François Brunault

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bio website perso.ens-lyon.fr/… location Lyon age 34 member for 3 years, 10 months seen Nov 10 '13 at 12:21 profile views 2,302

I am a number theorist working at the École normale supérieure de Lyon in France. My research interests are elliptic curves, L-functions and zeta functions, especially the study of their special values (conjectures by Beilinson, Bloch, Kato, Zagier...).

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 Mar18 awarded Notable Question Nov23 awarded Nice Answer Nov11 awarded Good Answer Nov10 revised calculate function from its divizor Updated link Oct31 comment calculate function from its divizor Dear Hicham, I edited my answer to give the link to the Pari/GP script. Oct31 revised calculate function from its divizor Added link to Pari/GP script. Jun25 awarded ag.algebraic-geometry Jun25 awarded nt.number-theory Jun25 awarded Revival Jun25 awarded Pundit Jun24 answered On Deligne's determinant of motives Jun19 comment modularity of elliptic curves with cm Note that if $\operatorname{Res}_{F/\mathbf{Q}} E$ appear inside $J_1(N)$ over $\mathbf{Q}$ then $E$ itself appears inside $J_1(N)$ over $F$. So in this case you can find a modular parametrization $X_1(N) \to E$ which is defined over $F$. Jun19 comment modularity of elliptic curves with cm What is true is that every elliptic curve over $\overline{\mathbf{Q}}$ with CM by $K$ is isomorphic over $\overline{\mathbf{Q}}$ to a $K$-curve (a curve which is isogenous over $\overline{\mathbf{Q}}$ to all its $\operatorname{Gal}(\overline{\mathbf{Q}}/K)$-conjugates), see Wortmann's article and the references. Jun19 comment modularity of elliptic curves with cm In the last sentence, I meant "isomorphic over $\overline{\mathbf{Q}}$"... Jun19 comment modularity of elliptic curves with cm There is a nice article where Shimura's result is generalized for a wider class of CM elliptic curves, see S. Wortmann, *Generalized Q-curves and factors of $J_1(N)$* dx.doi.org/10.1007/BF02940901 These CM elliptic curves are sometimes called of Shimura type. I'm not sure but I think these are exactly the CM elliptic curves whose restriction of scalars appear inside $J_1(N)$ over $\mathbf{Q}$. Moreover, every CM elliptic curve is isomorphic over $\mathbf{Q}$ to a curve of Shimura type. Jun15 comment Mersenne Prime Sequences @Dietrich : Eugène Catalan's footnote is here archive.org/stream/nouvellecorresp01mansgoog#page/n353/mode/2up In fact, he states this as an "empirical theorem" which holds for all terms "up to a certain limit". To me this seems far from conjecturing that all terms are prime. Jun15 comment Mersenne Prime Sequences @Barakman : The point is that there is no obvious bias towards primality arising from belonging to $A_n$. The exponents of the numbers in $A_n$ are very large, and I see no reason why they should be more prime than the Mersenne numbers of comparable size. So their primality becomes soon unlikely. Jun15 comment Mersenne Prime Sequences The Wagstaff heuristics primes.utm.edu/mersenne/heuristic.html assert that for large prime $p$, the probability of $2^p-1$ being prime is about $(\log p)/p$ (up to some multiplicative constant). So it seems unlike to me that $A_n$ contains only prime numbers. I would rather conjecture that any such sequence will contain a composite number. Jun15 comment How to explain the picturesque patterns in François Brunault's matrix? Thank you for spotting these nice patterns! Not a precise explanation, but it may not be surprising that the entries of the matrix have nice $p$-adic properties. Indeed, any generalized polynomial map (in the sense of the question you link to) extends to a map $\mathbf{Z}_p \to \mathbf{Z}_p$ which is continuous (and in fact, 1-Lipschitz). Jun15 comment Elementary tools for proving congruences of modular forms See E. Ghate, An introduction to congruences between modular forms math.tifr.res.in/%7Eeghate/basics.dvi