In this paper, based on an idea of Tian we establish a new sufficient condition for a positive integer n to be a congruent number in terms of the Legendre symbols for the prime factors of n. Our criterion generalizes previous results of Heegner, Birch–Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving the criterion is to give formulae for the analytic Tate–Shafarevich number $\mathcal{L}(n)$ in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the generalized Gross–Zagier formula of Yuan–Zhang–Zhang.

We study the geometry of the p‑adic analogues of the complex analytic period spaces first introduced by Griffiths. More precisely, we prove the Fargues–Rapoport conjecture for p‑adic period domains: for a reductive group G over a p‑adic field and a minuscule cocharacter μ of G, the weakly admissible locus coincides with the admissible one if and only if the Kottwitz set B(G,μ) is fully Hodge–Newton decomposable.

Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about periods.

No abstract uploaded!

The aim of the present paper is to suggest that statistical physics provides the correct language to understand the practical behavior of the LLL algorithm, most of which are left unexplained to this day. To this end, we propose sandpile models that imitate LLL with compelling accuracy, and prove for these models some of the most desired statements regarding LLL. We also formulate a few conjectures that formally capture our heuristics and would serve as milestones for further development of the theory.