Joint IAS/PU Number Theory

In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of nonlinear descent. Generalizing this, we consider families of log Calabi-Yau...

Let $E$ be a CM elliptic curves over rationals and $p$ an odd prime ordinary for $E$. If the $\mathbb Z_p$ corank of $p^\infty$ Selmer group for $E$ equals one, then we show that the analytic rank of $E$ also equals one. This is joint work with...

The generalised Kato classes of Darmon-Rotger arise as $p$-adic limits of diagonal cycles on triple products of modular curves, and in some cases, they are predicted to have a bearing on the arithmetic of elliptic curves over $Q$ of rank two. In...

Take \(E/Q\) to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of \(E\) have rank less than two, thus proving that the BSD conjecture...

The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes modulo $p$...

If two motives are congruent, is it the case that the special values of their respective $L$-functions are congruent? More precisely, can the formula predicting special values of motivic $L$-functions be interpolated in $p$-adic families of motives...

In mod-$p$ reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-$p$ reductions of more general Shimura varieties, there is a "Newton...

In a recent preprint with Sug Woo Shin (https://arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the...

Let $k$ be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of $k$ roots of unity can be. If the roots of unity have order dividing $N$, then an elementary argument shows that the modulus decreases at most...

Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface $X$ of general type over a number field $k$, the set $X(k)$, of $k$-rational points is not Zariski dense, and (ii) the conjecture of Lang that $X(k)$...