Mathematics

To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann zeta function in the half plane x>1 , produces the required exponential function. Unlike the method described in ([4], p.~278), which requires more advanced knowledge of the relationships between Dirichlet series and multiplicative arithmetic functions, our approach only needs mathematical induction on the total number of prime divisors of n , the Dirichlet product and the use of an analytic property characteristic of the exponential function in the complex plane.

An irreducible smooth representation of a p -adic group G is said to be distinguished with respect to a subgroup H if it admits a non-trivial H -invariant linear form. When H is the fixed group of an involution on G it is suggested by the works of Hervé Jacquet from the nineties that distinction can be characterized in terms of the principle of functoriality. If the involution is the Galois involution then a recent conjecture of Dipendra Prasad predicts a formula for the dimension of the space of invariant linear forms which once again involves base change. We will describe the proof of this conjecture (in the generic case) for SL(n) which is joint work with Dipendra Prasad. Then we describe one more newly discovered connection between distinction and base change which is that base change information appears in the constant of proportionality between two natural invariant linear forms on a distinguished representation. This latter result is for discrete series for GL(n) and is joint with Nadir Matringe. This paper is a report on the author's talk in the International Colloquium on Arithmetic Geometry held in January 2020 at TIFR Mumbai.

We study the zeta functions of curves over finite fields. Suppose C and C' are curves over a finite field K, with K-rational base points P and P', and let D and D' be the pullbacks (via the Abel-Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C,P) and (C',P') are *doubly isogenous* if Jac(C) and Jac(C') are isogenous over K and Jac(D) and Jac(D') are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than naive heuristics predict, and we provide an explanation for this phenomenon.

We construct overconvergent dual Eichler-Shimura maps as an application of the Higher Coleman's theory for the modular curve developed by Boxer-Pilloni. In the road, we redo the overconvergent Eichler- Shimura map of Andreatta-Iovita-Stevens. We give a new proof of Faltings Eichler-Shimura decompostion via a proétale dual BGG theory. Finally, we prove that the overconvergent Poincaré and Serre pairings are compatible with the overconvergent Eichler-Shimura morphisms.

As an algebraic surface, the equation of E 8 -singularity x 5 + y 3 + z 2 =0 can be obtained from a quotient C Y /SL(2,13) over the modular curve X(13) , where Y⊂ CP 5 is a complete intersection curve given by a system of SL(2,13) -invariant polynomials and C Y is a cone over Y . It is different from the Kleinian singularity C 2 /Γ , where Γ is the binary icosahedral group. This gives a negative answer to Arnol'd and Brieskorn's questions about the mysterious relation between the icosahedron and E 8 , i.e., the E 8 -singularity is not necessarily the Kleinian icosahedral singularity. In particular, the equation of E 8 -singularity possesses infinitely many kinds of distinct modular parametrizations, and there are infinitely many kinds of distinct constructions of the E 8 -singularity. They form a variation of the E 8 -singularity structure over the modular curve X(13) , for which we give its algebraic version, geometric version, j -function version and the version of Poincaré homology 3 -sphere as well as its higher dimensional lifting, i.e., Milnor's exotic 7 -sphere. Moreover, there are variations of Q 18 and E 20 -singularity structures over X(13) . Thus, three different algebraic surfaces, the equations of E 8 , Q 18 and E 20 -singularities can be realized from the same quotients C Y /SL(2,13) over the modular curve X(13) and have the same modular parametrizations.

In the proofs of most cases of the André-Oort conjecture, there are two different steps whose effectivity is unclear: the use of generalizations of Brauer-Siegel and the use of Pila-Wilkie. Only the case of curves in C 2 is currently known effectively (by other methods). We give an effective proof of André-Oort for non-compact curves in every Hilbert modular surface and every Hilbert modular variety of odd genus (under a minor generic simplicity condition). In particular we show that in these cases the first step may be replaced by the endomorphism estimates of Wüstholz and the second author together with the specialization method of André via G-functions, and the second step may be effectivized using the Q-functions of Novikov, Yakovenko and the first author.

Given a singular modulus j 0 and a set of primes S , we study the problem of finding effective bounds for the cardinality of the set of singular moduli j such that j??j 0 is an S -unit. For every j 0 ?? , we provide a way of finding these bounds for infinitely many sets S . The same is true if j 0 =0 and we assume the Generalized Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a "uniformity conjecture" for singular S -units.

Recently, Garunkštis, Laurin?ikas, Matsumoto, J. & R. Steuding showed an effective universality-type theorem for the Riemann zeta-function by using an effective multi-dimensional denseness result of Voronin. We will generalize Voronin's effective result and their theorem to the elements of the Selberg class satisfying some conditions.

Wooley ({\em J. Number Theory}, 1996) gave an elementary proof of a Bezout like theorem allowing one to count the number of isolated integer roots of a system of polynomial equations modulo some prime power. In this article, we adapt the proof to a slightly different setting. Specifically, we consider polynomials with coefficients from a polynomial ring F[t] for an arbitrary field F and give an upper bound on the number of isolated roots modulo t s for an arbitrary positive integer s . In particular, using s=1 , we can bound the number of isolated roots of a system of polynomials over an arbitrary field F .

We show that there is essentially a unique elliptic curve E defined over a cubic Galois extension K of Q with a K -rational point of order 13 and such that E is not defined over Q .