Title: On the frequency of primes preserving dynamical irredu cibility of polynomials

\nAbstract: In this talk we address an open question in arithmetic dynamics regarding the frequency of primes modulo w hich all the iterates of an integer polynomial remain irreducible. More pr ecisely\, for a class of integer polynomials $f$\, which in particular inc ludes all quadratic polynomials\, we show that\, under some natural condit ions\, the set of primes $p$ such that all iterates of $f$ are irreducible modulo $p$ is of relative density zero. Our results rely on a combination of analytic (Selberg's sieve) and Diophantine (finiteness of solutions to certain hyperelliptic equations) tools\, which we will briefly describe. Joint wok with Laszlo Mérai and Igor Shparlinski (2021\, 2024).

\nS piegelgasse 5\, Seminarraum 05.002

DTEND;TZID=Europe/Zurich:20241031T151500 END:VEVENT BEGIN:VEVENT UID:news1739@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241018T121606 DTSTART;TZID=Europe/Zurich:20241024T141500 SUMMARY:Number Theory Seminar: Michael Stoll (Universität Bayreuth) DESCRIPTION:Titel: Conjectural asymptotics of prime orders of points on ell iptic curves over number fields Abstract: Define\, for a positive integer ~$d$\, $S(d)$ to be the set of all primes $p$ that occur as the order of a point $P \\in E(K)$ on an elliptic curve $E$ defined over a number field $K$ of degree $d$. We discuss how some plausible conjectures on the sparsi ty of newforms with certain properties would allow us to deduce a fairly p recise result on the asymptotic behavior of $\\max S(d)$ as $d$ tends to i nfinity. This is joint work with Maarten Derickx.\\r\\nLocation: Spiegelg asse 5\, Seminarraum 05.002 X-ALT-DESC:Titel: Conjectural asymptotics of prime orders of points on e
lliptic curves over number fields

Abstract: Define\, for a po
sitive integer~$d$\, $S(d)$ to be the set of all primes $p$ that occur as
the order of a point $P \\in E(K)$ on an elliptic curve $E$ defined over a
number field $K$ of degree $d$. We discuss how some plausible conjectures
on the sparsity of newforms with certain properties would allow us to ded
uce a fairly precise result on the asymptotic behavior of $\\max S(d)$ as
$d$ tends to infinity.

This is joint work with Maarten Derick
x.

Location: Spiegelgasse 5\, Seminarraum 05.002

DTEND;TZID=Europe/Zurich:20241024T151500 END:VEVENT BEGIN:VEVENT UID:news1738@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20241009T144613 DTSTART;TZID=Europe/Zurich:20241010T104500 SUMMARY:Number Theory Seminar: Stefan Kebekus (Universität Freiburg) DESCRIPTION:Title: Extension Theorems for differential forms and applicatio ns\\r\\nAbstract: We present new extension theorems for differential forms on singular complex spaces and explain their use in the study of minimal varieties. We survey a number of applications\, pertaining to classificati on and characterisation of special varieties\, non-Abelian Hodge Theory in the singular setting\, and quasi-étale uniformization.\\r\\nLocation: H örsaal 114\, Kollegienhaus\\r\\nPlease carefully note the unusual time an d location. X-ALT-DESC:Title: Extension Theorems for differential forms and applicat ions

\nAbstract: We present new extension theorems for differential forms on singular complex spaces and explain their use in the study of min imal varieties. We survey a number of applications\, pertaining to classif ication and characterisation of special varieties\, non-Abelian Hodge Theo ry in the singular setting\, and quasi-étale uniformization.

\nLoca tion: Hörsaal 114\, Kollegienhaus

\n**Please carefully note t
he unusual time and location.**

More information on the website:

\nhttps://rhine-transcendence.github.io /meeting5

DTEND;TZID=Europe/Zurich:20240411T161000 END:VEVENT BEGIN:VEVENT UID:news1592@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20231016T134156 DTSTART;TZID=Europe/Zurich:20231110T140000 SUMMARY:Number Theory Days 2023 DESCRIPTION:Weitere Informationen zur Konferenz finden Sie hier: https://nu mbertheory.dmi.unibas.ch/ntd2023/index.html [https://numbertheory.dmi.unib as.ch/ntd2023/index.html]\\r\\nDie Registrierung ist kostenlos\, aber obli gatorisch. Registrieren Sie sich bitte hier: Registrierung [https://forms. gle/FWiTsAM5mP6MQhjFA]. X-ALT-DESC:Weitere Informationen zur Konferenz finden Sie hier: https://numbertheo ry.dmi.unibas.ch/ntd2023/index.html

\nDie Registrierung ist kost enlos\, aber obligatorisch. Registrieren Sie sich bitte hier: Registrierung.

DTEND;TZID=Europe/Zurich:20231111T113000 END:VEVENT BEGIN:VEVENT UID:news1371@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220519T114925 DTSTART;TZID=Europe/Zurich:20220526T170000 SUMMARY:Number Theory Web Seminar: Yunqing Tang (Princeton University) DESCRIPTION:In this talk\, we will discuss the proof of the unbounded denom inators conjecture on Fourier coefficients of SL_2(Z)-modular forms\, and the proof of irrationality of 2-adic zeta value at 5. Both proofs use an a rithmetic holonomicity theorem\, which can be viewed as a refinement of An dré’s algebraicity criterion. If time permits\, we will give a proof of the arithmetic holonomicity theorem via the slope method a la Bost.\\r\\n This is joint work with Frank Calegari and Vesselin Dimitrov.\\r\\nFor fur ther information about the seminar\, please visit this webpage [https://ww w.ntwebseminar.org/]. X-ALT-DESC:In this talk\, we will discuss the proof of the unb ounded denominators conjecture on Fourier coefficients of SL_2(Z)-modular forms\, and the proof of irrationality of 2-adic zeta value at 5. Both pro ofs use an arithmetic holonomicity theorem\, which can be viewed as a refi nement of André’s algebraicity criterion. If time permits\, we will giv e a proof of the arithmetic holonomicity theorem via the slope method a la Bost.

\nThis is joint work with Frank Calegari and Vessel in Dimitrov.

\nFor further information about the seminar\, please vi sit this webpage.

DTEND;TZID=Europe/Zurich:20220526T180000 END:VEVENT BEGIN:VEVENT UID:news1370@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220519T114626 DTSTART;TZID=Europe/Zurich:20220519T170000 SUMMARY:Number Theory Web Seminar: Jeffrey Vaaler (University of Texas at A ustin) DESCRIPTION:The abstract of the talk is here [https://drive.google.com/file /d/1VDQLDlcC3IDEMduR6H-X9Rf0jRxSZ_J-/view] available.\\r\\nFor further inf ormation about the seminar\, please visit this webpage [https://www.ntwebs eminar.org/]. X-ALT-DESC:The abstract of the talk is here available.

\n< p>For further information about the seminar\, please visit this webpage. DTEND;TZID=Europe/Zurich:20220519T180000 END:VEVENT BEGIN:VEVENT UID:news1363@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220502T112414 DTSTART;TZID=Europe/Zurich:20220512T170000 SUMMARY:Number Theory Web Seminar: Robert Charles Vaughan (Pennsylvania St ate University) DESCRIPTION:The abstract of the talk is here [https://drive.google.com/file /d/17K_PLvpAkfZ3S5nw2yOWQKC18MgLl2rC/view] available:\\r\\nFor further inf ormation about the seminar\, please visit this webpage [https://www.ntwebs eminar.org/]. X-ALT-DESC:The abstract of the talk is here available:

\n< p>For further information about the seminar\, please visit this webpage. DTEND;TZID=Europe/Zurich:20220512T180000 END:VEVENT BEGIN:VEVENT UID:news1306@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220502T111049 DTSTART;TZID=Europe/Zurich:20220505T170000 SUMMARY:Number Theory Web Seminar: Levent Alpöge (Harvard University) DESCRIPTION:It's easy that 0% of integers are the sum of two integral cubes (allowing opposite signs!).\\r\\nI will explain joint work with Bhargava and Shnidman in which we show:\\r\\n1. At least a sixth of integers are no t the sum of two rational cubes\,\\r\\nand\\r\\n2. At least a sixth of odd integers are the sum of two rational cubes!\\r\\n(--- with 2. relying on new 2-converse results of Burungale-Skinner.)\\r\\nThe basic principle is that "there aren't even enough 2-Selmer elements to go around" to contradi ct e.g. 1.\, and we show this by using the circle method "inside" the usua l geometry of numbers argument applied to a particular coregular represent ation. Even then the resulting constant isn't small enough to conclude 1.\ , so we use the clean form of root numbers in the family x^3 + y^3 = n and the p-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.\\r\\nFo r further information about the seminar\, please visit this webpage [https ://www.ntwebseminar.org/]. X-ALT-DESC:It's easy that 0% of integers are the sum of two in tegral cubes (allowing opposite signs!).

\nI will explain joint work with Bhargava and Shnidman in which we show:

\n1. At least a sixth of integers are not the sum of two rational cubes\,

\nand

\n2. At least a sixth of odd integers are the sum of two rational cu bes!

\n(--- with 2. relying on new 2-converse results of Burungale-S kinner.)

\nThe basic principle is that "there aren't even enough 2-Selmer elements to go around" to contradict e.g. 1.\, and we show this by using the circle method "inside" the usual geometry of numbers ar gument applied to a particular coregular representation. Even then the res ulting constant isn't small enough to conclude 1.\, so we use the clean fo rm of root numbers in the family x^3 + y^3 = n and the p-parity theorem of Nekovar/Dokchitser-Dokchitser to succeed.

\nFor further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220505T180000 END:VEVENT BEGIN:VEVENT UID:news1307@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220425T100745 DTSTART;TZID=Europe/Zurich:20220428T170000 SUMMARY:Number Theory Web Seminar: Andrew Granville (Université de Montré al) DESCRIPTION:In 1878\, in the first volume of the first mathematics journal published in the US\, Edouard Lucas wrote 88 pages (in French) on linear r ecurrence sequences\, placing Fibonacci numbers and other linear recurrenc e sequences into a broader context. He examined their behaviour locally as well as globally\, and asked several questions that influenced much resea rch in the century and a half to come.\\r\\nIn a sequence of papers in the 1930s\, Marshall Hall further developed several of Lucas' themes\, includ ing studying and trying to classify third order linear divisibility sequen ces\; that is\, linear recurrences like the Fibonacci numbers which have t he additional property that $F_m$ divides $F_n$ whenever $m$ divides $n$. Because of many special cases\, Hall was unable to even conjecture what a general theorem should look like\, and despite developments over the years by various authors\, such as Lehmer\, Morgan Ward\, van der Poorten\, Bez ivin\, Petho\, Richard Guy\, Hugh Williams\,... with higher order linear d ivisibility sequences\, even the formulation of the classification has rem ained mysterious.\\r\\nIn this talk we present our ongoing efforts to clas sify all linear divisibility sequences\, the key new input coming from a w onderful application of the Schmidt/Schlickewei subspace theorem from the theory of diophantine approximation\, due to Corvaja and Zannier.\\r\\nFor further information about the seminar\, please visit this webpage [https: //www.ntwebseminar.org/]. X-ALT-DESC:In 1878\, in the first volume of the first mathemat ics journal published in the US\, Edouard Lucas wrote 88 pages (in French) on linear recurrence sequences\, placing Fibonacci numbers and other line ar recurrence sequences into a broader context. He examined their behaviou r locally as well as globally\, and asked several questions that influence d much research in the century and a half to come.

\nIn a sequence of papers in the 1930s\, Marshall Hall further developed several of Lucas' themes\, including studying and trying to classify third order l inear divisibility sequences\; that is\, linear recurrences like the Fibon acci numbers which have the additional property that $F_m$ divides $F_n$ w henever $m$ divides $n$. Because of many special cases\, Hall was unable t o even conjecture what a general theorem should look like\, and despite de velopments over the years by various authors\, such as Lehmer\, Morgan War d\, van der Poorten\, Bezivin\, Petho\, Richard Guy\, Hugh Williams\,... w ith higher order linear divisibility sequences\, even the formulation of t he classification has remained mysterious.

\nIn this talk we present our ongoing efforts to classify all linear divisibility sequenc es\, the key new input coming from a wonderful application of the Schmidt/ Schlickewei subspace theorem from the theory of diophantine approximation\ , due to Corvaja and Zannier.

\nFor further information about the se minar\, please visit this webpage< /a>.

DTEND;TZID=Europe/Zurich:20220428T180000 END:VEVENT BEGIN:VEVENT UID:news1304@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220325T101357 DTSTART;TZID=Europe/Zurich:20220421T170000 SUMMARY:Number Theory Web Seminar: Joni Teräväinen (University of Turku) DESCRIPTION:I will discuss the short interval behaviour of the von Mangoldt and Möbius functions twisted by exponentials. I will in particular menti on new results on sums of these functions twisted by polynomial exponentia l phases\, or even more general nilsequence phases. I will also discuss co nnections to Chowla's conjecture. This is based on joint works with Kaisa Matomäki\, Maksym Radziwiłł\, Xuancheng Shao\, Terence Tao and Tamar Zi egler.\\r\\nFor further information about the seminar\, please visit this webpage [https://www.ntwebseminar.org/]. X-ALT-DESC:I will discuss the short interval behaviour of the von Mangoldt and Möbius functions twisted by exponentials. I will in part icular mention new results on sums of these functions twisted by polynomia l exponential phases\, or even more general nilsequence phases. I will als o discuss connections to Chowla's conjecture. This is based on joint works with Kaisa Matomäki\, Maksym Radziwiłł\, Xuancheng Shao\, Terence Tao and Tamar Ziegler.

\nFor further information about the seminar\, ple ase visit this webpage.

DTEND;TZID=Europe/Zurich:20220421T180000 END:VEVENT BEGIN:VEVENT UID:news1303@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220411T135326 DTSTART;TZID=Europe/Zurich:20220414T170000 SUMMARY:Number Theory Web Seminar: Ram Murty (Queen's University) DESCRIPTION:There is a probability distribution attached to the Riemann zet a function which allows one to formulate the Riemann hypothesis in terms o f the cumulants of this distribution and is due to Biane\, Pitman and Yor. The cumulants can be related to generalized Euler-Stieltjes constants and to Li's criterion for the Riemann hypothesis. We will discuss these resul ts and present some new results related to this theme.\\r\\nFor further in formation about the seminar\, please visit this webpage [https://www.ntweb seminar.org/]. X-ALT-DESC:There is a probability distribution attached to the Riemann zeta function which allows one to formulate the Riemann hypothesi s in terms of the cumulants of this distribution and is due to Biane\, Pit man and Yor. The cumulants can be related to generalized Euler-Stieltjes c onstants and to Li's criterion for the Riemann hypothesis. We will discuss these results and present some new results related to this theme.

\nFor further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220414T180000 END:VEVENT BEGIN:VEVENT UID:news1302@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220325T101305 DTSTART;TZID=Europe/Zurich:20220407T170000 SUMMARY:Number Theory Web Seminar: Ana Caraiani (Imperial College London) DESCRIPTION:Shimura varieties are certain highly symmetric algebraic variet ies that generalise modular curves and that play an important role in the Langlands program. In this talk\, I will survey recent vanishing conjectur es and results about the cohomology of Shimura varieties with torsion coef ficients\, under both local and global representation-theoretic conditions . I will illustrate the geometric ingredients needed to establish these re sults using the toy model of the modular curve. I will also mention severa l applications\, including to (potential) modularity over CM fields.\\r\\n For further information about the seminar\, please visit this webpage [htt ps://www.ntwebseminar.org/]. X-ALT-DESC:Shimura varieties are certain highly symmetric alge braic varieties that generalise modular curves and that play an important role in the Langlands program. In this talk\, I will survey recent vanishi ng conjectures and results about the cohomology of Shimura varieties with torsion coefficients\, under both local and global representation-theoreti c conditions. I will illustrate the geometric ingredients needed to establ ish these results using the toy model of the modular curve. I will also me ntion several applications\, including to (potential) modularity over CM f ields.

\nFor further information about the seminar\, please visit th is webpage.

DTEND;TZID=Europe/Zurich:20220407T180000 END:VEVENT BEGIN:VEVENT UID:news1339@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220325T101040 DTSTART;TZID=Europe/Zurich:20220331T170000 SUMMARY:Number Theory Web Seminar: William Chen (Institute for Advanced Stu dy) DESCRIPTION:In this talk we will show that the integral points of the Marko ff equation x^2 + y^2 + z^2 - xyz = 0 surject onto its F_p-points for all but finitely many primes p. This essentially resolves a conjecture of Bour gain\, Gamburd\, and Sarnak\, and a question of Frobenius from 1913. The p roof relates the question to the classical problem of classifying the conn ected components of the Hurwitz moduli spaces H(g\,n) classifying finite c overs of genus g curves with n branch points. Over a century ago\, Clebsch and Hurwitz established connectivity for the subspace classifying simply branched covers of the projective line\, which led to the first proof of t he irreducibility of the moduli space of curves of a given genus. More rec ently\, the work of Dunfield-Thurston and Conway-Parker establish connecti vity in certain situations where the monodromy group is fixed and either g or n are allowed to be large\, which has been applied to study Cohen-Lens tra heuristics over function fields. In the case where (g\,n) are fixed an d the monodromy group is allowed to vary\, far less is known. In our case we study SL(2\,p)-covers of elliptic curves\, only branched over the origi n\, and establish connectivity\, for all sufficiently large p\, of the sub space classifying those covers with ramification indices 2p. The proof bui lds upon asymptotic results of Bourgain\, Gamburd\, and Sarnak\, the key n ew ingredient being a divisibility result on the degree of a certain forge tful map between moduli spaces\, which provides enough rigidity to bootstr ap their asymptotics to a result for all sufficiently large p.\\r\\nFor fu rther information about the seminar\, please visit this webpage [https://w ww.ntwebseminar.org/]. X-ALT-DESC:In this talk we will show that the integral points of the Markoff equation x^2 + y^2 + z^2 - xyz = 0 surject onto its F_p-poi nts for all but finitely many primes p. This essentially resolves a conjec ture of Bourgain\, Gamburd\, and Sarnak\, and a question of Frobenius from 1913. The proof relates the question to the classical problem of classify ing the connected components of the Hurwitz moduli spaces H(g\,n) classify ing finite covers of genus g curves with n branch points. Over a century a go\, Clebsch and Hurwitz established connectivity for the subspace classif ying simply branched covers of the projective line\, which led to the firs t proof of the irreducibility of the moduli space of curves of a given gen us. More recently\, the work of Dunfield-Thurston and Conway-Parker establ ish connectivity in certain situations where the monodromy group is fixed and either g or n are allowed to be large\, which has been applied to stud y Cohen-Lenstra heuristics over function fields. In the case where (g\,n) are fixed and the monodromy group is allowed to vary\, far less is known. In our case we study SL(2\,p)-covers of elliptic curves\, only branched ov er the origin\, and establish connectivity\, for all sufficiently large p\ , of the subspace classifying those covers with ramification indices 2p. T he proof builds upon asymptotic results of Bourgain\, Gamburd\, and Sarnak \, the key new ingredient being a divisibility result on the degree of a c ertain forgetful map between moduli spaces\, which provides enough rigidit y to bootstrap their asymptotics to a result for all sufficiently large p.

\nFor further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220331T180000 END:VEVENT BEGIN:VEVENT UID:news1301@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220317T094916 DTSTART;TZID=Europe/Zurich:20220324T170000 SUMMARY:Number Theory Web Seminar: Winnie Li (Pennsylvania State University ) DESCRIPTION:The theme of this survey talk is zeta functions which count clo sed geodesics on objects arising from real and p-adic groups. Our focus is on PGL(n). For PGL(2)\, these are the Selberg zeta function for compact q uotients of the upper half-plane and the Ihara zeta function for finite re gular graphs. We shall explain the identities satisfied by these zeta func tions\, which show interconnections between combinatorics\, group theory a nd number theory. Comparisons will be made for zeta identities from differ ent background. Like the Riemann zeta function\, the analytic behavior of a group based zeta function governs the distribution of the prime geodesic s in its definition.\\r\\nFor further information about the seminar\, plea se visit this webpage [https://www.ntwebseminar.org/]. X-ALT-DESC:The theme of this survey talk is zeta functions whi ch count closed geodesics on objects arising from real and p-adic groups. Our focus is on PGL(n). For PGL(2)\, these are the Selberg zeta function f or compact quotients of the upper half-plane and the Ihara zeta function f or finite regular graphs. We shall explain the identities satisfied by the se zeta functions\, which show interconnections between combinatorics\, gr oup theory and number theory. Comparisons will be made for zeta identities from different background. Like the Riemann zeta function\, the analytic behavior of a group based zeta function governs the distribution of the pr ime geodesics in its definition.

\nFor further information about the seminar\, please visit this webpa ge.

DTEND;TZID=Europe/Zurich:20220324T180000 END:VEVENT BEGIN:VEVENT UID:news1300@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220310T151447 DTSTART;TZID=Europe/Zurich:20220317T170000 SUMMARY:Number Theory Web Seminar: Aaron Levin (Michigan State University) DESCRIPTION:The classical Weil height machine associates heights to divisor s on a projective variety. I will give a brief\, but gentle\, introduction to how this machinery extends to objects (closed subschemes) in higher co dimension\, due to Silverman\, and discuss various ways to interpret the h eights. We will then discuss several recent results in which these ideas p lay a prominent and central role.\\r\\nFor further information about the s eminar\, please visit this webpage [https://www.ntwebseminar.org/]. X-ALT-DESC:The classical Weil height machine associates height s to divisors on a projective variety. I will give a brief\, but gentle\, introduction to how this machinery extends to objects (closed subschemes) in higher codimension\, due to Silverman\, and discuss various ways to int erpret the heights. We will then discuss several recent results in which t hese ideas play a prominent and central role.

\nFor further informat ion about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220317T180000 END:VEVENT BEGIN:VEVENT UID:news1299@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220301T152844 DTSTART;TZID=Europe/Zurich:20220310T170000 SUMMARY:Number Theory Web Seminar: Dmitry Kleinbock (Brandeis University) DESCRIPTION:Let $\\psi$ be a decreasing function defined on all large posit ive real numbers. We say that a real $m \\times n$ matrix $Y$ is "$\\psi$- Dirichlet" if for every sufficiently large real number $T$ there exist non -trivial integer vectors $(p\,q)$ satisfying $\\|Yq-p\\|^m < \\psi(T)$ and $\\|q\\|^n < T$ (where $\\|\\cdot\\|$ denotes the supremum norm on vector s). This generalizes the property of $Y$ being "Dirichlet improvable" whic h has been studied by several people\, starting with Davenport and Schmidt in 1969. I will present results giving sufficient conditions on $\\psi$ t o ensure that the set of $\\psi$-Dirichlet matrices has zero (resp.\, full ) measure. If time allows I will mention a geometric generalization of the set-up\, where the supremum norm is replaced by an arbitrary norm. Joint work with Anurag Rao\, Andreas Strombergsson\, Nick Wadleigh and Shuchweng Yu.\\r\\nFor further information about the seminar\, please visit this we bpage [https://www.ntwebseminar.org/]. X-ALT-DESC:Let $\\psi$ be a decreasing function defined on all large positive real numbers. We say that a real $m \\times n$ matrix $Y$ is "$\\psi$-Dirichlet" if for every sufficiently large real number $T$ the re exist non-trivial integer vectors $(p\,q)$ satisfying $\\|Yq-p\\|^m < \; \\psi(T)$ and $\\|q\\|^n <\; T$ (where $\\|\\cdot\\|$ denotes the sup remum norm on vectors). This generalizes the property of $Y$ being "Dirich let improvable" which has been studied by several people\, starting with D avenport and Schmidt in 1969. I will present results giving sufficient con ditions on $\\psi$ to ensure that the set of $\\psi$-Dirichlet matrices ha s zero (resp.\, full) measure. If time allows I will mention a geometric g eneralization of the set-up\, where the supremum norm is replaced by an ar bitrary norm. Joint work with Anurag Rao\, Andreas Strombergsson\, Nick Wa dleigh and Shuchweng Yu.

\nFor further information about the seminar \, please visit this webpage.< /p> DTEND;TZID=Europe/Zurich:20220310T180000 END:VEVENT BEGIN:VEVENT UID:news1291@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220301T152634 DTSTART;TZID=Europe/Zurich:20220303T170000 SUMMARY:Number Theory Web Seminar: Ekin Özman (Boğaziçi University) DESCRIPTION:Understanding solutions of Diophantine equations over rationals or more generally over any number field is one of the main problems of nu mber theory. By the help of the modular techniques used in the proof of Fe rmat’s last theorem by Wiles and its generalizations\, it is possible to solve other Diophantine equations too. Understanding quadratic points on the classical modular curve play a central role in this approach. It is al so possible to study the solutions of Fermat type equations over number fi elds asymptotically. In this talk\, I will mention some recent results abo ut these notions for the classical Fermat equation as well as some other D iophantine equations.\\r\\nFor further information about the seminar\, ple ase visit this webpage [https://www.ntwebseminar.org/]. X-ALT-DESC:

Understanding solutions of Diophantine equations ov er rationals or more generally over any number field is one of the main pr oblems of number theory. By the help of the modular techniques used in the proof of Fermat’s last theorem by Wiles and its generalizations\, it is possible to solve other Diophantine equations too. Understanding quadrati c points on the classical modular curve play a central role in this approa ch. It is also possible to study the solutions of Fermat type equations ov er number fields asymptotically. In this talk\, I will mention some recent results about these notions for the classical Fermat equation as well as some other Diophantine equations.

\nFor further information about th e seminar\, please visit this webp age.

DTEND;TZID=Europe/Zurich:20220303T180000 END:VEVENT BEGIN:VEVENT UID:news1290@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220209T111633 DTSTART;TZID=Europe/Zurich:20220224T170000 SUMMARY:Number Theory Web Seminar: Igor Shparlinski (UNSW Sydney) DESCRIPTION:We present some old and more recent results which suggest that Kloosterman and Salie sums exhibit a pseudorandom behaviour similar to the behaviour which is traditionally attributed to the Mobius function. In pa rticular\, we formulate some analogues of the Chowla Conjecture for Kloost erman and Salie sums. We then describe several results about the non-corre lation of Kloosterman and Salie sums between themselves and also with some classical number-theoretic functions such as the Mobius function\, the di visor function and the sums of binary digits. Various arithmetic applicati ons of these results\, including to asymptotic formulas for moments of var ious L-functions\, will be outlined as well.\\r\\nFor further information about the seminar\, please visit this webpage [https://www.ntwebseminar.or g/]. X-ALT-DESC:We present some old and more recent results which suggest tha t Kloosterman and Salie sums exhibit a pseudorandom behaviour similar to t he behaviour which is traditionally attributed to the Mobius function. In particular\, we formulate some analogues of the Chowla Conjecture for Kloo sterman and Salie sums. We then describe several results about the non-cor relation of Kloosterman and Salie sums between themselves and also with so me classical number-theoretic functions such as the Mobius function\, the divisor function and the sums of binary digits. Various arithmetic applica tions of these results\, including to asymptotic formulas for moments of v arious L-functions\, will be outlined as well.

\nFor further informa tion about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220224T180000 END:VEVENT BEGIN:VEVENT UID:news1289@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220209T111403 DTSTART;TZID=Europe/Zurich:20220217T170000 SUMMARY:Number Theory Web Seminar: Harry Schmidt (University of Basel) DESCRIPTION:In this talk I will give an overview of the history of the Andr é-Oort conjecture and its resolution last year after the final steps were made in work of Pila\, Shankar\, Tsimerman\, Esnault and Groechenig as we ll as Binyamini\, Yafaev and myself. I will focus on the key insights and ideas related to model theory and transcendence theory.\\r\\nFor further i nformation about the seminar\, please visit this webpage [https://www.ntwe bseminar.org/]. X-ALT-DESC:In this talk I will give an overview of the history of the An dré-Oort conjecture and its resolution last year after the final steps we re made in work of Pila\, Shankar\, Tsimerman\, Esnault and Groechenig as well as Binyamini\, Yafaev and myself. I will focus on the key insights an d ideas related to model theory and transcendence theory.

\nFor furt her information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220217T180000 END:VEVENT BEGIN:VEVENT UID:news1288@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220131T140633 DTSTART;TZID=Europe/Zurich:20220210T170000 SUMMARY:Number Theory Web Seminar: Zeev Rudnick (Tel Aviv University) DESCRIPTION:The study of uniform distribution of sequences is more than a c entury old\, with pioneering work by Hardy and Littlewood\, Weyl\, van der Corput and others. More recently\, the focus of research has shifted to m uch finer quantities\, such as the distribution of nearest neighbor gaps a nd the pair correlation function. Examples of interesting sequences for wh ich these quantities have been studied include the zeros of the Riemann ze ta function\, energy levels of quantum systems\, and more. In this exposit ory talk\, I will discuss what is known about these examples and discuss t he many outstanding problems that this theory has to offer.\\r\\nFor furth er information about the seminar\, please visit this webpage [https://www. ntwebseminar.org/]. X-ALT-DESC:The study of uniform distribution of sequences is more than a century old\, with pioneering work by Hardy and Littlewood\, Weyl\, van d er Corput and others. More recently\, the focus of research has shifted to much finer quantities\, such as the distribution of nearest neighbor gaps and the pair correlation function. Examples of interesting sequences for which these quantities have been studied include the zeros of the Riemann zeta function\, energy levels of quantum systems\, and more. In this expos itory talk\, I will discuss what is known about these examples and discuss the many outstanding problems that this theory has to offer.

\nFor further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220210T180000 END:VEVENT BEGIN:VEVENT UID:news1287@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220131T135710 DTSTART;TZID=Europe/Zurich:20220203T170000 SUMMARY:Number Theory Web Seminar: Peter Humphries (University of Virginia) DESCRIPTION:A major area of study in analysis involves the distribution of mass of Laplacian eigenfunctions on a Riemannian manifold. A key result to wards this is explicit L^p-norm bounds for Laplacian eigenfunctions in ter ms of their Laplacian eigenvalue\, due to Sogge in 1988. Sogge's bounds ar e sharp on the sphere\, but need not be sharp on other manifolds. I will d iscuss some aspects of this problem for the modular surface\; in this sett ing\, the Laplacian eigenfunctions are automorphic forms\, and certain L^p -norms can be shown to be closely related to certain mixed moments of L-fu nctions. This is joint with with Rizwanur Khan.\\r\\nFor further informati on about the seminar\, please visit this webpage [https://www.ntwebseminar .org/]. X-ALT-DESC:A major area of study in analysis involves the distribution o f mass of Laplacian eigenfunctions on a Riemannian manifold. A key result towards this is explicit L^p-norm bounds for Laplacian eigenfunctions in t erms of their Laplacian eigenvalue\, due to Sogge in 1988. Sogge's bounds are sharp on the sphere\, but need not be sharp on other manifolds. I will discuss some aspects of this problem for the modular surface\; in this se tting\, the Laplacian eigenfunctions are automorphic forms\, and certain L ^p-norms can be shown to be closely related to certain mixed moments of L- functions. This is joint with with Rizwanur Khan.

\nFor further info rmation about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220203T180000 END:VEVENT BEGIN:VEVENT UID:news1286@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220119T084242 DTSTART;TZID=Europe/Zurich:20220127T170000 SUMMARY:Number Theory Web Seminar: Larry Guth (MIT) DESCRIPTION:The Vinogradov mean value conjecture concerns the number of sol utions of a system of diophantine equations. This number of solutions can also be written as a certain moment of a trigonometric polynomial. The con jecture was proven in the 2010s by Bourgain-Demeter-Guth and by Wooley\, a nd recently there was a shorter proof by Guo-Li-Yang-Zorin-Kranich. The de tails of each proof involve some intricate estimates. The goal of the talk is to try to reflect on the proof(s) in a big picture way. A key ingredie nt in all the proofs is to combine estimates at many different scales\, us ually by doing induction on scales. Why does this multi-scale induction he lp? What can multi-scale induction tell us and what are its limitations?\\ r\\nFor further information about the seminar\, please visit this webpage [https://www.ntwebseminar.org/]. X-ALT-DESC:The Vinogradov mean value conjecture concerns the number of s olutions of a system of diophantine equations. This number of solutions ca n also be written as a certain moment of a trigonometric polynomial. The c onjecture was proven in the 2010s by Bourgain-Demeter-Guth and by Wooley\, and recently there was a shorter proof by Guo-Li-Yang-Zorin-Kranich. The details of each proof involve some intricate estimates. The goal of the ta lk is to try to reflect on the proof(s) in a big picture way. A key ingred ient in all the proofs is to combine estimates at many different scales\, usually by doing induction on scales. Why does this multi-scale induction help? What can multi-scale induction tell us and what are its limitations?

\nFor further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220127T180000 END:VEVENT BEGIN:VEVENT UID:news1285@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220117T122020 DTSTART;TZID=Europe/Zurich:20220120T170000 SUMMARY:Number Theory Web Seminar: Jozsef Solymosi (University of British C olumbia) DESCRIPTION:We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Appl ying these bounds we show that the distance sets of finite pointsets in ℝ^d generate high rank multiplicative groups and that multiplicative gro ups of small rank cannot contain large sumsets. (Joint work with Noga Alon )\\r\\nFor further information about the seminar\, please visit this webpa ge [https://www.ntwebseminar.org/]. X-ALT-DESC:We establish lower bounds on the rank of matrices in which al l but the diagonal entries lie in a multiplicative group of small rank. Ap plying these bounds we show that the distance sets of finite pointsets in ℝ^d generate high rank multiplicative groups and that multiplicative gro ups of small rank cannot contain large sumsets. (Joint work with Noga Alon )

\nFor further information about the seminar\, please visit this webpage.

END:VEVENT BEGIN:VEVENT UID:news1310@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20220112T102603 DTSTART;TZID=Europe/Zurich:20220113T170000 SUMMARY:Number Theory Web Seminar: Péter Varjú (University of Cambridge) DESCRIPTION:Consider random polynomials of degree d whose leading and const ant coefficients are 1 and the rest are independent taking the values 0 or 1 with equal probability. A conjecture of Odlyzko and Poonen predicts tha t such a polynomial is irreducible in Z[x] with high probability as d grow s. This conjecture is still open\, but Emmanuel Breuillard and I proved it assuming the Extended Riemann Hypothesis. I will briefly recall the metho d of proof of this result and will discuss later developments that apply t his method to other models of random polynomials.\\r\\nFor further informa tion about the seminar\, please visit this webpage [https://www.ntwebsemin ar.org/]. X-ALT-DESC:Consider random polynomials of degree d whose leading and con stant coefficients are 1 and the rest are independent taking the values 0 or 1 with equal probability. A conjecture of Odlyzko and Poonen predicts t hat such a polynomial is irreducible in Z[x] with high probability as d gr ows. This conjecture is still open\, but Emmanuel Breuillard and I proved it assuming the Extended Riemann Hypothesis. I will briefly recall the met hod of proof of this result and will discuss later developments that apply this method to other models of random polynomials.

\nFor further in formation about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20220113T180000 END:VEVENT BEGIN:VEVENT UID:news1270@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211206T164028 DTSTART;TZID=Europe/Zurich:20211216T170000 SUMMARY:Number Theory Web Seminar: Sarah Zerbes (University College London\ , UK) DESCRIPTION:Euler systems are one of the most powerful tools for proving ca ses of the Bloch--Kato conjecture\, and other related problems such as the Birch and Swinnerton-Dyer conjecture. I will recall a series of recent wo rks (variously joint with Loeffler\, Pilloni\, Skinner) giving rise to an Euler system in the cohomology of Shimura varieties for GSp(4)\, and an ex plicit reciprocity law relating the Euler system to values of L-functions. I will then recent work with Loeffler\, in which we use this Euler system to prove new cases of the BSD conjecture for modular abelian surfaces ove r Q\, and modular elliptic curves over imaginary quadratic fields.\\r\\nFo r further information about the seminar\, please visit this webpage [https ://www.ntwebseminar.org/]. X-ALT-DESC:Euler systems are one of the most powerful tools for proving
cases of the Bloch--Kato conjecture\, and other related problems such as t
he Birch and Swinnerton-Dyer conjecture.

I will recall a series of r
ecent works (variously joint with Loeffler\, Pilloni\, Skinner) giving ris
e to an Euler system in the cohomology of Shimura varieties for GSp(4)\, a
nd an explicit reciprocity law relating the Euler system to values of L-fu
nctions. I will then recent work with Loeffler\, in which we use this Eule
r system to prove new cases of the BSD conjecture for modular abelian surf
aces over Q\, and modular elliptic curves over imaginary quadratic fields.

For further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20211216T180000 END:VEVENT BEGIN:VEVENT UID:news1269@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211201T103850 DTSTART;TZID=Europe/Zurich:20211209T170000 SUMMARY:Number Theory Web Seminar: Samir Siksek (University of Warwick) DESCRIPTION:The asymptotic Fermat conjecture (AFC) states that for a number field K\, and for sufficiently large primes p\, the only solutions to the Fermat equation X^p+Y^p+Z^p=0 in K are the obvious ones. We sketch recent work that connects the Fermat equation to the far more elementary unit eq uation\, and explain how this surprising connection can be exploited to pr ove AFC for several infinite families of number fields. This talk is based on joint work with Nuno Freitas\, Alain Kraus and Haluk Sengun.\\r\\nFor further information about the seminar\, please visit this webpage [https:/ /www.ntwebseminar.org/]. X-ALT-DESC:The asymptotic Fermat conjecture (AFC) states that for a numb er field K\, and for sufficiently large primes p\, the only solutions to t he Fermat equation X^p+Y^p+Z^p=0 in K are the obvious ones. We sketch rece nt work that connects the Fermat equation to the far more elementary unit equation\, and explain how this surprising connection can be exploited to prove AFC for several infinite families of number fields. This talk is bas ed on joint work with Nuno Freitas\, Alain Kraus and Haluk Sengun.

\nFor further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20211208T180000 END:VEVENT BEGIN:VEVENT UID:news1268@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211125T141706 DTSTART;TZID=Europe/Zurich:20211202T170000 SUMMARY:Number Theory Web Seminar: Kiran Kedlaya (University of California San Diego) DESCRIPTION:We describe several recent results on orders of abelian varieti es over $\\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\\mathbb{F}_2$ (joint with E. Howe)\; every positive integer occurs infinitely often as the order of a simple abelian variety over $\\mathbb{F}_2$\; the geometric decomposition of the simple abelian varieties over $\\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady)\; and the relative class number one problem for fu nction fields is reduced to a finite computation (work in progress). All o f these results rely on the relationship between isogeny classes of abelia n varieties over finite fields and Weil polynomials given by the work of W eil and Honda-Tate. With these results in hand\, most of the work is to co nstruct algebraic integers satisfying suitable archimedean constraints.\\r \\nFor further information about the seminar\, please visit this webpage [ https://www.ntwebseminar.org/]. X-ALT-DESC:We describe several recent results on orders of abelian varie ties over $\\mathbb{F}_2$: every positive integer occurs as the order of a n ordinary abelian variety over $\\mathbb{F}_2$ (joint with E. Howe)\; eve ry positive integer occurs infinitely often as the order of a simple abeli an variety over $\\mathbb{F}_2$\; the geometric decomposition of the simpl e abelian varieties over $\\mathbb{F}_2$ can be described explicitly (join t with T. D'Nelly-Warady)\; and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abel ian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand\, most of the work is to construct algebraic integers satisfying suitable archimedean constraints.< /p>\n

For further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20211202T180000 END:VEVENT BEGIN:VEVENT UID:news1267@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211125T141908 DTSTART;TZID=Europe/Zurich:20211125T170000 SUMMARY:Number Theory Web Seminar: Alexei Skorobogatov (Imperial College Lo ndon) DESCRIPTION:I will discuss logical links among uniformity conjectures conce rning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety\, the Néron–Severi lattice of a K3 surfac e\, and the Galois invariant subgroup of the geometric Brauer group. The t alk is based on a joint work with Martin Orr and Yuri Zarhin.\\r\\nFor fur ther information about the seminar\, please visit this webpage [https://ww w.ntwebseminar.org/]. X-ALT-DESC:I will discuss logical links among uniformity conjectures con cerning K3 surfaces and abelian varieties of bounded dimension defined ove r number fields of bounded degree. The conjectures concern the endomorphis m algebra of an abelian variety\, the Néron–Severi lattice of a K3 surf ace\, and the Galois invariant subgroup of the geometric Brauer group. The talk is based on a joint work with Martin Orr and Yuri Zarhin.

\nFo r further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20211125T180000 END:VEVENT BEGIN:VEVENT UID:news1265@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211116T092154 DTSTART;TZID=Europe/Zurich:20211118T170000 SUMMARY:Number Theory Web Seminar: Myrto Mavraki (Harvard University) DESCRIPTION:Inspired by an analogy between torsion and preperiodic points\, Zhang has proposed a dynamical generalization of the classical Manin-Mumf ord and Bogomolov conjectures. A special case of these conjectures\, for ` split' maps\, has recently been established by Nguyen\, Ghioca and Ye. In particular\, they show that two rational maps have at most finitely many c ommon preperiodic points\, unless they are `related'. Recent breakthroughs by Dimitrov\, Gao\, Habegger and Kühne have established that the classic al Bogomolov conjecture holds uniformly across curves of given genus. In t his talk we discuss uniform versions of the dynamical Bogomolov conjecture across 1-parameter families of certain split maps. To this end\, we estab lish an instance of a 'relative dynamical Bogomolov'. This is work in prog ress joint with Harry Schmidt (University of Basel).\\r\\nFor further info rmation about the seminar\, please visit this webpage [https://www.ntwebse minar.org/]. X-ALT-DESC:Inspired by an analogy between torsion and preperiodic points
\, Zhang has proposed a dynamical generalization of the classical Manin-Mu
mford and Bogomolov conjectures. A special case of these conjectures\, for
`split' maps\, has recently been established by Nguyen\, Ghioca and Ye. I
n particular\, they show that two rational maps have at most finitely many
common preperiodic points\, unless they are `related'. Recent breakthroug
hs by Dimitrov\, Gao\, Habegger and Kühne have established that the class
ical Bogomolov conjecture holds uniformly across curves of given genus.

In this talk we discuss uniform versions of the dynamical Bogomolov co
njecture across 1-parameter families of certain split maps. To this end\,
we establish an instance of a 'relative dynamical Bogomolov'. This is work
in progress joint with Harry Schmidt (University of Basel).

For f urther information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20211118T180000 END:VEVENT BEGIN:VEVENT UID:news1261@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20211111T170000 SUMMARY:Number Theory Web Seminar: Avi Wigderson (Institute for Advanced St udy) DESCRIPTION:Is the universe inherently deterministic or probabilistic? Perh aps more importantly - can we tell the difference between the two?\\r\\nHu manity has pondered the meaning and utility of randomness for millennia.\\ r\\nThere is a remarkable variety of ways in which we utilize perfect coin tosses to our advantage: in statistics\, cryptography\, game theory\, alg orithms\, gambling... Indeed\, randomness seems indispensable! Which of th ese applications survive if the universe had no (accessible) randomness in it at all? Which of them survive if only poor quality randomness is avail able\, e.g. that arises from somewhat "unpredictable" phenomena like the w eather or the stock market?\\r\\nA computational theory of randomness\, de veloped in the past several decades\, reveals (perhaps counter-intuitively ) that very little is lost in such deterministic or weakly random worlds. In the talk I'll explain the main ideas and results of this theory\, notio ns of pseudo-randomness\, and connections to computational intractability. \\r\\nIt is interesting that Number Theory played an important role throug hout this development. It supplied problems whose algorithmic solution mak e randomness seem powerful\, problems for which randomness can be eliminat ed from such solutions\, and problems where the power of randomness remain s a major challenge for computational complexity theorists and mathematici ans. I will use these problems (and others) to demonstrate aspects of this theory.\\r\\nFor further information about the seminar\, please visit thi s webpage [https://www.ntwebseminar.org/]. X-ALT-DESC:Is the universe inherently deterministic or probabilistic? Pe rhaps more importantly - can we tell the difference between the two?

\nHumanity has pondered the meaning and utility of randomness for millenn ia.

\nThere is a remarkable variety of ways in which we utilize perf ect coin tosses to our advantage: in statistics\, cryptography\, game theo ry\, algorithms\, gambling... Indeed\, randomness seems indispensable! Whi ch of these applications survive if the universe had no (accessible) rando mness in it at all? Which of them survive if only poor quality randomness is available\, e.g. that arises from somewhat "unpredictable" phenomena li ke the weather or the stock market?

\nA computational theory of rand omness\, developed in the past several decades\, reveals (perhaps counter- intuitively) that very little is lost in such deterministic or weakly rand om worlds. In the talk I'll explain the main ideas and results of this the ory\, notions of pseudo-randomness\, and connections to computational intr actability.

\nIt is interesting that Number Theory played an importa nt role throughout this development. It supplied problems whose algorithmi c solution make randomness seem powerful\, problems for which randomness c an be eliminated from such solutions\, and problems where the power of ran domness remains a major challenge for computational complexity theorists a nd mathematicians. I will use these problems (and others) to demonstrate a spects of this theory.

\nFor further information about the seminar\, please visit this webpage.

DTEND;TZID=Europe/Zurich:20211111T180000 END:VEVENT BEGIN:VEVENT UID:news825@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20190228T141500 SUMMARY:Number Theory Seminar: Yuri Bilu (Université de Bordeaux) DESCRIPTION:The celebrated André-Oort conjecture about special point on Sh imura varieties is now proved conditionally to the GRH in full generality and unconditionally in many important special cases. In particular\, Pi la (2011) proved it for products of modular curves\, adapting a method p reviously developed by Pila and Zannier in the context of the Manin-Mumfo rd conjecture. Unfortunately\, Pila's argument is non-effective\, using t he Siegel-Brauer inequality. Since 2012 various special cases of the Andr é-Oort conjecture has been proved effectively\, most notably in the work of Lars Kühne. In my talk I will restrict to the case of the "Shimura v ariety" C^n and will try to explain on some simple examples how the effec tive approach of Kühne works. No previous knowledge about André-Oort co njecture is required\, I will give all the necessary background. X-ALT-DESC: The celebrated André-Oort conjecture about special point on Sh imura varieties is now proved conditionally to the GRH in full generality and unconditionally in many important special cases. In particular\, Pi la (2011) proved it for products of modular curves\, adapting a method p reviously developed by Pila and Zannier in the context of the Manin-Mumfo rd conjecture. Unfortunately\, Pila's argument is non-effective\, using t he Siegel-Brauer inequality.Since 2012 various special cases of the André-Oort conjecture has been proved effectively\, most notably in the work of Lars Kühne. In my talk I will restrict to the case of the "\;Shimura variety"\; C^n and will try to explain on some simple examples how the effective approach of Kühne works.

No prev ious knowledge about André-Oort conjecture is required\, I will give all the necessary background. DTEND;TZID=Europe/Zurich:20190228T151500 END:VEVENT BEGIN:VEVENT UID:news321@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20181129T141500 SUMMARY:Number Theory Seminar: Ana Maria Botero (Univ. of Regensburg) DTEND;TZID=Europe/Zurich:20181129T151500 END:VEVENT BEGIN:VEVENT UID:news320@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20181122T141500 SUMMARY:Number Theory Seminar: Amador Martin-Pizarro (Univ. of Freiburg) DESCRIPTION:Lascar showed that the group of automorphisms of the complex fi eld which fix the algebraic closure of the prime field is simple. For this \, he first showed that there are no non-trivial bounded automorphisms. An automorphism is bounded if there is a finite set A such that the image of every element b is algebraic over A together with b. The same result hold s for a "universal" differentially closed field of characteristic zero\, w here we replace algebraic by differentially algebraic. Together with T. Bl ossier and C. Hardouin\, we provided in https://arxiv.org/abs/1505.03669 [ https://arxiv.org/abs/1505.03669] a complete classification of bounded aut omorphisms in various fields equipped with operators\, among others\, for generic difference fields in all characteristics or for Hasse-Schmidt diff erential fields in positive characteristic. X-ALT-DESC: Lascar showed that the group of automorphisms of the complex fi eld which fix the algebraic closure of the prime field is simple. For this \, he first showed that there are no non-trivial bounded automorphisms. An automorphism is bounded if there is a finite set A such that the image of every element b is algebraic over A together with b. The same result hold s for a "\;universal"\; differentially closed field of characteris tic zero\, where we replace algebraic by differentially algebraic. Togethe r with T. Blossier and C. Hardouin\, we provided in https://arxiv.org/abs/1505.03669 a complete class ification of bounded automorphisms in various fields equipped with operato rs\, among others\, for generic difference fields in all characteristics o r for Hasse-Schmidt differential fields in positive characteristic. DTEND;TZID=Europe/Zurich:20181122T151500 END:VEVENT BEGIN:VEVENT UID:news319@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20181108T141500 SUMMARY:Number Theory Seminar: David Masser (Univ. of Basel) DESCRIPTION:Inspired by Schanuel's Conjecture\, Boris Zilber has proposed a ``Nullstellensatz'' (also conjectural) asserting which sorts of polynomia l-exponential equations in several variables have a complex solution. Last year Dale Brownawell and I published a proof in the situation which can b e regarded as ``typical''. But it does not cover all situations for two va riables\, some of which involve simply stated problems in one variable lik e finding complex $z \\neq 0$ with $e^z+e^{1/z}=1$. Recently Vincenzo Mant ova and I have settled the general case of two variables. We describe our methods -- for example\, to solve $$e^z+e^{\\root 9 \\of {1-z^9}}=1$$ one approach uses theta functions on ${\\bf C}^{28}$. X-ALT-DESC: Inspired by Schanuel's Conjecture\, Boris Zilber has proposed a ``Nullstellensatz'' (also conjectural) asserting which sorts of polynomia l-exponential equations in several variables have a complex solution. Last year Dale Brownawell and I published a proof in the situation which can b e regarded as ``typical''. But it does not cover all situations for two va riables\, some of which involve simply stated problems in one variable lik e finding complex $z \\neq 0$ with $e^z+e^{1/z}=1$. Recently Vincenzo Mant ova and I have settled the general case of two variables. We describe our methods -- for example\, to solve

$$e^z+e^{\\root 9 \\of {1-z^9}}=1$ $

one approach uses theta functions on ${\\bf C}^{28}$. DTEND;TZID=Europe/Zurich:20181108T151500 END:VEVENT BEGIN:VEVENT UID:news318@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20181101T141500 SUMMARY:Number Theory Seminar: Shabnam Akhtari (Univ. of Oregon / MPIM Bonn ) DTEND;TZID=Europe/Zurich:20181101T151500 END:VEVENT BEGIN:VEVENT UID:news317@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20181004T141500 SUMMARY:Number Theory Seminar: Dijana Kreso (TU Graz) DESCRIPTION:In my talk I will present results that come from a joint work w ith M. Bennett and A. Gherga from The University of British Columbia. We s tudied Goormaghtigh's equation:\\begin{equation}\\label{eq}\\frac{x^m-1}{x -1} = \\frac{y^n-1}{y-1}\, \\\; \\\; y>x>1\, \\\; m > n > 2. \\end{equatio n}There are two known solutions $(x\, y\,m\, n)=(2\, 5\, 5\, 3)\, (2\, 90\ , 13\, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitely or infinitely many solutions\, and no t even if that is the case if we fix one of the variables. It is known tha t there are finitely many solutions if we fix any two variables. Moreover\ , there are effective results in all cases\, except when the two fixed var iables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionall y satisfy $\\gcd(m-1\, n-1)>1$\, then there is an effective finiteness r esult. My co-authors and me showed that if $n \\geq 3$ is a fixed integer\ , then there exists an effectively computable constant $c (n)$ such that $ \\max \\{ x\, y\, m \\} < c (n)$ for all $x\, y$ and $m$ that satisfy Goor maghtigh's equation with $\\gcd(m-1\,n-1)>1$. In case $n \\in \\{ 3\, 4\ , 5 \\}$\, we solved the equation completely\, subject to this non-coprima lity condition. X-ALT-DESC:In my talk I will present results that come from a joint work wi th M. Bennett and A. Gherga from The University of British Columbia. We st udied Goormaghtigh's equation:

\\begin{equation}\\label{eq}

\\fr ac{x^m-1}{x-1} = \\frac{y^n-1}{y-1}\, \\\; \\\; y>\;x>\;1\, \\\; m > \; n >\; 2.

\\end{equation}

There are two known solutions $(x \, y\,m\, n)=(2\, 5\, 5\, 3)\, (2\, 90\, 13\, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitel y or infinitely many solutions\, and not even if that is the case if we fi x one of the variables. It is known that there are finitely many solutions if we fix any two variables. Moreover\, there are effective results in al l cases\, except when the two fixed variables are the exponents $m$ and $n $. If the fixed $m$ and $n$ additionally satisfy \; $\\gcd(m-1\, n-1)& gt\;1$\, then there is an effective finiteness result. My co-authors and m e showed that if $n \\geq 3$ is a fixed integer\, then there exists an eff ectively computable constant $c (n)$ such that $\\max \\{ x\, y\, m \\} &l t\; c (n)$ for all $x\, y$ and $m$ that satisfy Goormaghtigh's equation wi th $\\gcd(m-1\,n-1)>\;1$. \; In case $n \\in \\{ 3\, 4\, 5 \\}$\, we solved the equation completely\, subject to this non-coprimality conditio n. DTEND;TZID=Europe/Zurich:20181004T151500 END:VEVENT BEGIN:VEVENT UID:news316@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;TZID=Europe/Zurich:20180927T141500 SUMMARY:Number Theory Seminar: David Belius (Univ. of Basel) DESCRIPTION:

I will describe how the Riemann Zeta function on t he critical line can be viewed as a pseudo-random Gaussian field with a co rrelation function with logarithmic growth. Such log-correlated random fie lds have recently attracted considerable interest in probability theory. F yodorv\, Hiary and Keating conjectured several striking results about the extreme values of the Riemann Zeta function based on this connection. In t his talk I will explain how a certain approximate tree structure in Dirich let polynomials can be used to prove one of their conjectures\, giving the asymptotics of the maximum of the magnitude of the function in a typical interval of length O(1).X-ALT-DESC: <\;pre wrap="\;"\;>\;I will describe how the Rieman n Zeta function on the critical line can be viewed as a pseudo-random Gaus sian field with a correlation function with logarithmic growth. Such log-c orrelated random fields have recently attracted considerable interest in p robability theory. Fyodorv\, Hiary and Keating conjectured several strikin g results about the extreme values of the Riemann Zeta function based on t his connection. In this talk I will explain how a certain approximate tree structure in Dirichlet polynomials can be used to prove one of their conj ectures\, giving the asymptotics of the maximum of the magnitude of the fu nction in a typical interval of length O(1).<\;/pre>\; DTEND;TZID=Europe/Zurich:20180927T151500 END:VEVENT BEGIN:VEVENT UID:news209@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20211104T143247 DTSTART;VALUE=DATE:20180713 SUMMARY:Donau–Rhein Modelltheorie und Anwendungen\, 3rd meeting DESCRIPTION:Link to schedule. [https://sites.google.com/site/drmta3/] X-ALT-DESC:Link to sch edule. \; END:VEVENT END:VCALENDAR