Related papers: BF path integrals for elliptic curves and $p$-adic…

Iwasawa's classical asymptotical formula relates the orders of the $p$-parts $X_n$ of the ideal class groups along a $\ZM_p$-extension $F_\infty/F$ of a number field $F$, to Iwasawa structural invariants $\la$ and $\mu$ attached to the…

We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective…

We propose and prove a mirror theorem for the elliptic quasimap invariants for smooth Calabi-Yau complete intersections in projective spaces. The theorem combined with the wall-crossing formula appeared in paper (arXiv:1308.6377) implies…

In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main…

Feynman's path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes $ K(x^{"},t^{"};x',t')$ for two-dimensional systems with quadratic Lagrangians are evaluated analytically and…

We recover a result of Iwasawa on the p-adic logarithm of principal units with the use of the value at 1 of p-adic L-functions. We deduce an Iwasawa-like result in the odd part of principal units.

Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{F}_\infty)$ consisting of a rational elliptic curve $E$ and a $p$-adic Lie extension $\mathcal{F}_\infty$ of $\mathbb{Q}$, is the $p$-primary Selmer group…

In this work we prove congruences between special values of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analogue for elliptic curves with CM…

We present a database of rational elliptic curves, up to Q-isomorphism, with good reduction outside {2,3,5,7,11,13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such…

We show that for a class of two-loop diagrams, the on-shell part of the integration-by-parts (IBP) relations correspond to exact meromorphic one-forms on algebraic curves. Since it is easy to find such exact meromorphic one-forms from…

We show that the $p$-adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts whose "normalized" action on $1$-forms preserves mod $p$ the space of invariant $1$-forms. We next show that, after removing…

Let $\Pi$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $\Pi$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic…

We provide a new interpretation of the Mazur-Tate Conjecture and then use it to obtain the first (unconditional) theoretical evidence in support of the conjecture for elliptic curves of strictly positive rank.

Let $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at a prime $p\geq 5$, and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ equals $-1$. When $p$ splits in $K$, Castella and Wan formulated the…

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…

In this note we consider questions about parametrisations of elliptic curves defined over number fields by quotients of the upper half-plane by finite index subgroups of SL_2(Z). We ask if we can choose such a parametrisation of an elliptic…

For small odd primes $p$, we prove that most of the rational points on the modular curve $X_0(p)/w_p$ parametrize pairs of elliptic curves having infinitely many supersingular primes. This result extends the class of elliptic curves for…

Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law. We present two methods of…

It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.

We propose three kinds of explicit formulas for the elliptic lambda function by the elliptic modular function. Further, we derive incredible cubic identities as a corollary of our explicit formulas and evaluate some singular values of the…