BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jakub Konieczny (Hebrew University of Jerusalem) DTSTART:20200525T111500Z DTEND:20200525T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/1 DESCRIPTION:Title: Automatic multiplicative sequences\nby Jakub Konieczny (Hebrew University of Jerusalem) as part of Warsaw Number Theory Seminar\n\nLectu re held in room 403 at IMPAN.\n\nAbstract\nAutomatic sequences - that is\, sequences computable by finite automata - give rise to one of the most ba sic models of computation. As such\, for any class of sequences it is natu ral to ask which sequences in it are automatic. In particular\, the questi on of classifying automatic multiplicative sequences has attracted conside rable attention in the recent years. In the completely multiplicative case \, such classification was obtained independently by S. Li and O. Klurman and P. Kurlberg. The main topic of my talk will be the resolution of the g eneral case\, obtained in a recent preprint with M. Lemańczyk and C. Mül lner.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Alex Youcis (IMPAN Warsaw) DTSTART:20200601T111500Z DTEND:20200601T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/2 DESCRIPTION:Title: An approach to the characterization of the $p$-adic local Langland s correspondence\nby Alex Youcis (IMPAN Warsaw) as part of Warsaw Numb er Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn 20 13 P. Scholze provided an alternative proof of the local Langlands corresp ondence (LLC) for GLn and\, in doing so\, Scholze gave a new characterizat ion of the LLC via a certain trace identity. In this talk the speaker will discuss joint work with A. Bertoloni Meli showing that a generalization o f this trace identity characterizes the LLC for much more general groups i f one assumes standard expected properties of such a correspondence.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Szumowicz (Sorbonne Université\, IMJ-PRG) DTSTART:20200608T111500Z DTEND:20200608T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/3 DESCRIPTION:Title: Equidistribution in number fields\nby Anna Szumowicz (Sorbonne Université\, IMJ-PRG) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nThe notion of $\\mathfrak{p}$-ord ering comes from the work of Bhargava on integer-valued polynomials. Let $ k$ be a number field and let $\\mathcal{O}_{k}$ be its ring of integers. A sequence of elements in $\\mathcal{O}_{k}$ is a simultaneous $\\mathfrak{ p}$-ordering if it is equidistributed modulo every prime ideal in $\\mat hcal{O}_{k}$ as well as possible. We prove that $\\mathbb{Q}$ the only num ber field $k$ such that $\\mathcal{O}_{k}$ admits a simultaneous $\\mathfr ak{p}$-ordering. It is a joint work with Mikołaj Frączyk.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Jolanta Marzec (TU Darmstadt) DTSTART:20200622T111500Z DTEND:20200622T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/4 DESCRIPTION:Title: Maass relations for Saito-Kurokawa lifts of higher levels\nby Jolanta Marzec (TU Darmstadt) as part of Warsaw Number Theory Seminar\n\nL ecture held in room 403 at IMPAN.\n\nAbstract\nIt is known that a Siegel m odular form is a (classical) Saito-Kurokawa\nlift of an elliptic modular f orm if and only if its Fourier coefficients\nsatisfy the so-called Maass r elations. The first construction of such a\nlift was given by Maass using correspondences between various modular\nforms. However\, in order to gene ralize this lift to higher levels it\nis easier to use a construction comi ng from representation theory. During\nthe talk we present history of thi s problem and briefly discuss the\naforementioned constructions. We also i ndicate how one can read off the\nMaass relations from the latter. This wo rk generalizes an approach\nof Pitale\, Saha and Schmidt from the classica l to a higher level case.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Adam Keilthy (University of Oxford) DTSTART:20200629T111500Z DTEND:20200629T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/5 DESCRIPTION:Title: The block filtration and motivic multiple zeta values\nby Adam Keilthy (University of Oxford) as part of Warsaw Number Theory Seminar\n\ nLecture held in room 403 at IMPAN.\n\nAbstract\nMultiple zeta values are a class of transcendental numbers\, going back to Euler in the 1700s\nand with ties to the Riemann zeta function. They are found arising naturally i n many areas of mathematics and physics\, from algebraic geometry to Feynm an amplitudes. Unlike in the case of single zeta values\, we know many alg ebraic relations satisfied by multiple zeta values: the double shuffle rel ations\, the associator relations\, the confluence relations. However it i s unknown if any of these sets of relations are complete. Assuming Grothen dieck's period conjecture\, a complete set of algebraic relations are give n by the motivic relations\, arising from a connection to P^1 minus three points. However these relations are inexplicit.\n\nIn this talk\, we intro duce a new filtration\, called the block filtration\, on the space of mult iple zeta values. By considering the associated graded\, we describe sever al new families of motivic relations\, that provide a complete description of relations in low block degree. A generalisation of these results would thus provide a complete description of relations among multiple zeta valu es and aid in settling several open problems about the motivic Galois grou p of mixed Tate motives.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Dorota Blinkiewicz (Adam Mickiewicz University\, Poznań) DTSTART:20201005T111500Z DTEND:20201005T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/6 DESCRIPTION:Title: Classes of extensions of commutative algebraic groups\nby Doro ta Blinkiewicz (Adam Mickiewicz University\, Poznań) as part of Warsaw Nu mber Theory Seminar\n\n\nAbstract\nDuring the lecture I give explicit char acterization of $n$-torsion elements in the group of extensions of commuta tive\, smooth algebraic groups.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Vesselin Dimitrov (University of Toronto) DTSTART:20201012T111500Z DTEND:20201012T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/7 DESCRIPTION:Title: Solution of the conjecture of Schinzel and Zassenhaus and some app lications\nby Vesselin Dimitrov (University of Toronto) as part of War saw Number Theory Seminar\n\n\nAbstract\nWe will detail the full proof of an explicit form of the Schinzel-Zassenhaus conjecture: an algebraic integ er of degree $n > 1$ is either a root of unity or else has at least one co njugate of modulus exceeding $2^{1/(4n)}$. We furthermore obtain an extens ion of the original conjecture over to the setting of holonomic functions\ , with an application to the smallest critical value for (certain) rationa l functions.\n\nIn another application\, we would like to take the occasio n to raise the apparently unsolved problem of the essential irreducibility (up-to the cyclotomic factor $X^2-X+1$ in degrees a multiple of 12) of $X ^{2g} - X^g(1+X+1/X) + 1$\, the characteristic polynomial of the integer r eciprocal Perron-Frobenius matrix of the smallest spectral radius in each given dimension. Our explicit Schinzel-Zassenhaus bound allows for at most $10$ factors of each of these polynomials.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Jakub Byszewski (Jagellonian University) DTSTART:20201019T111500Z DTEND:20201019T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/8 DESCRIPTION:Title: Finite order automorphisms of the ring of power series over a fini te field\nby Jakub Byszewski (Jagellonian University) as part of Warsa w Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\ nThe Nottingham group at a prime $p$ is the group of (formal) power series $t+a_2 t^2+ a_3 t^3+ \\cdots$ in the variable $t$ with coefficients $a_i$ from the field with $p$ elements with the group operation given by compos ition of power series. This group is known to contain elements of order be ing an arbitrary power of $p$. Elements of order $p$ have been classified by Klopsch and have a nice description. For higher orders\, however\, only a handful of examples have been known explicitly.\n\nIn the talk we will show how to describe such series in closed computational form through fini te automata. This allows us to construct many explicit examples and formul ate a number of questions. The talk is based on joint work with Gunther Co rnelissen and Djurre Tijsma.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Danylo Radchenko (ETH Zürich) DTSTART:20201026T121500Z DTEND:20201026T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/9 DESCRIPTION:Title: Universal optimality of the E8 and Leech lattices\nby Danylo R adchenko (ETH Zürich) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nWe look at the problem of arrangin g points in Euclidean space in order to minimize the potential energy of p airwise interactions. We show that the E8 lattice and the Leech lattice ar e universally optimal in the sense that they have the lowest energy for al l potentials that are given by completely monotone potentials of squared d istance. \nThe proof uses a new kind of interpolation formula for Fourier eigenfunctions\, which is intimately related to the theory of modular form s.\nThe talk is based on a joint work with Henry Cohn\, Abhinav Kumar\, St ephen D. Miller\, and Maryna Viazovska.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Julie Desjardins (University of Toronto) DTSTART:20201102T121500Z DTEND:20201102T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/10 DESCRIPTION:Title: Density of rational points on a family of del Pezzo surface of de gree 1\nby Julie Desjardins (University of Toronto) as part of Warsaw Number Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nL et k be a number field and X an algebraic variety over k. We want to study the set of k-rational points X(k). For example\, is X(k) empty? If not\, is it dense with respect to the Zariski topology? Del Pezzo surfaces are c lassified by their degrees d (an integer between 1 and 9). Manin and vario us authors proved that for all del Pezzo surfaces of degree >1 it is dense provided that the surface has a k-rational point (that lies outside a spe cific subset of the surface for d=2). For d=1\, the del Pezzo surface alwa ys has a rational point. However\, we don't know if the set of rational po ints is Zariski-dense. In this talk\, I present a result that is joint wit h Rosa Winter in which we prove the density of rational points for a speci fic family of del Pezzo surfaces of degree 1 over k.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Masha Vlasenko (IMPAN) DTSTART:20201109T121500Z DTEND:20201109T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/11 DESCRIPTION:Title: Dwork crystals\nby Masha Vlasenko (IMPAN) as part of Warsaw N umber Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn his work on rationality of zeta functions of algebraic varieties Bernard Dwork discovered a number of remarkable p-adic congruences. In this talk I will demonstrate some of these congruences and overview our recent work w ith Frits Beukers which explains their underlying mechanism.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Matija Kazalicki (University of Zagreb) DTSTART:20201116T121500Z DTEND:20201116T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/12 DESCRIPTION:Title: Congruences for sporadic sequences\, three fold covers of the ell iptic modular surfaces and modular forms for non-congruence subgroups\ nby Matija Kazalicki (University of Zagreb) as part of Warsaw Number Theor y Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nIn 1979\, in the course of the proof of the irrationality of $\\zeta(2)$\n Ap\\'ery int roduced numbers $b_n = \\sum_{k=0}^n {n \\choose k}^2{n+k\n\\choose k}$ th at are\, surprisingly\, integral solutions of recursive\nrelations $$(n+1 )^2 u_{n+1} - (11n^2+11n+3)u_n-n^2u_{n-1} = 0.$$\nZagier performed a compu ter search on first 100 million triples\n$(A\,B\,C)\\in \\mathbb{Z}^3$ and found that the recursive relation\ngeneralizing $b_n$\n$$(n+1)u_{n+1} - ( An^2+An+B)u_n + C n ^2 u_{n-1}=0\,$$\nwith the initial conditions $u_{-1}= 0$ and $u_0=1$ has (non-degenerate\ni.e. $C(A^2-4C)\\ne 0$) integral solut ion for only six more triples\n(whose solutions are so called sporadic seq uences) .\n\nStienstra and Beukers showed that the for prime $p\\ge 5$\n\\ begin{equation*}\nb_{(p-1)/2} \\equiv \\begin{cases} 4a^2-2p \\pmod{p} \\t extrm{ if } p =\na^2+b^2\,\\textrm{ a odd}\\\\ 0 \\pmod{p} \\textrm{ if } p\\equiv 3\n\\pmod{4}.\\end{cases}\n\\end{equation*}\n\nRecently\, Osburn and Straub proved similar congruences for all but one\nof the six Zagier's sporadic sequences (three cases were already known\nto be true by the wor k of Stienstra and Beukers) and we proved the\ncongruence for the sixth se quence.\n\nIn this talk we describe congruences for the Ap\\'ery numbers\n $b_{(p-1)/3}$ (and also for the other sporadic sequences).\nFor that we st udy Atkin and Swinnerton-Dyer type of congruences\nbetween Fourier coeffic ients of cusp forms for non-congruence\nsubgroups\, $L$-functions of three covers of elliptic modular surfaces\nand Galois representations attached to these covers.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Wadim Zudilin (Radboud University\, Nijmegen) DTSTART:20201123T121500Z DTEND:20201123T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/13 DESCRIPTION:Title: Dwork-type ($q$-)(super)congruences\nby Wadim Zudilin (Radbou d University\, Nijmegen) as part of Warsaw Number Theory Seminar\n\nLectur e held in room 403 at IMPAN.\n\nAbstract\nThe "microscope" principle is th is: If a rational function A(q) of variable q vanishes at every p-th root of unity (for p prime)\, then A(q) == 0 modulo Φ_p(q)\, the p-th cyclotom ic polynomial\; assuming that A(1) is a well-defined rational number with a p-free denominator and specialising the congruence at q=1 we conclude wi th A(1) == 0 modulo p.\nIn other words\, behaviour of rational functions a t p-th roots of unity may be instructive for gaining information about the ir values at 1 modulo p. With some "creative" extras\, we can further cons ider divisibility by higher powers of primes (and we can even deal with no t necessarily primes).\nIn my talk\, partly based on recent joint work wit h Victor Guo\, I plan to highlight some novel outcomes of this "creative m icroscope" methodology -- examples of Dwork-type supercongruences for trun cated hypergeometric sums.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Osburn (University College Dublin) DTSTART:20201130T121500Z DTEND:20201130T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/14 DESCRIPTION:Title: Generalized Fishburn numbers\, torus knots and quantum modularity \nby Robert Osburn (University College Dublin) as part of Warsaw Numbe r Theory Seminar\n\nLecture held in room 403 at IMPAN.\n\nAbstract\nThe Fi shburn numbers are a sequence of positive integers with numerous combinato rial interpretations and interesting asymptotic properties. In 2016\, Andr ews and Sellers initiated the study of arithmetic properties of these numb ers. In this talk\, we discuss a generalization of this sequence using kno t theory and the quantum modularity of the associated Kontsevich-Zagier se ries.\n\nThe first part is joint work with Colin Bijaoui (McMaster)\, Hans Boden (McMaster)\, Beckham Myers (Harvard)\, Will Rushworth (McMaster)\, Aaron Tronsgard (Toronto) and Shaoyang Zhou (Vanderbilt) while the second part is joint work with Ankush Goswami (RISC).\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Gunther Cornelissen (Utrecht University) DTSTART:20201207T121500Z DTEND:20201207T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/15 DESCRIPTION:Title: An analogy between number theory and spectral geometry\nby Gu nther Cornelissen (Utrecht University) as part of Warsaw Number Theory Sem inar\n\n\nAbstract\nSunada’s construction of non-isometric\, isospectral manifolds proceeds in the same way as Gassmann’s construction of non-is omorphic number fields with the same zeta function\, using a group G with two non-conjugate subgroups H and K such that the permutation representati ons given by G acting on their cosets are isomorphic. In Gassmann’s exam ple\, G was the permutation group on 6 letters and H and K the groups gene rated by (12)(34) and (13)(24)\, and (12)(34) and (12)(56)\, respectively. These can be realized as covering groups of a compact Riemann surface of genus 2. Recently\, the speaker and collaborators showed that isomophism o f number fields can be detected by equality of suitable L-series. This tal k is about the finding the analogous result for manifolds. The result says that if two manifolds are finite Riemannian covers of a developable orbif old\, and such that a certain homological condition is satisfied\, then th e manifolds are isometric if and only if the spectra of finitely many Lapl acians twisted by suitable unitary representations of the fundamental grou p are equal. The result is explicit: in the above example\, one needs 56 s pectral equalities corresponding to 180-dimensional representations. (Join t work with Norbert Peyerimhoff.)\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Tim Dokchitser (University of Bristol) DTSTART:20201214T121500Z DTEND:20201214T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/16 DESCRIPTION:Title: Curves with tame torsion\nby Tim Dokchitser (University of Br istol) as part of Warsaw Number Theory Seminar\n\nLecture held in currentl y online.\n\nAbstract\nIn this talk I will sketch the proof of the fact th at in every\ngenus and for every prime p there are curves over Q with tame p-torsion.\nIn genus 1\, this is something this is quite easy to deduce f rom the\ntheory of the Tate curve\, and I will explain how an explicit ver sion of\nthe theory of hyperelliptic Mumford curves gives this in arbitrar y\ngenus. This is joint work with Matthew Bisatt.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Bidisha Roy (IMPAN\, Warsaw) DTSTART:20201221T121500Z DTEND:20201221T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/17 DESCRIPTION:Title: Torsion groups of elliptic curves over number fields\nby Bidi sha Roy (IMPAN\, Warsaw) as part of Warsaw Number Theory Seminar\n\nLectur e held in currently online.\n\nAbstract\nComputing torsion groups of ellip tic curves defined over number fields is a classical topic and it has a va st literature in algebraic number theory. Any elliptic curve of the form y ^2 = x^3 + c is called a Mordell curve. Mordell curves are well studied e lliptic curves with complex multiplication.\n\nIn this talk\, for Mordell curves defined over the field of rational numbers we will discuss the clas sification of torsion groups over cubic and sextic fields. Also\, we prese nt the classification of torsion groups of Mordell curves defined over cub ic fields. For Mordell curves over sextic fields\, we provide all possible torsion groups. In the second part\, we briefly discuss torsion groups of Mordell curves over higher degree number fields.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Wojtek Wawrów (London School of Geometry and Number Theory) DTSTART:20210111T121500Z DTEND:20210111T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/18 DESCRIPTION:Title: Ranks of Jacobians over rational numbers\nby Wojtek Wawrów ( London School of Geometry and Number Theory) as part of Warsaw Number Theo ry Seminar\n\nLecture held in currently online.\n\nAbstract\nThe problem o f finding out the possible Mordell-Weil ranks of abelian varieties of give n dimension over a fixed number field has a long history. In this talk we shall survey various results which have appeared over the years bounding f rom below the maximal rank that those varieties can have\, with particular emphasis on Jacobians of special families of curves over the field of rat ional numbers.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Netan Dogra (King's College London) DTSTART:20210118T121500Z DTEND:20210118T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/19 DESCRIPTION:Title: Some new results in the nonabelian Chabauty method\nby Netan Dogra (King's College London) as part of Warsaw Number Theory Seminar\n\nL ecture held in currently online.\n\nAbstract\nIn this talk I will discuss the nonabelian Chabauty method\, which seeks to use p-adic analytic functi ons to determine the finite sets of rational points on higher genus curves \, and some new cases where it can be used to determine the solutions to D iophantine equations.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Oleksiy Klurman (University of Bristol) DTSTART:20210125T121500Z DTEND:20210125T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/20 DESCRIPTION:Title: Monotone chains of Hecke cusp forms\nby Oleksiy Klurman (Univ ersity of Bristol) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nWe discuss a general joint equidistribu tion result for the Fourier coefficients of Hecke cusp forms. One simple c onsequence of such a result is that there exist infinitely many integers n (in fact an upper density of this set is positive) such that \n$\\tau (n) <\\tau (n+1)<\\tau(n+2)$ where $\\tau$ is a Ramanujan $\\tau$-function. Th is is based on a joint work with A. Mangerel (CRM).\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Mikołaj Frączyk (University of Chicago) DTSTART:20210301T160000Z DTEND:20210301T170000Z DTSTAMP:20260404T111320Z UID:WarsawNT/22 DESCRIPTION:Title: Sarnak’s density hypothesis in horizontal families\nby Miko łaj Frączyk (University of Chicago) as part of Warsaw Number Theory Semi nar\n\nLecture held in currently online.\n\nAbstract\nLet $G$ be a real se mi simple Lie group with an irreducible unitary representation $\\pi$. The non-temperedness of $\\pi$ is measured by the real parameter $p(\\pi)$ wh ich is defined as the infimum of $p$ such that $\\pi$ has non-zero matrix coefficients in $L^p(G)$. Sarnak and Xue conjectured that for any arithmet ic lattice $\\Gamma\\subset G$ and principal congruence subgroup $\\Gamma( q)\\subset \\Gamma$\, the multiplicity of $\\pi$ in $L^2(G/\\Gamma(q))$ is at most $O(V(q)^{2/p(\\pi) +\\varepsilon})$\, where $V(q)$ is the covolum e of $\\Gamma(q)$. Sarnak and Xue proved this conjecture for $G=SL(2\,\\ma thbb R)\,SL(2\,\\mathbb C)$. I will talk about the joint work with Gergely Harcos\, Peter Maga and Djordje Milicevic where we prove bounds of the sa me quality that hold uniformly for families of pairwise non-commensurable lattices in $G=SL(2\,\\mathbb R)^a\\times SL(2\,\\mathbb C)^b$. These fami lies of lattices\, which we call horizontal\, are given as unit groups of maximal orders of quaternion algebras over number fields.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Sudhir Pujahari (University of Warsaw) DTSTART:20210308T121500Z DTEND:20210308T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/23 DESCRIPTION:Title: Arithmetic and statistics of sums of eigenvalues of Hecke operato rs\nby Sudhir Pujahari (University of Warsaw) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn the f irst part of the talk\, we will study about the distribution of gaps betwe en eigenvalues of Hecke operators in both horizontal and vertical settings . As an application of this we will obtain a strong multiplicity one theor em and evidence towards Maeda conjecture. The horizontal setting is a join t work with M. Ram Murty. In the second part of the talk\, using recent de velopments in the theory of l-adic Galois representations we will study th e normal number of prime factors\nof sums of Fourier coefficients of eigen forms. Moreover\, we will see the distribution of distinct prime factors o f sums of Fourier coefficients of eigenforms. The final part is a joint wo rk with M. Ram Murty and V. Kumar Murty.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Carlo Verschoor (University of Utrecht) DTSTART:20210315T121500Z DTEND:20210315T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/24 DESCRIPTION:Title: Bailey Type Factorizations of Horn Functions\nby Carlo Versch oor (University of Utrecht) as part of Warsaw Number Theory Seminar\n\nLec ture held in currently online.\n\nAbstract\nA well-known identity by Baile y states that Appell’s F4 function can be written as the product of two Gauss hypergeometric functions under a suitable specialization of its para meters. Other identities of this type are known for Appell’s F2 and F3\, which are closely related to Bailey’s identity. The aim of this talk is to show that the same can be done for Horn’s H1\, H4 and H5 functions.\ n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Enis Kaya (University of Groningen) DTSTART:20210322T121500Z DTEND:20210322T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/25 DESCRIPTION:Title: Explicit Vologodsky Integration for Hyperelliptic Curves\nby Enis Kaya (University of Groningen) as part of Warsaw Number Theory Semina r\n\nLecture held in currently online.\n\nAbstract\nLet $X$ be a curve ove r a $p$-adic field with semi-stable reduction and let $\\omega$ be a merom orphic $1$-form on $X$. There are two notions of $p$-adic integration one may associate to this data: the Berkovich–Coleman integral which can be performed locally\; and the Vologodsky integral with desirable number-theo retic properties. In this talk\, we present a theorem comparing the two\, and describe algorithms for computing these integrals in the case that $X$ is a hyperelliptic curve. We also illustrate our algorithm with a numeric al example computed in Sage. This talk is partly based on joint work with Eric Katz.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Boris Adamczewski (Institut Camille Jordan & CNRS) DTSTART:20210329T111500Z DTEND:20210329T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/26 DESCRIPTION:Title: Furstenberg's conjecture\, Mahler's method\, and finite automata< /a>\nby Boris Adamczewski (Institut Camille Jordan & CNRS) as part of Wars aw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\ nIt is commonly expected that expansions of numbers in multiplicatively in dependent bases\, such as 2 and 10\, should have no common structure. Howe ver\, it seems extraordinarily difficult to confirm this naive heuristic p rinciple in some way or another. In the late 1960s\, Furstenberg suggested a series of conjectures\, which became famous and aim to capture this heu ristic. The work I will discuss in this talk is motivated by one of these conjectures. Despite recent remarkable progress by Shmerkin and Wu\, it re mains totally out of reach of the current methods. While Furstenberg’s c onjectures take place in a dynamical setting\, I will use instead the lang uage of automata theory to formulate some related problems that formalize and express in a different way the same general heuristic. I will explain how the latter can be solved thanks to some recent advances in Mahler’s method\; a method in transcendental number theory initiated by Mahler at t he end of the 1920s. This a joint work with Colin Faverjon.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Anna Szumowicz (Caltech) DTSTART:20210412T140000Z DTEND:20210412T150000Z DTSTAMP:20260404T111320Z UID:WarsawNT/27 DESCRIPTION:Title: Cuspidal types on GL_p(O)\nby Anna Szumowicz (Caltech) as par t of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\n Abstract\nLet $F$ be a non-Archimedean local field and let $O$ be its ring of\nintegers. We describe the cupidal types on $\\mathrm{GL}_p(O)$ (where $p$ is a prime\nnumber) using Clifford theory. This gives some informatio n and\ninvariants attached to cuspidal types called orbits. We give an\nex ample which shows that the orbit of a representation does not give\nenough information to determine whether a representation is a cuspidal\ntype or not.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Rusen Li (Shandong University) DTSTART:20210419T111500Z DTEND:20210419T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/28 DESCRIPTION:Title: Summation formulas of q-hyperharmonic numbers\nby Rusen Li (S handong University) as part of Warsaw Number Theory Seminar\n\nLecture hel d in currently online.\n\nAbstract\nIn 1990\, Spieß gave some identities including the types of $\\sum_{\\ell=1}^n\\ell^k H_\\ell$\, $\\sum_{\\ell= 1}^n\\ell^k H_{n-\\ell}$ and $\\sum_{\\ell=1}^n\\ell^k H_\\ell H_{n-\\ell} $. In this talk\, based upon a certain type of $q$-harmonic numbers $H_n^{ (r)}(q)$\, several formulas of $q$-hyperharmonic numbers are derived as $q $-generalizations. The main tools used in the talk are Abel’s identity a nd a q-version of the relation by Spieß.\nThis is based on a joint work w ith Takao Komatsu.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Valentijn Karemaker (Universiteit Utrecht) DTSTART:20210426T111500Z DTEND:20210426T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/29 DESCRIPTION:Title: Polarisations of abelian varieties over finite fields via canonic al liftings\nby Valentijn Karemaker (Universiteit Utrecht) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstr act\nIn this talk we will give a widely applicable and computable descript ion of polarisations of abelian varieties over finite field. More precisel y\, we will describe all polarisations of all abelian varieties over a fin ite field in a fixed isogeny class corresponding to a squarefree Weil poly nomial\, when one variety in the isogeny class admits a canonical lifting to characteristic zero. The computability of the description relies on app lying categorical equivalences between abelian varieties over finite field s and fractional ideals in étale algebras. \nThis is joint work with Jona s Bergström and Stefano Marseglia.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Robert Slob (Ulm University) DTSTART:20210510T111500Z DTEND:20210510T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/30 DESCRIPTION:Title: Primitive divisors of sequences associated to elliptic curves ove r function fields\nby Robert Slob (Ulm University) as part of Warsaw N umber Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nIn the first part of the talk\, we give a gentle introduction into the subjec t of divisibility sequences over the rational numbers and discuss the noti on of a primitive divisor/Zsigmondy bound. We then explain how these notio ns can be extended to number fields and function fields\, and how to obtai n a divisibility sequence from a non-torsion point on an elliptic curve ov er any of these fields. There will also be plenty of nice examples.\n\nIn the second part of the talk\, we discuss the typical methods that are used to prove the existence of a Zsigmondy bound for a divisibility sequence o btained from a non-torsion point on an elliptic curve $E$ over a number or function field $K$. Let $P$ be this non-torsion point in $E(K)$\, and sup pose Q is a torsion point in $E(K)$. We can also associate a sequence of d ivisors $\\{D_{nP+Q}\\}$ on $K$ to the sequence of points $\\{nP+Q\\}$. In my preprint\, we proved the existence of a Zsigmondy bound for this seque nce $\\{D_{nP+Q}\\}$ for $K$ a function field (under some minor conditions )\, extending the analogous result of Verzobio over number fields. I will provide the crucial ideas to apply the existing methods of the case $\\{nP \\}$ to my case $\\{nP+Q\\}$. Additionally\, I will highlight the differen ces with the number field case.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Marco Streng (Universiteit Leiden) DTSTART:20210517T111500Z DTEND:20210517T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/31 DESCRIPTION:Title: Obtaining modular units via a recurrence relation\nby Marco S treng (Universiteit Leiden) as part of Warsaw Number Theory Seminar\n\nLec ture held in currently online.\n\nAbstract\nThe modular curve $Y^1(N)$ par ametrises pairs $(E\,P)$\, where $E$ is an elliptic curve and $P$ is a poi nt of order $N$ on $E$. One tool for studying this curve is the group of m odular units on it\, that is\, the group of algebraic functions with no po les or zeroes.\n\nWe first review how a recurrence relation (related to el liptic divisibility sequences) gives rise to defining equations for the cu rves $Y^1(N)$. We then show that the same recurrence relation also gives e xplicit algebraic formulae for a basis of the group of units on $Y^1(N)$.\ n\nThis proves a conjecture of Maarten Derickx and Mark van Hoeij.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Eric Delaygue (University Lyon 1) DTSTART:20210524T111500Z DTEND:20210524T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/32 DESCRIPTION:Title: On primary pseudo-polynomials and around Ruzsa's Conjecture\n by Eric Delaygue (University Lyon 1) as part of Warsaw Number Theory Semin ar\n\nLecture held in currently online.\n\nAbstract\nEvery polynomial $P(X )$ with integer coefficients satisfies the congruences $P(n+m)=P(n) \\mod m$ for all integers $n$ and $m$. An integer valued sequence is called a ps eudo-polynomial when it satisfies these congruences. Hall characterized ps eudo-polynomials and proved that they are not necessarily polynomials. A l ong standing conjecture of Ruzsa says that a pseudo-polynomial $a(n)$ is a polynomial as soon as $\\limsup |a_n|^{1/n} < e$. A primary pseudo-polyno mial is an integer valued sequence $a(n)$ such that $a(n+p)=a(n) \\mod p$ for all integers $n ≥ 0$ and all prime numbers $p$. The same conjecture has been formulated for them\, which implies Ruzsa’s\, and this talk wil l revolve around this conjecture.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Vargas-Montoya (University Lyon 1) DTSTART:20210531T111500Z DTEND:20210531T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/33 DESCRIPTION:Title: Algebraicity modulo $p$ of G functions\, hypergeometric series an d strong Frobenius structure\nby Daniel Vargas-Montoya (University Lyo n 1) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nB. Dwork in his work about zeta function of a hypersu rface over finite fields introduced the notion of strong Frobenius structu re. In this talk we are going to take up this notion for the study of alge braicity modulo $p$ of Siegel G functions\, where $p$ is a prime number. \nFirstly\, we are going to see that if $f(t)$ is a power series (or Siege l G function) with coefficients in the ring of integers $\\mathbb{Z}$ and if $f(t)$ is solution of a differential operator $L$ having strong Frobeni us structure for $p$ of period $h$\, then the reduction of $f$ modulo $p$ is algebraic over $\\mathbb{F}_p(t)$ and its algebraicity degree is bounde d by $p^{n^2h}$\, where $n$ is the order of L and $\\mathbb{F}_p$ is the f ield of $p$ elements. Secondly\, we are going to show that\, under reasona ble hypotheses\, rigid differential operators have a strong Frobenius str ucture for almost every prime number $p$.\nFinally\, we are going to illu strate our results with several examples coming of hypergeometric series o f type ${}_nF_n-1$.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Michael J. Schlosser (Unviersity of Vienna) DTSTART:20210607T140000Z DTEND:20210607T150000Z DTSTAMP:20260404T111320Z UID:WarsawNT/34 DESCRIPTION:Title: On the infinite Borwein product raised to a real power\nby Mi chael J. Schlosser (Unviersity of Vienna) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nWe study the $q$ -series coefficients appearing in the expansion of $\\prod_{n\\ge 1}[(1-q^ n)/(1-q^{pn})]^\\delta$\, the infinite Borwein product for an arbitrary pr ime $p$\, raised to an arbitrary positive real power $\\delta$. Applicatio n of the Hardy-Ramanujan-Rademacher circle method gives an asymptotic form ula for the coefficients. For $p=3$ we give an estimate of their growth wh ich enables us to partially confirm an earlier conjecture we made concerni ng an observed sign pattern of the coefficients when the exponent $\\delta $ is within a specified range of positive real numbers. We then take a clo ser look at the cube of the infinite Borwein product\, for arbitrary $p$ ( now a positive integer)\, and establish some vanishing and divisibility pr operties of the respective coefficients.\nThis is joint work with Nian Hon g Zhou.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Jason Bell (University of Waterloo) DTSTART:20210614T111500Z DTEND:20210614T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/35 DESCRIPTION:Title: Effective isotrivial Mordell-Lang in positive characteristic\ nby Jason Bell (University of Waterloo) as part of Warsaw Number Theory Se minar\n\nLecture held in currently online.\n\nAbstract\nThe Mordell-Lang c onjecture (now a theorem\, proved by Faltings\, Vojta\, McQuillan\, …) a sserts that if $G$ is a semiabelian variety $G$ defined over an algebraica lly closed field of characteristic zero\, $X$ is a subvariety of $G$\, and $\\Gamma$ is a finite rank subgroup of $G$\, then $X\\cap \\Gamma$ is a f inite union of cosets of $\\Gamma$. In positive characteristic\, the naiv e translation of this theorem does not hold\, however Hrushovski\, using m odel theoretic techniques\, showed that in some sense all counterexamples arise from semiabelian varieties defined over finite fields (the isotrivia l case). This was later refined by Moosa and Scanlon\, who showed in the isotrivial case that the intersection of a subvariety of a semiabelian var iety $G$ with a finitely generated subgroup $\\Gamma$ of $G$ that is invar iant under the Frobenius endomorphism $F:G\\to G$ is a finite union of set s of the form $S+A$\, where $A$ is a subgroup of $\\Gamma$ and $S$ is a su m of orbits under the map $F$. We show how how one can use finite-state automata to give a concrete description of these intersections $\\Gamma\\c ap X$ in the isotrivial setting\, by constructing a finite machine that id entifies all points in the intersection. In particular\, this allows us to give decision procedures for answering questions such as: is $X\\cap \\Ga mma$ empty? finite? does it contain a coset of an infinite subgroup? In ad dition\, we are able to read off coarse asymptotic estimates for the numbe r of points of height $\\le H$ in the intersection from the machine. This is joint work with Dragos Ghioca and Rahim Moosa.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Lars Kühne (University of Copenhagen) DTSTART:20210621T111500Z DTEND:20210621T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/36 DESCRIPTION:Title: Equidistribution and Uniformity in Families of Curves\nby Lar s Kühne (University of Copenhagen) as part of Warsaw Number Theory Semina r\n\nLecture held in currently online.\n\nAbstract\nIn the talk\, I will p resent an equidistribution result for fa milies of (non-degenerate) subvar ieties in a (general) family of abelian varieties. This extends a result o f DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result\, one can deduce a uniform version of the classical Bog omolov conjecture for curves embed ded in their Jacobians\, namely that th e number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in a few select cases b y work of David–Philippon and DeMarco–Krieger–Ye. Finally\, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a re sult of Dimitrov–Gao–Habegger: The number of ration al points on a smooth algebraic curve defined over a number field can be b ounded solely in terms of its genus and the Mordell-Weil rank of its Jacob ian. Again\, this was previously known only under additional assumptions ( Stoll\, Katz–Rabinoff–Zureick-Brown).\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Vasily Golyshev (Steklov Institute) DTSTART:20210628T111500Z DTEND:20210628T121500Z DTSTAMP:20260404T111320Z UID:WarsawNT/37 DESCRIPTION:Title: Modularity proofs via fibered motives.\nby Vasily Golyshev (S teklov Institute) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nI will explain how techniques of fibered hypergeometric motives can be used to provide `opportunistic' modularity proofs for conifold fibers in Calabi-Yau families. This is a report on joi nt work with Don Zagier\, and work in progress with Kilian Bönisch and Al brecht Klemm.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Maciej Ulas (Jagiellonian University\, Kraków) DTSTART:20220307T120000Z DTEND:20220307T130000Z DTSTAMP:20260404T111320Z UID:WarsawNT/38 DESCRIPTION:Title: On solutions of certain meta-Fibonacci recurrences\nby Maciej Ulas (Jagiellonian University\, Kraków) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n\nAbstract\nDuring the talk I will speak about recent findings concerning the solutions of the recurre nce sequence $h(n)=h(n-h(n-1))+h(n-2)$. This is a member of a class of so called meta-Fibonacci (or exotic) sequences. We show that for a broad clas s of initial conditions the behavior of the solutions is easy\, i.e.\, gov erned by sequences satisfying linear recurrence with constant coefficients or is closely related to certain functions counting binary partitions of special type. The talk is based on a joint work with Bartosz Sobolewski.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Neelam Saikia (University of Virginia) DTSTART:20220307T131500Z DTEND:20220307T141500Z DTSTAMP:20260404T111320Z UID:WarsawNT/39 DESCRIPTION:Title: Traces of pth Hecke operators and p-adic hypergeometric functions \nby Neelam Saikia (University of Virginia) as part of Warsaw Number T heory Seminar\n\nLecture held in currently online.\n\nAbstract\nMcCarthy d efined hypergeometric functions in the p-adic setting using p-adic gamma f unctions. This function can be described as p-adic analogue of classical h ypergeometric function. In this talk we discuss the traces of pth Hecke op erators acting on spaces of cusp forms of weight k and level 1 and their r elations with p-adic hypergeometric functions. As a consequence of this re sult we establish relations of Ramanujan’s tau-function and p-adic hyper geometric functions. This is a joint work with Sudhir Pujahari.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Pierre Colmez (Institut de Mathématiques de Jussieu-Paris Rive Ga uche) DTSTART:20220404T110000Z DTEND:20220404T120000Z DTSTAMP:20260404T111320Z UID:WarsawNT/40 DESCRIPTION:Title: The upper half-planes\nby Pierre Colmez (Institut de Mathéma tiques de Jussieu-Paris Rive Gauche) as part of Warsaw Number Theory Semin ar\n\nLecture held in IMPAN\, Warsaw\, conference room 6\, ground floor.\n \nAbstract\nWe will give a short introduction to the geometric part of the p-adic Langlands program.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Piotr Miska (Jagiellonian University) DTSTART:20220404T121500Z DTEND:20220404T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/41 DESCRIPTION:Title: (R)-dense and (N)-dense subsets of positive integers and generali zed quotient sets\nby Piotr Miska (Jagiellonian University) as part of Warsaw Number Theory Seminar\n\nLecture held in currently online (via Zoo m).\n\nAbstract\nA subset $A$ of the set of positive integers is (R)-dens e if its quotient set $ R(A)=\\{a/b \\colon a\, b \\in A\\}$ is dense in t he positive real half-line (with respect to natural topology on real numbe rs). It is a classical result that the set of prime numbers is (R)-dense. The proof of this fact is based on the property of counting function of pr ime numbers. Actually\, this proof shows something more. Namely\, for each infinite subset $B$ of the set of positive integers\, the set $R(P\,B)=\\ {p/b \\colon p \\in P\, b \\in B\\}$ is dense in the set of positive real numbers. This motivates to introduce the notion of (N)-denseness. We say t hat a set $A$ of positive integers is (N)-dense if the set $R(A\,B)$ is de nse in the set of positive real numbers for every set $B$ of positive inte gers. During the talk we will consider characterizations of (N)-dense sets and connections between (N)-denseness of a given set.\nIn 2019 Leonetti a nd Sanna introduced the notion of direction sets $D^k(A)=\\{(a_1/\\|a\\|^2 \, \\ldots\, a_k/\\|a\\|^2)\\colon a=(a_1\,\\ldots\, a_k) \\in A^k\\}$ tha t allows us to generalize the property of (R)-denseness. Indeed\, $A$ is ( R)-dense if and only if $D^2(A)$ is dense in the set of points of unit cir cle with all the coordinates positive. We will see that denseness of $D^k( A)$ in the set of points of unit sphere with all the coordinates positive is equivalent to denseness of the generalized quotient set $R^k(A)=\\{(a_1 /a_k\,\\ldots\, a_{k-1}/a_k)\\colon a_1\,\\ldots\, a_k \\in A\\}$ in the s et of points of $R^{k-1}$ with all the coordinates positive.\nWe will also show some connections between (N)-denseness of a given set $A$ and densen ess of sets $R^k(A)$ with the counting function of $A$ and its dispersion. \nThe talk is based on a joint work with János T. Tóth.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Gwladys Fernandes (Université de Versailles Saint-Quentin-en-Yvel ines) DTSTART:20220509T110000Z DTEND:20220509T120000Z DTSTAMP:20260404T111320Z UID:WarsawNT/42 DESCRIPTION:Title: Hypertranscendence of solutions of linear difference equations\nby Gwladys Fernandes (Université de Versailles Saint-Quentin-en-Yvelin es) as part of Warsaw Number Theory Seminar\n\nLecture held in currently o nline.\n\nAbstract\nThe general question of this talk is the classificatio n of differentially algebraic solutions of linear difference equations of the following type: \n\n$$ (\\ast) \\qquad a_0(z)f(z) + a_1(z)f(R(z)) + . .. + a_n(z)f(R_n(z))=0$$\n\nwhere\, for every $i$\, $a_i(z) \\in \\C(z)$\, $R(z) \\in \\C(z)$ and $R_n(z)$ is the $n$-th composition of $R(z)$ with itself. We say that such a function is differentially algebraic over $\\C( z)$ if there exist an non-zero integer $n$ and a non-zero polynomial $P \\ in \\C(z)[X_0\,...\, X_n]$ such that $P(f(z)\,...\, f^{(n)}(z))=0$\, where $f^{(i)}$ is the $i$-th derivative of $f$ with respect to $z$. Otherwise\ , it is hypertranscendental over $\\C(z)$.\n\nThe classification of differ entially algebraic solutions is known for three types of non-linear differ ence equations : the Schröder's\, Böttcher's and Abel's equations : $f(q z)=R(f(z))$\, $f(z^d)=R(f(z))$\, $f(R(z))=f(z)+1$\, respectively\, where $ q \\in \\C^{\\ast}$\, $d \\in \n$\, $d \\geq 2$. A classification of the d ifferential algebraicity of solutions of linear difference equations of th e above type $(\\ast)$ is made in an article of B. Adamczewski\, T. Dreyfu s\, C. Hardouin\, for these same operators : q-differences $z \\to qz$\, m ahlerian $z \\to z^d$\, and shift $z \\to z+1$\, by the means of an adapt ed difference Galois theory.\n\nIn this talk\, we discuss the generalisati on of these results to any function $R$ (rational or algebraic over $\\C(z )$)\, in the case where $(\\ast)$ is of order $1$. This is a work in progr ess with L. Di Vizio. Natural applications appear in examples of generatin g series of random walks\, which satisfy this kind of equation of order $1 $.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Matthias Storzer (MPIM Bonn) DTSTART:20220509T121500Z DTEND:20220509T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/43 DESCRIPTION:Title: Modularity of Nahm sums\nby Matthias Storzer (MPIM Bonn) as p art of Warsaw Number Theory Seminar\n\nLecture held in currently online.\n \nAbstract\nThe modularity properties of so-called Nahm sums are known to be related to certain elements in the Bloch group. Nevertheless\, a first conjecture about the characterisation of modular Nahm sums in terms of the se elements turned out to be false. In this talk\, we will review the moti vation behind the conjecture and discuss why it fails\, which could lead t o a refined version.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Eugenia Rosu (TU Darmstadt and University​ of Leiden) DTSTART:20220606T110000Z DTEND:20220606T120000Z DTSTAMP:20260404T111320Z UID:WarsawNT/44 DESCRIPTION:Title: Twists of elliptic curves with CM\nby Eugenia Rosu (TU Darmst adt and University​ of Leiden) as part of Warsaw Number Theory Seminar\n \nLecture held in IMPAN Warsaw\, ground floor\, room 6.\n\nAbstract\nWe co nsider certain families of sextic twists of the elliptic curve $y^2=x^3+1$ that are not defined over $\\mathbb{Q}$\, but over $\\mathbb{Q}[\\sqrt{-3 }]$. We compute a formula that relates the central value of their L-functi ons $L(E\, 1)$ to the square of a trace of a modular function evaluated at a CM point. Assuming the Birch and Swinnerton-Dyer conjecture\, when the value above is non-zero\, we should recover the order of the Tate-Shafarev ich group\, and we show that the value is indeed an integer square.\n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Vargas Montoya (IMPAN) DTSTART:20220606T121500Z DTEND:20220606T131500Z DTSTAMP:20260404T111320Z UID:WarsawNT/45 DESCRIPTION:Title: q-strong Frobenius structure\nby Daniel Vargas Montoya (IMPAN ) as part of Warsaw Number Theory Seminar\n\nLecture held in IMPAN\, Warsa w\, conference room 6\, ground floor\,.\n\nAbstract\nThe notion of strong Frobenius structure is classically studied in the theory of $p$-adic diffe rential equations. This notion was introduced by B.Dwork in his study of z eta functions. Recently\, we propose a new definition of this notion for q-difference operators. The relevance of this definition is supported for two results. The first one deals with confluence and the second one deals with congruence modulo the cyclotomic polynomial. So the first part of the talk is devoted to presenting our definition of q-strong Frobenius struct ure and the second part we are going to present the two previous results.\ n LOCATION:https://stable.researchseminars.org/talk/WarsawNT/45/ END:VEVENT END:VCALENDAR