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Kapil Paranjape

Publications and source records attributed to Kapil Paranjape.

4 recordsLinked to original sources
arXiv2026

Bounds on the Mordell-Weil rank of elliptic fibrations

We prove that the Mordell-Weil group of a higher dimensional elliptic fibration naturally embeds into that of a suitable elliptic surface. We give sufficient conditions for the existence of such surfaces. We apply our result to obtain explicit and uniform bounds for the Mordell-Weil rank of elliptic threefolds of Kodaira dimension zero, including Calabi-Yau threefolds, confirming predictions from physics. We prove new explicit bounds for a broad class of elliptic fourfolds. These results suggest a general linear bound for the Mordell-Weil rank in terms of the dimension of the elliptic fibration, which we formulate as a conjecture.

math.AG↗
arXiv2014

Modular Forms and Calabi-Yau Varieties

Given a holomorphic newform $f$ of weight $k$ and with rational coefficients, a question of Mazur and van Straten asks if there is an associated Calabi-Yau variety $X$ over ${\mathbb Q}$ of dimension $k-1$ such that the $\ell$-adic Galois representation of $f$ occurs in the cohomology of $X$ in degree $k-1$. We provide some explicit examples giving a positive answer, and show moreover that such $X$ come equipped with an involution $τ$ acting by $-1$ on $H^0(X, Ω^{k-1})$. We also raise a general question regarding the regular algebraic, (essentially) selfdual cusp forms $π$ on GL$(n)$ with ${\mathbb Q}$-coefficients, asking for associated Calabi-Yau varieties $X=X_π$ (with an involution $τ$ on each such $X$ such that the quotient variety $X/τ$ is rational) carrying the (conjectural) motive of $π$. We then investigate the compatibility of this with Rankin-Selberg products of modular forms.

math.NT↗
arXiv2005

Quotients of E^n by A_{n+1} and Calabi-Yau manifolds

We give a simple construction, starting with any elliptic curve E, of an n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering the quotient Y of the n-fold self-product of E by a natural action of the alternating group A_{n+1} (in n+1 variables). The vanishing of H^m(Y, O_Y) for 0<m<n follows from the non-existence of (non-zero) fixed points in certain representations of A_{n+1}. For n<4 we provide an explicit crepant resolution X in characteristics different from 2,3. The key point is that Y can be realized as a double cover of P^n branched along a hypersurface of degree 2(n+1).

math.AG↗