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BEGIN:VEVENT
SUMMARY:Filip Najman (University of Zagreb)
DTSTART;VALUE=DATE-TIME:20200831T160000Z
DTEND;VALUE=DATE-TIME:20200831T163000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/1
DESCRIPTION:Title: Q-curves over odd degree number fields\nby Filip Najman (Universi
ty of Zagreb) as part of BIRS workshop: Modern Breakthroughs in Diophantin
e 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 degrees wh
ich 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 elliptic curv
es over $\\mathbb Q$. This is joint work with\nJohn Cremona.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Matschke (Boston University)
DTSTART;VALUE=DATE-TIME:20200831T164000Z
DTEND;VALUE=DATE-TIME:20200831T171000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/2
DESCRIPTION:Title: A general S-unit equation solver and tables of elliptic curves over n
umber fields\nby Benjamin Matschke (Boston University) as part of BIRS
workshop: Modern Breakthroughs in Diophantine Problems\n\n\nAbstract\nIn
this talk we present work in progress on a new highly optimized solver for
general and constraint S-unit equations over number fields. It has diopha
ntine applications including asymptotic Fermat theorems\, Siegel's method
for computing integral points\, and most strikingly for computing large ta
bles of elliptic curves over number fields with good reduction outside giv
en sets of primes S. For the latter\, we improved on the method of Koutsia
nas (Parshin\, Shafarevich\, Elkies).\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Abbey Bourdon (Wake Forest University)
DTSTART;VALUE=DATE-TIME:20200831T172000Z
DTEND;VALUE=DATE-TIME:20200831T175000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/3
DESCRIPTION:Title: On Isolated Points of Odd Degree\nby Abbey Bourdon (Wake Forest U
niversity) as part of BIRS workshop: Modern Breakthroughs in Diophantine P
roblems\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 $C$ of degre
e $d$ is isolated if it does not belong to an\ninfinite family of degree d
points parametrized by the projective line or a\npositive rank abelian su
bvariety of the curve's Jacobian. In this talk we will\nidentify the non-C
M elliptic curves with rational $j$-invariant which give rise to\nan isola
ted 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://researchseminars.org/talk/BIRS_20w5005/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Rachel Pries (Colorado State University)
DTSTART;VALUE=DATE-TIME:20200901T160000Z
DTEND;VALUE=DATE-TIME:20200901T163000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/4
DESCRIPTION:Title: Principal polarizations and Shimura data for families of cyclic cover
s of the projective line\nby Rachel Pries (Colorado State University)
as part of BIRS workshop: Modern Breakthroughs in Diophantine Problems\n\n
\nAbstract\nConsider a family of degree m cyclic covers of the projective
line\, with any number of branch points and inertia type. The Jacobians of
the curves in this family are abelian varieties having an automorphism of
order m with a prescribed signature. For each such family\, the signatur
e determines a PEL-type Shimura variety. Under a condition on the class n
umber of m\, we determine the Hermitian form and Shimura datum of the comp
onent of the Shimura variety containing the Torelli locus. For the proof\
, we study the boundary of Hurwitz spaces\, investigate narrow class numbe
rs of real cyclotomic fields\, and build on an algorithm of Van Wamelen ab
out principal polarizations on abelian varieties with complex multiplicati
on. This is joint work with Li\, Mantovan\, and Tang.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Lajos Hajdu (University of Debrecen)
DTSTART;VALUE=DATE-TIME:20200901T164000Z
DTEND;VALUE=DATE-TIME:20200901T171000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/5
DESCRIPTION:Title: Powers in arithmetic progressions\nby Lajos Hajdu (University of
Debrecen) as part of BIRS workshop: Modern Breakthroughs in Diophantine Pr
oblems\n\n\nAbstract\nThe question that at most how many squares one can f
ind among $N$ consecutive terms of an arithmetic progression\, has attract
ed a lot of attention. An old conjecture of Erd\\H{o}s predicted that 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 predicts a
much stronger behavior of $P_N(2)$\, namely\, that $P_N(2)=O(\\sqrt{N})$
should be valid. An even stronger form of this conjecture says that we hav
e\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 arithmetic
progression $24n+1$ for $0 \\leq n < N$. This stronger form has been rece
ntly 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 cha
racterize those arithmetic progressions which contain the most $\\ell$-th
powers asymptotically. In fact\, we can give a complete description\, and
it turns out that basically the 'best' arithmetic progression is unique fo
r any $\\ell$. Then we formulate analogues of Rudin's conjecture for gener
al powers $\\ell$\, and we prove these conjectures for $\\ell=3$ and $4$ u
p to $N=19$ and $5$\, respectively.\nThe new results presented are joint w
ith Sz. Tengely.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Loughran (University of Bath)
DTSTART;VALUE=DATE-TIME:20200902T160000Z
DTEND;VALUE=DATE-TIME:20200902T163000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/6
DESCRIPTION:Title: Hasse principle for a family of K3 surfaces\nby Daniel Loughran (
University of Bath) as part of BIRS workshop: Modern Breakthroughs in Diop
hantine Problems\n\n\nAbstract\nIn this talk we study the Hasse principle
for the family of "diagonal K3 surfaces of degree 2"\, given by the explic
it equations:\n\n$$w^2 = A_1 x_1^6 + A_2 x_2^6 + A_3 x_3^6.$$\n\nI will ex
plain how many such surfaces\, when ordered by their coefficients\, have a
Brauer-Manin obstruction to the Hasse principle. This is joint work with
Damián Gvirtz and Masahiro Nakahara.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Logan (Government of Canada)
DTSTART;VALUE=DATE-TIME:20200902T164000Z
DTEND;VALUE=DATE-TIME:20200902T171000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/7
DESCRIPTION:Title: Explicit coverings of K3 surfaces by the square of a curve\nby Ad
am Logan (Government of Canada) as part of BIRS workshop: Modern Breakthro
ughs in Diophantine Problems\n\n\nAbstract\nParanjape showed that K3 surfa
ces that are double covers of $P^2$\nbranched along six lines are dominate
d by the square of a curve of genus 5. In\nthis talk\, we describe a somew
hat analogous construction and use it to show that\nK3 surfaces in $P^4$ w
ith 15 nodes are dominated by the square of a curve of genus\n7. We will
explain a birational equivalence between the moduli space of a\nrelated fa
mily of K3 surfaces and a moduli space of covers of rational curves\nwith
additional data. This is joint work with Colin Ingalls and Owen\nPatashni
ck.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Victoria Cantoral Farfán (Katholieke Universiteit Leuven)
DTSTART;VALUE=DATE-TIME:20200902T172000Z
DTEND;VALUE=DATE-TIME:20200902T175000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/8
DESCRIPTION:Title: Fields of definition of elliptic fibrations on covers of certain extr
emal rational elliptic surfaces\nby Victoria Cantoral Farfán (Katholi
eke Universiteit Leuven) as part of BIRS workshop: Modern Breakthroughs in
Diophantine Problems\n\n\nAbstract\nK3 surfaces have been extensively stu
died over the past decades for\nseveral reasons. For once\, they have a ri
ch and yet tractable geometry and they\nare the playground for several ope
n arithmetic questions. Moreover\, they form\nthe only class which might a
dmit more than one elliptic fibration with section.\nA natural question is
to ask if one can classify such fibrations\, and indeed\nthat has been do
ne by several authors\, among them Nishiyama\, Garbagnati and\nSalgado. Th
e particular setting that we were interested in studying is when a K3\nsur
face arises as a double cover of an extremal rational elliptic surface wit
h a\nunique reducible fiber. This K3 surface will have a non-symplectic in
volution τ\nfixing two smooth Galois-conjugate genus 1 curves. In this jo
int work we provide\na list of all elliptic fibrations on those K3 surface
s together with the degree\nof a field extension over which each genus one
fibration is defined and admits a\nsection. We show that the latter depen
ds\, 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 Garbagnati\, Ce
cília Salgado\, Antonela Trbovíc and Rosa Winter.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tim Browning (Institute of Science and Technology Austria)
DTSTART;VALUE=DATE-TIME:20200903T160000Z
DTEND;VALUE=DATE-TIME:20200903T163000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/9
DESCRIPTION:Title: The geometric sieve for quadrics and applications\nby Tim Brownin
g (Institute of Science and Technology Austria) as part of BIRS workshop:
Modern Breakthroughs in Diophantine Problems\n\n\nAbstract\nWe discuss a v
ersion of Ekedahl's geometric sieve for integral\nquadratic forms of 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://researchseminars.org/talk/BIRS_20w5005/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Marta Pieropan (Utrecht University)
DTSTART;VALUE=DATE-TIME:20200903T164000Z
DTEND;VALUE=DATE-TIME:20200903T171000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/10
DESCRIPTION:Title: Campana points\, a new number theoretic challenge\nby Marta Pier
opan (Utrecht University) as part of BIRS workshop: Modern Breakthroughs i
n Diophantine Problems\n\n\nAbstract\nThis talk introduces Campana points\
, an arithmetic notion\, first\nstudied by Campana and Abramovich\, that i
nterpolates between the notions of\nrational and integral points. Campana
points are expected to satisfy suitable\nanalogs of Lang's conjecture\, Vo
jta's conjecture and Manin's conjecture\, and\ntheir study introduce new n
umber theoretic challenges of a computational nature.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Josha Box (University of Warwick)
DTSTART;VALUE=DATE-TIME:20200904T160000Z
DTEND;VALUE=DATE-TIME:20200904T163000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/12
DESCRIPTION:Title: Modularity of elliptic curves over totally real quartic fields not c
ontaining the square root of 5\nby Josha Box (University of Warwick) a
s part of BIRS workshop: Modern Breakthroughs in Diophantine Problems\n\n\
nAbstract\nFollowing Wiles's breakthrough work\, it has been shown in rece
nt years that\nelliptic curves over each totally real field of degree 2 (F
reitas-Le\nHung-Siksek) or 3 (Derickx-Najman-Siksek) are modular. We study
the degree 4\ncase and show that if K is a totally real quartic field in
which 5 is not a\nsquare\, then every elliptic curve over K is modular. Th
anks to strong results of\nThorne and Kalyanswami\, this boils down to the
determination of all quartic\npoints on a few modular curves. Some of the
se curves have infinitely many\nquartic points. In this talk I will discus
s how Chabauty's method and sieving\ncan nevertheless be used to describe
such points.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hwajong Yoo (Seoul National University)
DTSTART;VALUE=DATE-TIME:20200904T164000Z
DTEND;VALUE=DATE-TIME:20200904T171000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/13
DESCRIPTION:Title: Rational torsion points on J_0(N)\nby Hwajong Yoo (Seoul Nationa
l University) as part of BIRS workshop: Modern Breakthroughs in Diophantin
e Problems\n\n\nAbstract\nFor any positive integer N\, we propose a conjec
ture on the rational\ntorsion points on J_0(N). Also\, we prove this conje
cture up to finitely many\nprimes. More precisely\, we prove that the prim
e-to-m parts of the rational\ntorsion subgroup of J_0(N) and the rational
cuspidal divisor class group of\nX_0(N) coincide\, where m is the largest
perfect square dividing 12N.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hector Pasten (Pontificia Universidad Catolica de Chile)
DTSTART;VALUE=DATE-TIME:20200904T172000Z
DTEND;VALUE=DATE-TIME:20200904T175000Z
DTSTAMP;VALUE=DATE-TIME:20220528T190052Z
UID:BIRS_20w5005/14
DESCRIPTION:Title: A Chabauty-Coleman bound for surfaces in abelian threefolds\nby
Hector Pasten (Pontificia Universidad Catolica de Chile) as part of BIRS w
orkshop: Modern Breakthroughs in Diophantine Problems\n\n\nAbstract\nWe wi
ll give a bound for the number of rational points in a hyperbolic surface
contained in an abelian threefold of Mordell-Weil rank $1$ over $\\mathbb{
Q}$. The form of the estimate is analogous to the classical Chabauty-Colem
an bound for curves\, although the proof uses a completely different appro
ach. The new method concerns w-integral schemes\, especially in positive c
haracteristic. This is joint work with Jerson Caro.\n
LOCATION:https://researchseminars.org/talk/BIRS_20w5005/14/
END:VEVENT
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