The HUJI-BGU Workshop in Arithmetic meets semiregularly,
alternating locations save for pandemics etc., starting from the
year 5780. All are welcome.

The workshop is currently organized by Ari Shnidman
(HUJI) and Daniel
Disegni (BGU).

10.15-10.40 Welcome (in Manchester building faculty lounge)

*
*

**Abstracts of talks**

Eran Assaf (10.45-12.00)

We consider spaces of modular forms attached to positive definite quadratic forms in four variables, and make explicit their connection to Hilbert modular forms using the even Clifford functor. By relating it to the theory of theta lifts we obtain an explicit theta correspondence, and gain a full understanding of the systems of Hecke eigenvalues that can arise. We will discuss further applications to non-vanishing and Eisenstein congruences, and touch upon the picture in higher rank.

This is joint work with Dan Fretwell, Colin Ingalls, Adam Logan, Spencer Secord and John Voight.

Ishai Dan-Cohen (13.30-14.45)

The rational points of a smooth curve over a number field map to the set of augmentations of the associated motivic algebra. An expectation, closely related to Kim's conjecture, is that the set of augmentations which are locally geometric is equal to the set of rational points. We provide evidence for this expectation by extending aspects of the "Weil height machine" to the set of locally geometric augmentations. This is ongoing joint work with L. Alexander Betts.

Ari Shnidman (15.15-16.30)

Selmer showed that the cubic plane curve 3x^3 + 4y^3 + 5z^3 = 0 has no rational points over Q but has points everywhere locally (i.e. over each Q_p). It therefore gives an element of order 3 in the Tate-Shafarevich group Sha(E) of a certain elliptic curve E. It is still not known whether for every prime p there exists an elliptic curve E over Q with elements of order p in Sha(E). In work with Ariel Weiss last year, we showed that for each prime p, there exist simple abelian varieties A over Q with elements of order p in Sha(A). In this talk, I'll discuss recent work with Victor Flynn, which shows that the p-part of Sha(A) can in fact be arbitrarily large. Moreover, we construct the torsors violating the local-to-global principle explicitly, as mu_p-covers of Jacobians of superelliptic curves.

**Dates to be saved**

TBA

** Past meetings**

**
**4th meeting, June 29th-July 1st, 2020
(Zoom)

3rd meeting, March 30th,
2020 (Zoom)

2nd meeting, January 13th, 2020 (BGU)

1st meeting,
December 16th, 2019 (HUJI)

Locations

HUJI: Einstein Institute of Mathematics, Hebrew University
Giv'at Ram Campus, Jerusalem.

BGU: Deichmann Building for Mathematics (building 58), Ben-Gurion University of the Negev, Be'er Sheva.

Zoom: www.zoom.us

Directions to the physical locations: you may use Google Maps
or Moovit. (Please note that Google is sometimes optimistic about
the travel time of local buses.)

10.45-12.00 Eran Assaf (Dartmouth), *Definite orthogonal
modular forms * (slides)

13.30-14.45 Ishai Dan-Cohen (BGU),*A motivic Weil height
machine for curves* (link to watch live)

15.15-16.30 Ari Shnidman (HUJI),*Explicit counterexamples to the local-to-global principle *(link to watch live)

13.30-14.45 Ishai Dan-Cohen (BGU),

15.15-16.30 Ari Shnidman (HUJI),

Eran Assaf (10.45-12.00)

We consider spaces of modular forms attached to positive definite quadratic forms in four variables, and make explicit their connection to Hilbert modular forms using the even Clifford functor. By relating it to the theory of theta lifts we obtain an explicit theta correspondence, and gain a full understanding of the systems of Hecke eigenvalues that can arise. We will discuss further applications to non-vanishing and Eisenstein congruences, and touch upon the picture in higher rank.

This is joint work with Dan Fretwell, Colin Ingalls, Adam Logan, Spencer Secord and John Voight.

Ishai Dan-Cohen (13.30-14.45)

The rational points of a smooth curve over a number field map to the set of augmentations of the associated motivic algebra. An expectation, closely related to Kim's conjecture, is that the set of augmentations which are locally geometric is equal to the set of rational points. We provide evidence for this expectation by extending aspects of the "Weil height machine" to the set of locally geometric augmentations. This is ongoing joint work with L. Alexander Betts.

Ari Shnidman (15.15-16.30)

Selmer showed that the cubic plane curve 3x^3 + 4y^3 + 5z^3 = 0 has no rational points over Q but has points everywhere locally (i.e. over each Q_p). It therefore gives an element of order 3 in the Tate-Shafarevich group Sha(E) of a certain elliptic curve E. It is still not known whether for every prime p there exists an elliptic curve E over Q with elements of order p in Sha(E). In work with Ariel Weiss last year, we showed that for each prime p, there exist simple abelian varieties A over Q with elements of order p in Sha(A). In this talk, I'll discuss recent work with Victor Flynn, which shows that the p-part of Sha(A) can in fact be arbitrarily large. Moreover, we construct the torsors violating the local-to-global principle explicitly, as mu_p-covers of Jacobians of superelliptic curves.

TBA

2nd meeting, January 13th, 2020 (BGU)

Locations

BGU: Deichmann Building for Mathematics (building 58), Ben-Gurion University of the Negev, Be'er Sheva.

Zoom: www.zoom.us