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BEGIN:VEVENT
SUMMARY:Filip Najman (University of Zagreb)
DTSTART:20200831T160000Z
DTEND:20200831T163000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/1
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/1/">Q-curves over odd degree number fields</a>\nby Filip Najman (U
 niversity of Zagreb) as part of BIRS workshop: Modern Breakthroughs in Dio
 phantine Problems\n\n\nAbstract\nBy reformulating and extending results of
  Elkies\, we prove some\nresults on $\\mathbb Q$-curves over number fields
  of odd degree. We show that\,\nover such fields\, the only prime isogeny 
 degrees~$\\ell$ which an\nelliptic curve without CM may have are those deg
 rees which are already\npossible over~$\\mathbb Q$ itself (in particular\,
  $\\ell\\le37$)\, and we show\nthe existence of a bound on the degrees of 
 cyclic isogenies between\n$\\mathbb Q$-curves depending only on the degree
  of the field.  We also prove\nthat the only possible torsion groups of $\
 \mathbb Q$-curves over number fields\nof degree not divisible by a prime $
 \\ell\\leq 7$ are the $15$ groups that appear\nas torsion groups of ellipt
 ic curves over $\\mathbb Q$. This is joint work with\nJohn Cremona.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Matschke (Boston University)
DTSTART:20200831T164000Z
DTEND:20200831T171000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/2
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/2/">A general S-unit equation solver and tables of elliptic curves
  over number fields</a>\nby Benjamin Matschke (Boston University) as part 
 of BIRS workshop: Modern Breakthroughs in Diophantine Problems\n\n\nAbstra
 ct\nIn this talk we present work in progress on a new highly optimized sol
 ver for general and constraint S-unit equations over number fields. It has
  diophantine applications including asymptotic Fermat theorems\, Siegel's 
 method for computing integral points\, and most strikingly for computing l
 arge tables of elliptic curves over number fields with good reduction outs
 ide given sets of primes S. For the latter\, we improved on the method of 
 Koutsianas (Parshin\, Shafarevich\, Elkies).\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbey Bourdon (Wake Forest University)
DTSTART:20200831T172000Z
DTEND:20200831T175000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/3
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/3/">On Isolated Points of Odd Degree</a>\nby Abbey Bourdon (Wake F
 orest University) as part of BIRS workshop: Modern Breakthroughs in Diopha
 ntine Problems\n\n\nAbstract\nLet C be a curve defined over a number field
  $k$\, and suppose $C(k)$ is nonempty. We\nsay a closed point $x$ on $C$ o
 f degree $d$ is isolated if it does not belong to an\ninfinite family of d
 egree d points parametrized by the projective line or a\npositive rank abe
 lian subvariety of the curve's Jacobian. In this talk we will\nidentify th
 e non-CM elliptic curves with rational $j$-invariant which give rise to\na
 n isolated point of odd degree on $X_1(N)$ for some positive integer $N$. 
 This is\njoint work with David Gill\, Jeremy Rouse\, and Lori D. Watson.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Pries (Colorado State University)
DTSTART:20200901T160000Z
DTEND:20200901T163000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/4
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/4/">Principal polarizations and Shimura data for families of cycli
 c covers of the projective line</a>\nby Rachel Pries (Colorado State Unive
 rsity) as part of BIRS workshop: Modern Breakthroughs in Diophantine Probl
 ems\n\n\nAbstract\nConsider a family of degree m cyclic covers of the proj
 ective line\, with any number of branch points and inertia type. The Jacob
 ians of the curves in this family are abelian varieties having an automorp
 hism of order m with a prescribed signature.  For each such family\, the s
 ignature determines a PEL-type Shimura variety.  Under a condition on the 
 class number of m\, we determine the Hermitian form and Shimura datum of t
 he component of the Shimura variety containing the Torelli locus.  For the
  proof\, we study the boundary of Hurwitz spaces\, investigate narrow clas
 s numbers of real cyclotomic fields\, and build on an algorithm of Van Wam
 elen about principal polarizations on abelian varieties with complex multi
 plication.  This is joint work with Li\, Mantovan\, and Tang.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lajos Hajdu (University of Debrecen)
DTSTART:20200901T164000Z
DTEND:20200901T171000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/5
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/5/">Powers in arithmetic progressions</a>\nby Lajos Hajdu (Univers
 ity of Debrecen) as part of BIRS workshop: Modern Breakthroughs in Diophan
 tine Problems\n\n\nAbstract\nThe question that at most how many squares on
 e can find among $N$ consecutive terms of an arithmetic progression\, has 
 attracted a lot of attention. An old conjecture of Erd\\H{o}s predicted th
 at this number $P_N(2)$ is at most $o(N)$\; it was proved by Szemer\\'edi.
  Later\, using various deep tools\, Bombieri\, Granville and Pintz showed 
 that $P_N(2) < O(N^{2/3+o(1)})$\, which bound was refined to $O(N^{3/5+o(1
 )})$ by Bombieri and Zannier. There is a conjecture due to Rudin which pre
 dicts a much stronger behavior of $P_N(2)$\, namely\, that $P_N(2)=O(\\sqr
 t{N})$ should be valid. An even stronger form of this conjecture says that
  we have\n$$ P_2(N)=P_{24\,1\;N}(2)=\\sqrt{\\frac{8}{3}N}+O(1) $$\nfor $N\
 \geq 6$\, where $P_{24\,1\;N}(2)$ denotes the number of squares in the ari
 thmetic progression $24n+1$ for $0 \\leq n < N$. This stronger form has be
 en recently proved for $N \\leq 52$ by Gonz\\'alez-Jim\\'enez and Xarles.\
 nIn the talk we take up the problem for arbitrary $\\ell$-th powers. First
  we characterize those arithmetic progressions which contain the most $\\e
 ll$-th powers asymptotically. In fact\, we can give a complete description
 \, and it turns out that basically the 'best' arithmetic progression is un
 ique for any $\\ell$. Then we formulate analogues of Rudin's conjecture fo
 r general powers $\\ell$\, and we prove these conjectures for $\\ell=3$ an
 d $4$ up to $N=19$ and $5$\, respectively.\nThe new results presented are 
 joint with Sz. Tengely.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Loughran (University of Bath)
DTSTART:20200902T160000Z
DTEND:20200902T163000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/6
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/6/">Hasse principle for a family of K3 surfaces</a>\nby Daniel Lou
 ghran (University of Bath) as part of BIRS workshop: Modern Breakthroughs 
 in Diophantine Problems\n\n\nAbstract\nIn this talk we study the Hasse pri
 nciple for the family of "diagonal K3 surfaces of degree 2"\, given by the
  explicit equations:\n\n$$w^2 = A_1 x_1^6 + A_2 x_2^6 + A_3 x_3^6.$$\n\nI 
 will explain how many such surfaces\, when ordered by their coefficients\,
  have a Brauer-Manin obstruction to the Hasse principle. This is joint wor
 k with Damián Gvirtz and Masahiro Nakahara.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Logan (Government of Canada)
DTSTART:20200902T164000Z
DTEND:20200902T171000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/7
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/7/">Explicit coverings of K3 surfaces by the square of a curve</a>
 \nby Adam Logan (Government of Canada) as part of BIRS workshop: Modern Br
 eakthroughs in Diophantine Problems\n\n\nAbstract\nParanjape showed that K
 3 surfaces that are double covers of $P^2$\nbranched along six lines are d
 ominated by the square of a curve of genus 5. In\nthis talk\, we describe 
 a somewhat analogous construction and use it to show that\nK3 surfaces in 
 $P^4$ with 15 nodes are dominated by the square of a curve of genus\n7.  W
 e will explain a birational equivalence between the moduli space of a\nrel
 ated family of K3 surfaces and a moduli space of covers of rational curves
 \nwith additional data.  This is joint work with Colin Ingalls and Owen\nP
 atashnick.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victoria Cantoral Farfán (Katholieke Universiteit Leuven)
DTSTART:20200902T172000Z
DTEND:20200902T175000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/8
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/8/">Fields of definition of elliptic fibrations on covers of certa
 in extremal rational elliptic surfaces</a>\nby Victoria Cantoral Farfán (
 Katholieke Universiteit Leuven) as part of BIRS workshop: Modern Breakthro
 ughs in Diophantine Problems\n\n\nAbstract\nK3 surfaces have been extensiv
 ely studied over the past decades for\nseveral reasons. For once\, they ha
 ve a rich and yet tractable geometry and they\nare the playground for seve
 ral open arithmetic questions. Moreover\, they form\nthe only class which 
 might admit more than one elliptic fibration with section.\nA natural ques
 tion is to ask if one can classify such fibrations\, and indeed\nthat has 
 been done by several authors\, among them Nishiyama\, Garbagnati and\nSalg
 ado. The particular setting that we were interested in studying is when a 
 K3\nsurface arises as a double cover of an extremal rational elliptic surf
 ace with a\nunique reducible fiber. This K3 surface will have a non-symple
 ctic involution τ\nfixing two smooth Galois-conjugate genus 1 curves. In 
 this joint work we provide\na list of all elliptic fibrations on those K3 
 surfaces together with the degree\nof a field extension over which each ge
 nus one fibration is defined and admits a\nsection. We show that the latte
 r depends\, in general\, on the action of the cover\ninvolution τ on the 
 fibers of the genus 1 fibration. This is a joint work with\nAlice Garbagna
 ti\, Cecília Salgado\, Antonela Trbovíc and Rosa Winter.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Browning (Institute of Science and Technology Austria)
DTSTART:20200903T160000Z
DTEND:20200903T163000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/9
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/9/">The geometric sieve for quadrics and applications</a>\nby Tim 
 Browning (Institute of Science and Technology Austria) as part of BIRS wor
 kshop: Modern Breakthroughs in Diophantine Problems\n\n\nAbstract\nWe disc
 uss a version of Ekedahl's geometric sieve for integral\nquadratic forms o
 f rank at least five.  This can be used to address some natural\nquestions
  to do with strong approximation and local solubility in families.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marta Pieropan (Utrecht University)
DTSTART:20200903T164000Z
DTEND:20200903T171000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/10
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/10/">Campana points\, a new number theoretic challenge</a>\nby Mar
 ta Pieropan (Utrecht University) as part of BIRS workshop: Modern Breakthr
 oughs in Diophantine Problems\n\n\nAbstract\nThis talk introduces Campana 
 points\, an arithmetic notion\, first\nstudied by Campana and Abramovich\,
  that interpolates between the notions of\nrational and integral points. C
 ampana points are expected to satisfy suitable\nanalogs of Lang's conjectu
 re\, Vojta's conjecture and Manin's conjecture\, and\ntheir study introduc
 e new number theoretic challenges of a computational nature.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josha Box (University of Warwick)
DTSTART:20200904T160000Z
DTEND:20200904T163000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/12
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/12/">Modularity of elliptic curves over totally real quartic field
 s not containing the square root of 5</a>\nby Josha Box (University of War
 wick) as part of BIRS workshop: Modern Breakthroughs in Diophantine Proble
 ms\n\n\nAbstract\nFollowing Wiles's breakthrough work\, it has been shown 
 in recent years that\nelliptic curves over each totally real field of degr
 ee 2 (Freitas-Le\nHung-Siksek) or 3 (Derickx-Najman-Siksek) are modular. W
 e study the degree 4\ncase and show that if K is a totally real quartic fi
 eld in which 5 is not a\nsquare\, then every elliptic curve over K is modu
 lar. Thanks to strong results of\nThorne and Kalyanswami\, this boils down
  to the determination of all quartic\npoints on a few modular curves. Some
  of these curves have infinitely many\nquartic points. In this talk I will
  discuss how Chabauty's method and sieving\ncan nevertheless be used to de
 scribe such points.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hwajong Yoo (Seoul National University)
DTSTART:20200904T164000Z
DTEND:20200904T171000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/13
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/13/">Rational torsion points on J_0(N)</a>\nby Hwajong Yoo (Seoul 
 National University) as part of BIRS workshop: Modern Breakthroughs in Dio
 phantine Problems\n\n\nAbstract\nFor any positive integer N\, we propose a
  conjecture on the rational\ntorsion points on J_0(N). Also\, we prove thi
 s conjecture up to finitely many\nprimes. More precisely\, we prove that t
 he prime-to-m parts of the rational\ntorsion subgroup of J_0(N) and the ra
 tional cuspidal divisor class group of\nX_0(N) coincide\, where m is the l
 argest perfect square dividing 12N.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hector Pasten (Pontificia Universidad Catolica de Chile)
DTSTART:20200904T172000Z
DTEND:20200904T175000Z
DTSTAMP:20260404T042248Z
UID:BIRS_20w5005/14
DESCRIPTION:Title: <a href="https://stable.researchseminars.org/talk/BIRS_
 20w5005/14/">A Chabauty-Coleman bound for surfaces in abelian threefolds</
 a>\nby Hector Pasten (Pontificia Universidad Catolica de Chile) as part of
  BIRS workshop: Modern Breakthroughs in Diophantine Problems\n\n\nAbstract
 \nWe will give a bound for the number of rational points in a hyperbolic s
 urface contained in an abelian threefold of Mordell-Weil rank $1$ over $\\
 mathbb{Q}$. The form of the estimate is analogous to the classical Chabaut
 y-Coleman bound for curves\, although the proof uses a completely differen
 t approach. The new method concerns w-integral schemes\, especially in pos
 itive characteristic. This is joint work with Jerson Caro.\n
LOCATION:https://stable.researchseminars.org/talk/BIRS_20w5005/14/
END:VEVENT
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