improve Mu's result [loc. cit.] in the following way. If \(1<c<6/5\) then the Diophantine inequality \(|p^c_1+ p^c_2+ p^c_3+ p^c_4- N|<\log^{-1} N\) is solvable in prime variables \(p_1\), \(p_2\), \(p_3\), \(p_4\) for sufficiently large \(N\). The authors also present a lower bound for the number of solutions. The improvement comes from using the methods develop by \textit{A. Kumchev} [Acta Arith. 89, No. 4, 311--330 (1999; Zbl 0980.11044)] with more delicate sieving techniques and various bounds for exponential sums, combining with a version of Harman's sieve (see \textit{G. Harman} [J. Lond. Math. Soc., II. Ser. 27, 9--18 (1983; Zbl 0504.10018); Proc. Lond. Math. Soc. (3) 72, No. 2, 241--260 (1996; Zbl 0874.11052)]).
Reviewer: Mihály Szalay (Budapest)On descriptions of all translation invariant \(p\)-adic Gibbs measures for the Potts model on the Cayley tree of order three.https://www.zbmath.org/1456.821812021-04-16T16:22:00+00:00"Saburov, Mansoor"https://www.zbmath.org/authors/?q=ai:saburov.mansoor"Khameini Ahmad, Mohd Ali"https://www.zbmath.org/authors/?q=ai:ahmad.mohd-ali-khameiniSummary: Unlike the real number field, a set of \(p\)-adic Gibbs measures of \(p\)-adic lattice models of statistical mechanics has a complex structure in a sense that it is strongly tied up with a Diophantine problem over \(p\)-adic fields. Recently, all translation-invariant \(p\)-adic Gibbs measures of the \(p\)-adic Potts model on the Cayley tree of order two were described by means of roots of a certain quadratic equation over some domain of the \(p\)-adic field. In this paper, we consider the same problem on the Cayley tree of order three. In this case, we show that all translation-invariant \(p\)-adic Gibbs measures of the \(p\)-adic Potts model can be described in terms of roots of some cubic equation over \(\mathbb Z_p\setminus\mathbb Z_p^\ast\). In own its turn, we also provide a solvability criterion of a general cubic equation over \(\mathbb Z_p\setminus\mathbb Z_p^\ast\) for \(p>3\).Some family of Diophantine pairs with Fibonacci numbers.https://www.zbmath.org/1456.110322021-04-16T16:22:00+00:00"Park, Jinseo"https://www.zbmath.org/authors/?q=ai:park.jinseo"Lee, June Bok"https://www.zbmath.org/authors/?q=ai:lee.june-bokLet \( (F_n)_{n\ge 0} \) be the sequence of Fibonacci numbers defined by the linear recurrence: \( F_0=0 \), \( F_1=1 \), and \( F_{n+2}=F_{n+1}+F_n \) for all \( n\ge 0 \). A Diophantine \( m \)-tuple is a set of \( m \) positive integers such that the product of any two of them increased by \( 1 \) gives a perfect square. For any Diophantine triple \( \{a,b, c\} \), the set \( \{a,b,c, d_{\pm}\} \) is a Diophantine quadruple, where \[ d_{\pm}=a+b+c+2abc\pm 2rst, \] and \( r, s, t \) are the positive integers satisfying \[ ab+1=r^{2}, ~ ac+1=s^{2}, ~ bc+1=t^{2}. \]
An important conjecture in this area of research states that if \( \{a,b,c,d\} \) is a Diophantine quadruple and \( d> \max\{a,b,c\} \), then \( d=d_{+} \). Such a Diophantine quadruple is called a regular Diophantine quadruple. In the paper under review, the authors prove the following theorem, which is the main result in the paper.
Theorem 1. Let \( k \) be a positive integer. If the set \( \{F_{2k}, F_{2k+4}, c, d\} \) is a Diophantine qaudruple with \( c<d \), then \( d=d_{+} \).
The proof of Theorem 1 follows from a clever combination of techniques in Diophantine number theory, the results from solving a system of simultaneous Pellian equations, the usual properties of the Fibonacci numbers, the theory of linear forms in logarithms of algebraic numbers á la Baker, and the reduction techniques involving the theory of continued fractions. Computations are done in \texttt{PARI/GP}.
Reviewer: Mahadi Ddamulira (Kampala)An algorithm for solving a quartic Diophantine equation satisfying Runge's condition.https://www.zbmath.org/1456.112372021-04-16T16:22:00+00:00"Osipov, N. N."https://www.zbmath.org/authors/?q=ai:osipov.nikolay-n|osipov.nikolai-n"Dalinkevich, S. D."https://www.zbmath.org/authors/?q=ai:dalinkevich.s-dThe classical method of \textit{ C. Runge} [J. Reine Angew. Math. 100, 425--435 (1887; JFM 19.0076.03)] is well known in diophantine number theory, see also \textit{ P. G. Walsh} [Acta Arith. 62, No. 2, 157--172 (1992; Zbl 0769.11017)]. The scope of equations that can be solved using this method is although restricted, but it extends also to surprising cases. Moreover, the resolution of these equations might be very efficient.
For cubic equations \textit{ N. N. Osipov} and \textit{B. V. Gulnova} [J. Sib. Fed. Univ. Math. Phys. 11, No. 2, 137--147 (2018; Zbl 07325400)] gave a practical algorithm which is extended in the present paper to certain quartic equations by the authors. Moreover, the algorithm is implemented in the computer algebra system PARI/GP.
For the entire collection see [Zbl 1428.68016].
Reviewer: István Gaál (Debrecen)The finite subgroups of \(\mathrm{SL}(3,\overline{F})\).https://www.zbmath.org/1456.200562021-04-16T16:22:00+00:00"Flicker, Yuval Z."https://www.zbmath.org/authors/?q=ai:flicker.yuval-zThe \textit{special linear group} \(\mathrm{SL}(n,F)\) of degree \(n\) over a field \(F\) is the set of all \(n\times n\) matrices with determinant \(1\), with the group operations of ordinary matrix multiplication and matrix inversion. In the paper under review, the author gives a complete exposition of the finite subgroups of \(\mathrm{SL}(3,\bar{F})\), where \(\bar{F}\) is a separably closed field of characteristic not dividing the order of the finite subgroup. This completes the earlier work of \textit{H. F. Blichfeldt} [Math. Ann. 63, 552--572 (1907; JFM 38.0192.03)]. Several elementary examples are illustrated by the author in the exposition.
Reviewer: Mahadi Ddamulira (Kampala)Linearly dependent powers of binary quadratic forms.https://www.zbmath.org/1456.110442021-04-16T16:22:00+00:00"Reznick, Bruce"https://www.zbmath.org/authors/?q=ai:reznick.bruceSummary: Given an integer \(d \ge 2\), what is the smallest \(r\) so that there is a set of binary quadratic forms \(\{f_1,\dots,f_r\}\) for which \(\{f_j^d\}\) is nontrivially linearly dependent? We show that if \(r \le 4\), then \(d \le 5\), and for \(d \ge 4\), construct such a set with \(r = \lfloor d/2\rfloor + 2\). Many explicit examples are given, along with techniques for producing others.